SAIMEHDHAAPTITUDE

APTI-INDEX

01. Number Systems-0302. L.C.M & H.C.F-1103. Percentage-1304. Profit & Loss-1805. Ratio & Proportion-2306. Alligation or Mixtures-2707. Problems on Ages-3008. Partnership-3409. Average-3610. Simple Interest-4011. Compound Interest-4312. Time & Work-4713. Pipes & Cisterns-5114. Time & Distance-5415. Trains, Boats & Streams-57

01. NUMBER SYSTEMS

Whole Numbers: the set of whole numbers is denoted by W where W={0, 1, 2, 3, 4, 5, ---}

Natural Numbers: The numbers which are used in counting are known as Natural Numbers or Positive Integers. Their set is denoted by N. thus, N = { 1, 2, 3, 4 -----}

Set of integers : All counting numbers, their negatives and O, when combined from the set of integers. This is denoted by 1. Thus I = ( -3, -2, -1, 0, 1, 2, 3 .)

Even Numbers: Those numbers which are exactly divisible by 2 are known as even numbers. For example 2, 4, 6, 8 .

Odd Numbers : The numbers which are not exactly divisible by 2 are called odd numbers. Ex : 1, 3, 5, 7, 9 .. are odd numbers

Prime Numbers: A number is said to be a Prime number if its factors are 1 and the number itself only. Ex : 2, 3, 5, 7, There are 25 prime numbers below 100.

Composite Numbers : The numbers which are not prime are called Composite Numbers For example 4, 6, 8, 9, 12, ..

Rational Numbers : A number of the form a/b where a and b are integers prime to each other b 0 is called a Rational Numbers.

Irrational Numbers: The numbers whose exact value cannot be determined are called Irrational Numbers. For example , , etc

Real Numbers: Both Rational and Irrational Numbers are collectively known as Real Numbers.

Important results :(1) Sum of first n natural numbers = where n= last term(2) If a = first term, I= last term then sum of n natural numbers = (a + l)(3) Sum of squares of first n natural numbers = (4) Sum of cubes of first n natural numbers = (5) Sum of first n even numbers = n( n+1)

(6) Sum of first n odd numbers = n2

Test of Divisibility:

1. A numbers is divisible by 2, if its units digit is 0 or even.

2. A number is divisible by 3, if the sum of the digits can be divided by 3.

3. A number is divisible by 4, if the number composed of the last two digits can be divided by 4 or the last two digits are zeros.

4. A numbers is divisible by 5, if the last digit is either 0 or 5 5. A numbers is divisible by 6, if is divisible both 2 and 3.

6. A numbers is divisible by 8, if the number composed of the last three digits can be divided by 8 or the last three digits are zeros.

7. A number is divisible by 9, if the sum of the of the digits can be divided by 9.8. A number is divisible by 11, if the sum of the digits in the odd place of the number and the sum of the digits in the even places are equal or the difference of these two suns is divisible by 11.

9. A number is divisible by 12, if it is divisible by both 3 and 4.

10. A number is divisible by 25, if it ends in 25, 75, 00.

EXERCISE1. What is the numbers of prime factors contained in the product 307 x 225 x 3411 (1) 23 (2) 44(3) 46 (4) 53

2. A man engaged a servant on the condition that he would pay him Rs. 710 and a uniform after a year service. He served only for 8 months and received uniform and Rs. 460. Find the price of the uniform? (1) 40 (2) 30(3) 25 (4) 20

3. In a division sum, the divisor is 12 times the quotient and 5 times the remainder. If the remainder is 48, then the divided is?(1) 8484 (2)4488 (3)8844 (4) 4848

4. A certain number when divided by 81 leaves a remainder 53. What is the remainder if the same number be divided by 27?(1) 24 (2) 25 (3) 26 (4)27

5. A certain number when divided by 69 leaves a remainder 19. What is the remainder when the same number is divided by 23.(1) 19 (2)20 (3) 21 (4) 22

6. Find the total number of prime numbers contained in (15)13 (14)21x (62)7(1) 81 (2) 82 (3) 83 (4) 84

7. What is the number in the unit place of ( 347)305? (1) 4 (2) 5(3)6 (4)7

8. A number is as much greater than 3421 as it is less than 530n. the number is (1) 4364 (2) 4414(3) 4519 (4) 4424

9. How many numbers, lying between 1 and 500, are divisible by 13? (1) 40 (2) 38 (3) 41 (4) 46

10. How many numbers lying between 300 and 900 are divisible by 17? (1) 35 (2) 34 (3) 33 (4)32

11. How many numbers between 1 and 300 are divisible by 3 and 5 together?(1) 16(2) 18(3) 20 (4) 100

12. Find the sum of first 20 odd counting numbers? (1) 20 (2) 100 (3) 400 (4) 313

13. How many figures (digits) are required to number a book containing 200 pages? (1) 200 (2) 600 (3) 492 (4) 372

14. The sum of two digits of a number is 9. If the digits are reversed, the number is increased by 63. Find the number. (1) 18 (2) 27 (3) 36 (4) 72

15. A number when divided by 5 leaves a remainder 3. What is the remainder when the square of the same number is divided by 5? (1) 9(2) 3 (3) 0 (4)4

Fraction: A fraction is a part of something. m/n means considering m parts out of n parts.

Proper Fraction: When the numerator is less than denominator, the fraction is said to be a proper fraction. For example 2/3, 4/6, 5/7 etc.,

Improper fraction: When the numerator is greater than the denominator, the fraction is said to be an improper fraction. For example 15/4, 13/11, 3/2 etc.,

Mixed fraction: A mixed fraction consists of a whole number and a proper fraction. For example means 7 + where 7 is the whole part and is the fractional part.

Decimal Fraction: A fraction whose denominator is 10 or a multiple of 10 is called a decimal fraction. For example, 2/10, 4/100, 6/1000 etc..

Vulgar Fraction: A fraction whose denominator is other than 10 is called a vulgar fraction. For example. 2/7, 7/6 etc.

Recurring Decimals: In a decimal, some figure or a set of figures is repeated continually, we call it a recurring decimal.Ex: 0. 333.. 0.2454545

The set of repeated figures is called a period. We denote the recurring decimal by putting a bar or a dot on the period.0.3333. = 0. 0.2454545 .. = 0. 2

EXERCISE

16. A man spends 1/3 of his income on food, of the rest on house rent and 1/5 of the rest on clothes. He still has Rs. 1, 760 left with him. find his income? (1) Rs. 4, 000(2) Rs. 4,400 (3) Rs. 3, 400 (4) Rs. 2, 400

17. A lamp post has half of its length in mud, of its length in water and 3 m above the water. Find the total length on the post. (1) 20 meter(2) 19 meter (3) 18 mete (4) 17 mete

18. A man spends 1/7 of his salary of food and of the remaining on clothing and 1/3 of the remaining on entertainment. He is still left with Rs. 600. Find the salary. (1) Rs. 1, 200(2) Rs. 3, 100 (3) Rs 2, 100 (4) Rs, 4, 100

19. A man while returning from factory, travels 2/3 of the distance by bus and of the rest, partly by car, and partly by foot. If he travels 2 km on foot, find the distance covered by him. (1) 24 km(2) 25 km(3) 26 km(4) 27 km

20. Which of the following fraction is the smallest? (1) 7/13(2) 14/33 (3) 11/25(4) 8/15

21. Which of the following fraction is the Largest ? (1) 16/21(2) 11/14(3) 16/19(4) 16/23

22. Simplify (1) 2500(2) 2600 (3) 2700 (4) 2800

23. Simplify 1 (1) 1(2) 2(3) 3 (4) 4

24. I red of a book on one day and of the remainder another day. If there were now 30 pages unread, how many pages did the book contain. (1) 239(2) 240 (3) 241 (4) 242

25. A tractor and a car together cost Rs. 85000. The tractor cost 4 times as much as the car. What is the cost of tractor? (1) Rs. 66000(2) Rs. 67000(3) Rs. 68000(4) Rs. 60000

26. The simplification of gives (1) 0. 57(2) 0.66 (3) 0. 68 (4) 0.77

27. If the number 258*4 is divisible by 11, the digit in place of * should be (1) 4(2)7 (3) 8 (4) 9

Square: If a number is multiplied by itself, then the resulting Number is called Square of the 1st number and the 1st number is called the square root of the resulting number.

Ex: 5 x 5= 25 25 is called the Square of 5 and 5 is called the Square root.

= 1. 732; = 1. 414; = 3.162; = 2.235

Properties of a Perfect Square:

No perfect square should end with 2, 3, 7 and 8

No perfect square ends with an odd number of zeroes.

The perfect square consisting of (n-1) or n digits will have n/2 digits in their root n being an even positive integer.

Cube : If a number is multiplied 3 times by itself then the resulting number is called Cube of the given number.

Ex : 5 x 5 x 5 = 125 125 is called the Cube of 5 and 5 is called the Cube root Cubes of the numbers from 1 to 10 13 = 1: 23= 8; 33= 27; 43= 64; 53 = 125; 63= 216; 73 = 343; 83= 512; 93 = 729; 103= 1000

From above it follows that When any on. is cubed units place will be as follows. 1 1 ; 2 8 ; 3 7; 4 4 ; 5 5 ; 6 6; 7 3; 8 2 ; 9 9 ; 0 0. EXERCISE

28. I collected some money by raising subscription for opening a library. If the whole mount collected be 1600 currency notes of 100 denomination and each person subscribed as many rupees as there were subscribers, find the amount paid by each subscriber. (1) Rs. 200(2) Rs. 300 (3) Rs. 400 (4) Rs. 500

29. By what least number, 2450 be multiplied, so that the resulting number is a perfect square? (1) 8(2)10(3) 5(4) 2

30. The square root of is (1) 24. 0(2) 2.40 (3) 0. 24 (4) 0.024

31. If = , then x equals to (1) 0(2) 12(3) 13(4) 25

32. A general wishing to draw up his 5180 men in the form of a solid square found that he had 4 men less if he may get four more men and form solid square, the number of men in the from row is? (1) 68(2) 72 (3) 78(4) 82

= ? (1) 0.02(2)0.2(3)2(4) None

is equal to: (1) 0.3 (2) 0.03(3) 0. 42(4) None

BASIC LAWS OF INDICES:

am x an = am+nan bn = (ab)nam/ an= am-nan/bn = (a/b)n(am)n = amn a-n= 1/an= (1/a)n = a(1/2) = a(1/n)a1= a a0= 1, if a 0

VBODMAS In resolving the value of a given expression the various operations must be performed in the given order.

1. Viniculum or Bar V2. Removal of brackets in the order ( ), { }, [ ] B3. Of O4. Division D5. Multiplication xM6. Addition +A7. Subtraction -S

EXAMPLE

01. Simplify 1+ of (6+ 8 x ) +

Sol: 1+ + of (6+ 8 x 1)) +

=1 + of 14 +

= 1 + 6 + =

EXERCISE

35. On simplification, the expression 1 [1 {1 (1 )}] yields (1) 0(2) 1(3) 2(4) 3

36. How many are there in 37 ? (1) 300(2) 400(3) 500(4) cant be determined

37. The value of is (1) (2) (3) (4) 38. The value of 4 - is : (1) (2) (3) (4) 39. If , the value of is : (1) 7(2) 2(3) (4)

40. In a certain office , of the workers are women, of the women are married and of the married women have children. If of the men are married and of the married men have children what part of workers are without children ? (1) (2) (3) (4)

02. L.C.M & H.C.F

Factors and Multiples: if a number m divides another number n exactly, then m is called the factor of n and n is called the multiple of m

H.C.F (Highest common factor): The H.C.F of two or more Numbers is the greatest Number that divides each one of the numbers exactly (fully) Ex: Hence H.C.F of 24 and 36 to 6 18 and 54 is 18

If H is H.C.F of two numbers a and b then H divides a and H divides b H.C.F is also called G.C.D or G.C.M

Co-Promes: If the HCF of two numbers is one then the numbers are sed to be Co-prime Ex: 10 and 21, 12 and 17 etc---

Co-prime need not be prime numbers

L.C.M (Least Common multiple): The least number which is exactly divided by each on of the given numbers is called their L.C.M. Here each number should divid the L.C.M Ex: L.C.M of 20, 30 and 40 is 120

L.C.M of Co-Primes is the product of the numbers

H.C.F always divides L.C.M fully

H.C.F and L.C.M. of Fractions:

H.C.F of fractions =

L.C.M of Fractions =

Relation between two numbers and their H.C.F and L.C.F: a and b are two numbers, then product of the numbers is equal to the product of H.C.F and L.C.M

a x b = L.C.M x H.C.F

EXERCISE

1. Find the greatest number that will divide 148, 246 and 623 leaving remainders 4, 6 and11 respectively (1)12(2) 13(3) 14(4) 15

2. Find the least number which divided by 27, 35, 45 and 49 leaves the remainder 6 in each case. (1) 6621(2) 6216(3) 2166(4) 1266

3. Find the least number which divided 36, 48 and 64 leaves the remainders 25, 37 and 53 respectively (1) 565(2) 556(3) 655(4) 555

4. Find the greatest possible length of a scale can be used to measure exactly the following lengths of cloth; 3m, 5m, 10cm and 12m, 90cm (1) 30m(2) 20m(3) 10m(4) 5m

5. Four bells first begin to toll together and then at intervals of 6, 7, 8 and 9 seconds respectively Find how many times the bells toll together in two hours and at what intervals they toll together? (1) 14(2) 13(3) 12(4) 11

6. H.C.F and L.C.F of two numbers are 16 and 240 respectively. If one of the numbers is 48, find the other number. (1) 80(2) 90(3) 100(4) 110

7. Fins out the H.C.F of 11, 0.121 and 0.1331 (1) 0.0011(2) 0.011(3) 0.010(4) 0.000011

8. Find out the H.C.F of 35, 39 and 314 (1) 35(2) 36(3) 37(4) 38

9. Find out the L.C.M of 45, 4-81, 412 and 47 (1) 412(2) 512(3) 612(4) 712

10. Find the least number which when divided by 20, 25, 30, 36 and 48 leaves the remainders 15, 20, 25, 31 and 43 respectively (1) 3595(2) 5593(3) 3955(4) 5359

11. The least number, which when divided by 2, 3, 4, 5 and 6, leaves in each case, a remainder 1, but when divided by 7 leaves no remainder. The number is (1) 121(2) 181(3) 241(4) 301

12. Find the least number of soldiers in a regiment, such that they stand in rows of 10, 15, and 25 and form a perfect square ? (1) 900(2) 1600(3) 2500(4) 40013. The product of two numbers is 12960 and their H.C.F is 36. How many pairs of such numbers can be formed ? (1) 3(2) 4(3) 5(4) None

03. PERCENTAGE Percent : By a certain percent, we mean that many hundredths. Thus, x percent means x hundredths, written as x%.

To express x% as a fraction : x % = .

To express a fraction as a percent : %

Thus. 15% = . And, %= 80%

Formula: If A is x% more than B, then B is % less than A.

If A is x% less than B, then B is % more than A.

EXAMPLES

1. A is 25 % more than B, By how much percent B is less than A.Sol: B is less than A by % = 20 %

Formula: A is x% of B, then B is % of A.

2. A is 40% of B, then what percent is B of A.

Sol: B is % of A. i.e 250% Formula: A increases by s% then resulting value is x A.

A decreases by s%, then the resulting value is x A.

0.3 . A number is decreased by 40% and then it is equal to 180. What is the original number

Sol. Let the original number be P, then P x = 180; P x = 300.

Formula: A is increases to its x% (x > 100), then the resulting value = A .

0.4.A number is increased to its 150%, then its value is 240. What is that number? Sol: Let the original number be x, then = 240; = 160.

ON POPULATION

Formula: The population of a town increases uniformly at r%. the present population of the town is p.(1) The population after years = P and increase in population P

(2) Population n years ago=

0.5. The present population of a town is 60, 000. The population increases annually at 10%. Find the population after 3 years. Sol: The required population : 60000 = 79, 860.

RESULTS ON DEPRECIATION: Let the present value of a machine be P and let it depreciate at R % per annum. Then:

(1) Value of machine after n years = P

(2) Value of machine n years ago =

06. The value of a machine depreciates at, the rate of 10% per annum. If its present value is Rs. 81000. What will be its worth after 2 years? What was the value of the machine 2 years ago?

Sol: Value of the machine after 2 years = Rs.

= Rs. = Rs. 65610.

Value of the machine 2 years ago = Rs. = Rs. 100000.

Formula ; x% of y= y % of x.

07. 20 % of 250 = 250% of 20 = 50.

Formula: To fine x % of a number, the number should be multiplied by x/100.

08. To find 33 % of a number, the number should be multiplied by which fraction? Sol. The answer = 33 % 100 or 1/3

Formula : x % of y- y% of x =0. x% of y + y% of x = 2% of (xy).

09. The difference between 5% of 900% of 5 is how much ? Sol. The answer is 0.

Formula : The tax on a commodity is diminished by x% and the sale increased by y% then the net effect on the revenue derived is %

10. If the tax on a product is decreased by 20% and the sales increased by 30% then the net effect on the revenue is % = (10- 6) %= 4% increase.

Formula: To convert percentage into fraction divide by 100.

11. 10% is = Formula: To convert a fraction into percentage multiply with 100.

12. X 100 = 16 %.

Formula: If the length of the rectangle is increased by x% then the breadth should be decreased by what % in order to maintain the same area is %

13. If the length of a rectangle is increased by 25 % then the breadth should be decreased by % to maintain the same area.

Sol. %= 20 %.

EXERCISE1. Convert into equivalent smallest fraction and equivalent decimal : 105%= ? (1) 10. 5 (2) 1.5 (3)105 (4) none

2. Convert into equivalent percentage: = ? (1)6 % (2) 8. 25 % (3)8 % (4) 8. 5%

3. 75% of 480 = ? x15 (1)48 (2) 24 (3) 42 (4) 120

4. The price of 1 kg. of rice is increased from Rs. 6 to 7. 50. By what % should be family reduce its consumption of rice so as not to get any change in the expenditure on rice? (1)25 % (2) 20% (3) 60% (4)40%

5. The wages of a man are increased by 20 % and afterwards decreased by 20 %. Find the overall percentage change? (1) 4% increase (2) 10% decrease (3) no change (4) 4% decrease

6. The production of a company is decreased by 10% and then increased 30%. Find the overall % change ? (1) 17% increase (2) 17% decrease (3) 20 % increase (4) 19% decrease

7. The length of a rectangle is increased by 20% and its breadth is decreased by 40%. Find the % charge in its area? (1)12% decrease (2) 14% increase (3) 28 % decrease (4) 22%decrease

8. By using a scale a boy measured the length and the breadth of a room. The area according to him is 400 sq.m. Later on he should that his scale is 10% less in length. Find the actual area of the room? (1) 500 sq. m (2) 493. 8 sq. m (3) 384. 9 sq. m (4) 298. 3 sq. m

9. In measuring the side of a square, on error of 10% in excess is made. The error percent in the calculated area is ? (1)20% (2) 10% (3) 21% (4) 100%

10. A man spend 10% on raw materials, 20% on buildings, 25% on machinery and now has Rs. 9 lakhs with him. find the total capital? (1) 18 lakhs(2) 10 lakhs(3) 20 lakhs(4) 24 lakhs

11. In a library 40% of the books are in English, 50% of the remaining are in Hindi, 60 % of the remaining are in Sanskrit and the remaining 7,200 books are in all other languages. Find the total number of books in that library? (1) 60, 000(2) 36, 000(3) 52, 000(4) 48, 000

12. A student has to get 40% of the maximum marks to pass examination. Amar got 150 marks and failed by a shortage of 10 marks. Find the maximum marks. (1) 320(2) 420(3) 300(4) 400

13. A students has to get 35% of the maximum marks to pass on examination Madhav got 172 marks and failed by 3 marks. Find the maximum marks? (1) 480(2) 510(3) 500(4) 460

14. P got 32% of the maximum marks and failed by 12 marks. Q secured 36% of the maximum marks which are 4 marks above the pass mark. Find the pass % and maximum marks? (1) 70%(2) 35%(3) 39%(4)53%

15. 1, 200 candidates out of which 1, 000 are boys appeared for a test. 60% of the boys and 72% of the girls passed. Find the overall failed %? (1) 38%(2) 62 %(3) 42%(4) 50%

04. PROFIT & LOSS

By selling at a cost more than the cost price, a trader gets gain and by selling at a cost less than the cost price, a trader losses. The profits and losses are calculated on cost price. Thus Profit = Selling price Cost price;Loss= Cost Price- selling Price.

Profit and Losses can be expressed as percentages also PROFIT %= ; LOSS % = 100 Formula : SP= x cost Price

CP= x Selling Price

= x Cost Price = x selling Price

EXAMPLES

1. A man bought an article for Rs. 150 and sold if for Rs. 165. What is the gain %?

Sol: Cost price = Rs 150, selling Price = Rs 165, profit = ( 165- 150)= Rs. 15. Profit %= x 100 = x 100= 10%

Formula: Cost Price of x articles is equal to the selling Price of y articles.

Then the loss or gain percent is : x 100 according as or + sign.

2. The cost price of 10 articles is equal to the selling price 8 articles. Find loss or gain %?

Sol: Gain or Loss x 100= 25, hence gain %= 25%

3. The cost of 15 articles is equal to the selling price of 18 articles. Find gain or loss %? Sol: Here y > x. Hence loss

Loss %=

Note: The sign is called modulus. Modulus means only positive value i.e = 5

4. A man sold an article for Rs. 400 and gets some loss. He sold it for Rs. 625 and gained double the previous loss. What is Cost Price?

Sol: A man sold an article for Rs. 400 Loss in 1st case= Cost Price 400 Gain in 2nd case = 625 Cost Price Hence 625 Cost Price = 2 (Cost Price- 400) = 2 (Cost price ) 800 625 + 800= 2 (Cost price) + selling price = 1425 = 3 (Cost Price) Cost Price = = 475 Cost Price = Rs. 475.

5. A man sold 1/4th of his good at 30% profit, remaining at 40% profit. What is the net gain %? Sol: Net gain % =

6. A sold an article to B at a profit of 20% B sold to C at a lloss of 10%. If C purchased the article for Rs.216. What is the Cost Price of A ? Sol: Cost price of A = Formula: Cost Price = Difference of selling Prices x

7. A man sold an article at a loss of 10%. If he had sold for excess amount of Rs. 120. He would have gained 20%. What is the C.P ? Sol: CP = difference of SPs x = 120 x

8. A man sold an article for 20% profit, if he had sold it for Rs. 60 less he would have gained only 10% Find the Cost Price? Sol: Cost Price = 60 x = 60 x 10 = Rs.600

9. By selling an article for Rs. 240 a man losses 20%. What is the Cost Price ? Sol: Selling Price = Rs. 240, Loss = 20% Cost Price = selling Price = = 240 x = Rs.300

Formula : By selling an article for Rs. Y, a man gains x% of Selling Price, then the Cost Price = Selling Price = Rs If it is loss, then cost Price =

10. By selling an article for Rs. 150, a man losses 10% of selling Price. Find the Cost Price? Sol: Cost Price = Selling Price x

Formula : x% profit on selling Price would mean gain on cost Price x % loss on selling Price means loss on Cost Price

11. By selling an article for a certain amount a man gains 20% on selling Price. What is the actual gain %?

Sol: Actual gain % = gain % on Cost Price =

12. By selling on article for a certain amount a man loss 1.3rd of selling Price. What is actual loss %? Sol: Actual loss = =

Formula: A man sold two articles each for Rs z on one he gained x% and on another he gained y% . The net gain % ?

If on second one it is x% loss, then it is always net loss of

13. A man sold two horses each for Rs. 600 on one he gained 20% and on another he gained 10% . What is the net gain? Sol: x = 20, y = 10

Net gain =

=

14. A man sold two houses each for Rs 1,00,000 one at a gain of 20% and another at a loss of 20% What is the net gain or loss?

Sol: Here always net loss of

DISHONET MILK MAN: A dishonest milk man mixes water with pure milk and sells the mixture at the cost price of milk. Thus if he mixes x liters of water, then the gains x times the cost price if 1 liter of milk, if he gains y% then he had mixed y liters of water with 100 liters of milk The ratio of water and milk in the mixture s y : 100 and % of water in the mixture =

15. A milk gains 25% by mixing water with milk. What is the ratio of milk and water in the mixture Sol: Ratio = 100 : 25 = 4 : 1

16. In a mixture of milk and water, water is 20% of the mixture , what is the gain on selling the mixture at the cost price of milk?

Sol: Water % in the mixture = 20%; milk % = 80% Amount of water for 1 liter of milk = Gain = x 100% = 25%

EXERCISE

1. An article is bought for Rs. 1600 and sold for Rs. 1840. Find the gain % (1) 15%(2) 12%(3) 16%(4) None

2. A cycle is bought for Rs. 900 and sold for Rs. 720. Find the loss % (1) 15%(2) 18%(3) 20%(4) 24%

3. A book is bought for Rs. 150 and sold at a gain of 50%. Find the selling price? (1) Es.200(2) Rs. 225(3) Rs. 230(4) Rs.None

4. A table is bought for Rs.540 and sold at a loss 15%. Find out the selling price ? (1) Rs.420(2) Rs. 525(3) Rs. 555(4) 459

5. A radio was sold for Rs. 528 at a loss 12%. Find the cost price? (1) Rs. 700(2) Rs. 900(3) Rs.750(4) None

6. An article was sold Rs. 289 at a gain at . Find the cost price? (1) Rs. 372(2) Rs. 274(3) Rs. 374(4) Rs.272

7. The profit obtained when an article is sold for Rs. 520 is same as the loss obtained when the same article is sold for Rs. 480. Find the cost price of the article? (1) Rs. 500(2) Rs. 600(3) Rs. 800(4) Rs. 300

8. A man sells two horses for Rs. 4000 each. On one he gains 5% and on the other he losses 5% Find the overall loss or gain % ? (1) 2/4%(2) %(3) %(4) None of these

9. A merchant sells two articles for Rs. 1920 each. On one he gains 20% and on the other he losses 20%. Find the total loss or gain %? (1) 4%(2) 6%(3) 3%(4) 7%

10. By selling a Radio for Rs.340 a man losses 15% For how much should he sell it to gain 15% (1) Rs. 640(2) Rs. 360(3) Rs. 460(4) Rs. 240

11. A publisher allows a book seller a discount of . If the book sells the book at the publishes price, who is the gain %? (1) (2) (3) (4)

12. By selling an article for Rs. 8000 a man losses 1/5 of its cost price. Find the cost price ? (1) Rs. 1,00,000(2) Rs.20,000(3) Rs.1000(4) Rs.10,000

13. The cost price of 10 articles is same as the selling price of 8 articles. What is the gain %? (1) 20%(2) 35%(3) 30%(4) 25%

14. The cost price of 12 articles is same as the selling price of 15 articles. What is the loss %? (1) 21%(2) 23%(3) 20%(4) 26%

15. A man purchases 11 articles at Rs . 10 and sells 10 articles at Rs. 11. Find the overall loss or gain % ? (1) 23%(2) 21%(3) 26%(4) 20%

05. RATIO & PROPORTION

RATIO : Comparison of two quantities in same units is called ratio. Thus the ratio between Rs 10 and Rs.5 is 10 : 5 or 2 : 1. The ratio is a constant. It has no units. The ratio can be expressed in the from of a fraction also. Thus a : b is also denoted by a/b In the ratio a : b; a is called antecedent and b called consequent.

In the ratio if both antecedent and consequent are multiplied or divided by a constant K, the ratio is not changed.

EXAMPLES01. Find the ratio between 1 hour 30 minutes and 2 hours 40 minutes The ratio = 1 hour 30 min : 2hours 40min = 90 min : 160 min = 9 : 16 Note: If an = bm, then a : b = m : n

02. What is A : B if A is 40% of B Sol: A = 40% of B = B = , 5A = 2B; A : B = 2 : 5 Note: xA = yB = zC, then A : B : C = :

03. 3A = 4B = 5C, A : B: C = ? Sol: 3A = 4B = 5C, A : B : C = : = = 20 : 15 : 12

Formula : derive Ratios from a : b

1. Duplicate ratio = a2 : b22. Triplicate ratio = a3 : b33. Sub Duplicate ratio = : 4. Sub Triplicate ratio = : 5. Inverse ratio of a : b is b : a

04. The areas of two circles are in the ratio 9 : 1. Find the ratio of their radii

Sol: The radii are in Sub Duplicate ratio = : = 3:1

05. The sides of two squares are in 1 : 4 then the areas are in which ratio

Sol: The areas are in Duplicate ratio = 12 : 42 = 1 : 16

PROPORTION: If two ratios are equal , then the four terms are said to be in Proportion. Thus if a b = c : d, then the terms a, b, c d are in Proportion. In this case, the terms a and d are called extremes or ends and b and c are called means And also the product of ends is equal to product of means, Thus ad = bc, we can assume = b = ar, c = ar2, d = ar3 in same cases.

If a : b = c :d then (1) = (2) = (3) = (4)

The last relation is called compoendo & dividend useful in solving some problems1. The mean proportional of a and b is 2. The third proportional of a and b is 3. The fourth proportional of a, b and c is bc/a1. If two terms are in a : b and x added to each term of the ratio and new ratio is c : d , then 1st term is and 2nd term is

06. Two terms are in 3 : 4 if 10 is added to each of the terms, they will be 4 : 5. Find the second term. Sol: 2nd term =

07. Find the mean proportional of 0.9 and 0.4.

Sol: the mean proportion is

08. Find the fourth proportional of 6, 13 and 78

Sol: The fourth proportional =

09. If the terms 10, 15, x, 135 are in proportion, find x ?

Sol: Product of means = Product of end

10. Arrange the terms 14,78, 195, 35 in order, so that they may be in proportion

Sol: Order is 35, 14, 195, 78

11. The Third proportion of two numbers is 8. First of the numbers is 18. Find the other ? Sol: Third proportion = Other number = 12

12. How much be subtracted from each of the numbers 17, 22, 32, 42 to set then the proportion

Sol: Let x be subtracted, then (17 x): (22 x) = (32 x) : (42 x) = x2 59x + 714 = x2 54x + 704 -5x = 704 714 = -10 x = 2

13. If A : B = 3 : 4, B : C = 5 : 6 , A : C ? Sol: =

Formula : If A : B = x : y , B : C = m : n then A : B : C = mx : my : ny

14. If A : B = 2:3, B : C = 4 : 5 then A : B : C = 8 : 12 : 15

Formula : if A : B : C = x : y: z; C : D = m: n then A : B : C : D = m(x : y) : z(m : n)

15. If A : B : C = 2 : 3 : 4, C : D = 2 : 3 then A : B : C : D = 2(2 : 3) : 4(2: 3) = 4 : 6 : 8 : 12 = 2: 3 : 4:6

EXERCISE

1. A : B =2 : 3 and B : C = 2 : 3 Find A : B : C + ? (1) 2:6:3(2) 4:6:9(3) 4:6:3(4) 2:3:2

2. If A : B = 3: 4 and B : C = 5 : 6 then A : B : C equals (1) 3:15:24(2)3:9:10(3) 15:20:24(4) 3:20:24

3. A, B and C enter into partnership with investments in the ratio of 5 : 7 : 8 . If at the end of the year As share of profit is Rs. 42,360. How much is the total profit (1) 1,59,440(2) Rs. 1,69,404(3) 1,59,404(4) Rs. 1,69,440

4. Rs. 5,625 is divided among A, B and C so that A receives as much as B and C together receive and B receive of what A and C together receive. The share of A is more than that of B by Rs ? (1) Es. 750(2) Rs.725(3) 675(4) Rs. 705

5. P : Q = 3 : 4 : Q : R = 12 : 13 and R : S = 26 : 45, Find P : S ? (1) 2:5(2) 2:3(3) 2:4(4) 2:2

6. Find the fourth proportion of 45,135 and 80? (1) 235(2) 180(3) 240(4) 220

7. Find the third proportion of 24 and 36 ? (1) 45(2) 34(3) 59(4) 27

8. Find the mean proportion of 4 and 121? (1) 33(2) 44(3) 11(4) 22

9. Mean proportional between 7 and 28 is : (1) 12(2) 14(3) 7(4) 28

10. In box, there are 50 paise, 25 paise and 10 paise coins in the ratio of 3 : 6 : 5. If the total value of the amount is Rs. 189, find the total number of coins ? (1) 836(2) 736(3) 756(4) 765

11. A bag contains Rs 600 in the form of one-rupee, 50-paise and 25-paise coins in the ratio 3 : 4 : 12. The number of 25-paise coins is : (1) 899(2) 905(3) 895(4) 900

12. In a bag for every 2 notes of Rs.100 there are 5 notes of Rs .50. For every 3 notes of Rs.50 there are 4 notes of Rs.10. If the value of Rs. 50 notes is Rs .2250, find the total value of the money in the bag ? (1) Rs 4550(2) Rs 4450(3)Rs 4475(4) Rs.4650

13. Vinay got thrice as many marks in mathematics as in English. The ratio of his marks Mathematics and History is 4 : 3. If his total marks in these three subjects are 250, what are his marks in English ? (1) 40(2) 42(3) 44(4) 38

14. Rs. 1050 is divided among P, Q and R. The share of P is 2/5 of the combined share of Q and R Thus, P gets Rs? (1) Rs. 320(2) Rs. 305(3) Rs. 300(4) Rs. 285

15. A garrison of 1200 men has provisions for 36 days. After 6 days 400 men left. How long will the provisions last now ? (1) 30(2) 45(3) 40(4) 35

06. ALLIGATION OR MIXTURE

Rule of allegation given us the ratio in which two quantities, one at price and another lower price, should be mixed to produce a mixture given price. This rule can be applicable in case of solutions of hence strengths also. Cost of higher price is Rs. x, the cost of lower price is Rs. y, these varieties are mixed in certain ratio to give another variety of price.

Then those two varieties should be mixed in the ratio z- y; x-z following figure explains the same.

EXAMPLE01. Two varieties of rice costing Rs. 7 a kg. and Rs. 4 a. kg. are fixed in a certain ratio and sold at Rs. 6a . kg. Without on loss or gain

Sol. Ratio of mixing = 2:1

EXERCISE

1. The quantity of water, which should be added to 100 liters of milk worth Rs. 5 per liters to reduce the price to Rs. 4 a liter.(1)20 (2) 25(3) 50(4) 80

2. I sold 100 articles, some at a gain of 20% and rest at a gain of 10% but on the whole I gained 16%. Article at gain of 10%. I sold were (1)Rs. 30(2) Rs. 20(3) Rs. 40(4) Rs. 50

3. A sum of Rs. 100 consists of 140 coins, of which some are Re, I and some 50 ps. Coin. One rupee coins are:(1)120 (2) 30(3) 90(4) 60

4. 110 litters of mixture contains 13% water. How much water must be added to get 20% water in the resulting mixture? (1) 8.6 liters(2) 9.6 litters(3) 3.2 liters(4)6.9 litters

5. Milk and Water are mixed for A grade as 4: 1 and for B grade 3: 2. A man takes equal quantities of A and B grades and mixes them. Now Milk: Water is as : (1) 9:7(2) 5: 7(3) 3: 5(4) 7:3

6. Three varieties of rice worth Rs. 4, Rs. 6, and Rs. 8 per kg. should be mixed in some ratio to obtain rice at Rs. 7 perk. Ratio is: (1) 1: 2: 3(2) 1: 4: 9(3) 1: 1: 4(4) 1:1: 2

7. In a mixture 80% is syrup; How much syrup should be added to get 90% syrup of the syrup initially present in the mixture? (1) 75%(2) 125%(3) 115%(4) 85%

8. A cup of milk contains 3 parts of pure milk and one part of water. How much of the mixture must be drawn and water substituted in order that the resulting mixture may be milk water? (1)1/3(2) 1/2(3) 3/4 (4) 1/4

9. Two Kinds of rice at Rs. 7.20 per kg. and Rs. 6.40 per kg. be mixed in a some ratio so that mixture becomes worth Rs. 7 per kg. What is that ratio ? (1) 1:1(2) 3:1(3) 1:3(4) 3:2

10. 70 litters of medicine has 10% of water. How much water should be added to get 37% water in the resulting mixture? (1) 30 litters(2) 60 liters(3) 80 liters(4) 40 liters

11. 18 litters are taken out (and replaced with water) from a 54 liters cash full of milk. It is done two times again. What is the quantity of milk now left? (1) 32 liters(2) 36 liters(3) 16 liters(4) 26 liters

12. A mixture of 80 gallons of wine and water contains 20% water. How much water must be added to it to increase the percentage of water to 24? (2) 5 (3) (4) 3

13. At an examination 280 candidates appeared and 60% passed. Pass percentage of boys was 70 and that of girls 35. What was the number of girl candidates? (1) 126(2) 60(3) 110(4) 80

14. The ratio in which the two qualities of rice at Rs. 6 and Rs. 4 per kg. be mixed respectively in order the mixture may cost Rs. 4.80 paisa per Kg? (1) 3:2(2) 6:8(3) 2:3(4) 4:3

15. What is the proportion in which a gricer must rice at Rs. 102 p. a kg. with the other quality of rice at Rs. 144 p. so as to make a mixture wart hrs. 126 p. a kg? (1) 2:3(2) 3:4(3) 3:6(4)4:3

07. PROBLEMS ON AGES

INTRODUCTION:

One of the practical use of linear simultaneous equations, is the PROBLEMS BASED ON AGES. While doing the problems, consider the present age of a man be xyears. If we have to find his the age before y years than subtract y years ti x i.e, x + y.

If the difference between the ages of two persons is x years, life long the difference between them would be x years only. If A is elder to B, this happens throughout their life. B never exceeds A (which is impossible)

EXAMPLES

1. A man after 15 years will be four time of that age which was 15 years back. Find his present age

Sol: let his present age be x years, his age 15 years back = x 15 , and age after 15 years = x + 15 Then according to the problem age after 15 years = 4 (age 15 years back) i.e (x + 15) = 4( x 15) or x + 15 = 4x 60 or 3x = 75 X = 25 years

2. My brother is three years older than me. At the time of birth of my sister my fathers age was 28 years and at the time of my birth my mothers age was 26. At the time of the my brothers birth my sisters age was four years. What was my fathers and mothers age at the time of my bothers birth?

Sol: let present age of my father is x years and that of my mother is y years Fathers age at the time of birth my sister = 28 years Mothers age at the time of my birth = 26 years According to the problem x 28 = 4x = 32 years y 26 = 3y = 29 years

3. The sum of ages of A and b is now 110 years and their ages 20 years ago were in the ratio of 4 : 3 .Find their ages.

Sol: Let present ages of A and B are x and y years respectively. Then X + y = 110------(1) According to second condition (x 20) : (y 20) = 4 : 3 or or 3x - 60 = 4y - 80 or 4y = 3x + 20 ----(2)

From (1), x = (110 y) , putting the value in (2) we have 4y = 3(100 y) + 20 + 350 3y or 7y = 350 y = 50 then from (1)x = 60, Hence age of A = 60 years, and age of B = 50 years

4. The sum of the ages of father and his son is 88 years. If the ratio between their ages is 7 : 4 then find their ages?

Sol: Ratio = 7 : 4 Let their ages be 7x and 4x and their sum = 11x but 11x = 88 x = 8 then Fathers age = 7 x 8 = 56 years Sons age = 4 x 8 = 32 years

5. The ratio between the ages of Ravi and Karuna after 5 years will be 4: 5 if the sum of their present ages is 17 years. What is the present age of Ravi?

Sol: Ratio in the ages of Ravi and Karuna after 5 years is 4 x : 5x Ratio in the ages of Ravi and Karuna at present = 4x 5 : 5x 5 but 4x 5 + 5x 5 = 17 9x 10 = 17 x = 3 Present age of Ravi = 4 x 3 5 = 7 years (or) Present age of Ravi =

6. At the time of marriage a man was 6 years elder to his wife but 12 years after the marriage his age was 6/5 time the age of his wife. What were their ages at the time of marriage ?

Sol: Let the age of the wife at the time of the marriage be x years then the age of the husband at the time of the marriage = x + 6 years but after 12 years x + 6 + 12 = 6/5 (x + 12) 5x + 90 = 6x + 72 x = 90 72 = 18 At the time of the marriage, their ages were 24 & 18 years respectively

7. If from the present ages of Ravi, 6 years is subtracted and the remainder is divided by 18, the result is the present age of his grqadson Gopi. If Gopi is 2 years younger to Praksh and Paksh at present is 5 years old, What is the present age of Ravi ?

Sol: Let the present ages of Ravi & Gopi be x & y years respectively According to the first condition According to the second condition y = 5 2 = 3 X = 18(3) + 6 x = 60

EXERCISE1. A mans age after fifteen years will be four time of his that age which was fifteen years ago. His present age is : (1) 25 years(2) 30 years(3) 45 years(4) 35 years

2. Ten years before Neerus age was four time of Rashmis age. After ten years Neerus age will be double of Rashmis age. What is Neerus present age? (1) 38 years(2) 47 years(3) 50 years(4) 45 years

3. Ratio between the present ages of Praksh and his daughter is 3 : 1 after five years, this ratio will change to 7 : 3. what is daughters present age ? (1) 7 years(2) 10 years(3) 9 years(4) 12 years

4. Ratio between Rams present age and Shyams present age is 7 : 9 Nine years ago, their ages were in the ratio of 2 : 3, the present age of Shyams is : (1) 19 years(2) 23 years(3) 18 years(4) 27 years

5. Four years ago, the ratio of the ages of Mahesh and Shalilesh was 3 : 5 , If the sum of their present ages is 40 years, What is Shaleshs present age in years? (1) 22 years(2) 18 years(3) 24 years(4) 28 years

6. The ratio between the present ages of Sudhir and praksh is 4 : 5 Seven years ago, the ratio of their ages was 3 : 4 . What is the present age of Prakash ? (1) 37 years(2) 40 years(3) 35 years(4) 33 years

7. The sum of ages of A and B is 110 years and either ages 20 years ago were in the ratio of 4 The age of A in years is : (1) 48 years(2) 54 years(3) 60 years(4) 55 years

8. The age of a man is three times the sum of the ages two children. Five years hence his age will be double the sum of their ages. What is the present age in years ? (1) 40 years(2) 45 years(3) 30 years(4) 43 years

9. The sum of the ages of son and father is 56 years. After 4 years, the age of the father will be 3 times that of the son. Their present ages respectively are: (1) 12 years&44years(2) 12 years & 48 years (3)22 years & 44years(4) 18 years & 35 years

10. Five years hence the ratio between the ages of Gopi and Sudha will be 4: 5. The total of their ages at present is 17 years. What is the present age of Gopi ? (1) 8 years(2) 6 years(3)4 years(4) 2 years

11. Four years ago a mother was six and a half times as old as her son. After three years she will become thrice as old as her son. What is her sons present age? (1) 7 years(2) 8 years(3) 9 years(4) 10 years

12. The age of the father 10 years ago was thrice the age of his son. The years hence, the fathers age will be twice that of his son. What is the fathers age ? (1) 70 years(2) 68 years(3) 74 years(4) 65 years

13. The age of Arvinds father is 4 times his age. If 5 years ago Father age was7 time of the age of his son at that time .What is Arvinds fathers present age? (1) 41 years(2) 38 years(3) 44 years(4) 40 years

14. The sum of ages f father and son is 45 years. Five years ago, the product of their ages was four time the fathers age at that time. The present age of the father is : (1) 28 years(2) 36 years(3) 35 years(4) 37 years

15. 10 years ago Chadraravatis mother was four times older than her daughter. After 1o years , the mother will be twice older than the daughter. The present ago of chandravati is : (1) 25 years(2) 20 years(3) 15 years(4) 22 years

08. PARTNERSHIP

Two or more partners start a business jointly such business is called Partnership business and they are called Partners. The divid the profits at the end of the year in the ratio of their investments.

The partner who manages the business is called managing or working partner. Other partner is known as sleeping partner.

Money received by managing partner= profit + salary

Money received by sleeping partner = profit ( only)

If A, B invest for different periods of time, the profits are divided in the ratio of their capital & time products.

=

EXERCISE

1. In a business, A, B and C invested Rs. 50, 000 Rs. 60, 000 and Rs. 75, 000 respectively. Find the share of C in the total profit of Rs. 9, 250. (1) Rs . 3000(2) Rs. 3570(3) Rs. 3750(4) None

2. Investments of A, B and C are Rs. 2, 500 Rs. 5, 000 and Rs. 4, 000 . If the share of C is Rs. 1, 200 more than that of A in the annual profit, find the share of Bin the annual profit.(1) Rs. 3500(2) Rs. 3800(3) Rs. 4000(4) Rs. 4200

3. In a business, A contributes Rs. 2, 500 for 8 months, B contributes Rs. 16, 000 for 6 months and C contributes Rs. 20, 000 for 4 months. If C s share in the profit is Rs. 5, 000 find the total profit (1) Rs. 17 000(2) Rs. 17250(3) Rs. 18200(4) None 4. Raju and Krisha entered into a partnership with Rs. 20, 000 and Rs. 30,000 respectively. After 4 months Raju put in Rs. 5, 000 more. Find the difference between their share in the annual profit of Rs. 24,000 (1) Rs. 3 000(2) Rs. 3200(3) Rs. 3500(4) None

5. A and Bentered into a partnership with Rs. 30, 000 and Rs. 60, 000 respectively. After 4 months A. invested Rs. 15,000 more while B withdrew Rs. 30, 000. Find the share of A in annual profit Rs.1, 00, 000. (1) Rs. 45 000(2) Rs. 5000(3) Rs. 48000(4) None

6. A and B entered into a partnership with Rs. 40, 000 and Rs. 60, 000 respectively. After 4 months they invested Rs, 10, 000 each. Find the difference between their shares in the annual profit of Rs. 1, 36, 000 (1) Rs. 2,400(2) Rs. 2,500(3) Rs. 2600(4) None

7. A and B entred into a partnership with Rs. 12, 000 and Rs. 20, 000. B was a sleeping partner. At the end of the year they received Rs. 12, 000 and Rs. 10, 000 respectively. Find the monthly salary of A. (1) Rs. 400(2) Rs. 500(3) Rs. 600(4) None

8. A and B started a business by investing Rs. 15, 000 nad Rs. 20,000 respectively. After 3 months A invested Rs. 10, 000, while B withdrew Rs. 5, 000. If As share in the annual profit is Rs. 4, 050, find the share of B in the annual profit? (1) Rs. 3240(2) Rs. 3675(3) Rs. 3800(4) None

9. A, B and C started a business by investing Rs. 20, 000 Rs. 25, 000 and Rs 40, 000. They decided to receive 10% interest on their capitals and the balance to be divided equally. If they got Rs. 16,000 as the annual profit, find the share of C including the interest. (1) Rs. 7 000(2) Rs. 6500(3) Rs. 6000(4) None

10. A and B started a joint firm. As investment was thrice the investment of B and the period of his investment was two times the period of investment of B. If B got Rs. 4, 000 as profit, then their total profit is Rs.? (1) Rs. 25 000(2) Rs. 27000(3) Rs.28000(4) None

11. A, B and C invest Rs. 2, 000 Rs. 3, 000 and Rs. 4000 in a business. After one year, A removed his money but B and C continued for one more year. If the net profit after 2 years be Rs. 3, 200, then As share in the profit is Rs.? (1) Rs. 700(2) Rs. 750(3) Rs. 800(4) None

12. In a business A, B and C invested Rs. 36. 000 Rs. 63, 000 and Rs. 48, 000 respectively. Find the shares of C in the total profit of Rs. 3, 43, 000. (1) Rs. 112000(2) Rs. 111000(3) Rs. 112800(4) None

13. In a business A, B and C invested Rs. 16, 000 Rs. 20. 000 and Rs. 28, 000 respectively. Find the difference between the shares of A and C in the total profit of Rs. 1, 44, 000. (1) Rs. 4,00, 000 (2) Rs. 4, 32,000 (3) Rs. 4, 60,200 (4) None

14. A, B and C enter into a partnership. A contributes Rs. 36, 000 for 8 months, B contributes Rs. 48, 000 for 6 months and C contributes Rs. 72, 000 for 4 months. The share of C in the total profit is Rs. 1, 44, 000. Find the total profit. (1) Rs. 4, 00,000 (2) Rs. 4, 32, 000(3) Rs. 4, 60, 200(4) None

15. A started a business by investing Rs. 25, 000. After 3 months B joined. At the end of 9 months from the beginning they shared the profit in the ratio of 15: 16. What was the capital of B? (1) Rs. 1,12, 000(2) Rs. 1,22,000(3) Rs. 1, 24,000(4) None

09. AVERAGE

Formula: The average of a observations = or A= where A= Averages ; T= Total ; n= number of things:

Formula : If each observations is increased by K, then the average also increases by K. If each observation is decreased by K, then the average also decreases by K.If each observation is multiplied or divided by K, the average is also multiplied or divided by K

Formula: The average of first n natural numbers is ( n+ 1)/2. The average of n consecutive integers including and between a and b is (a+b)/2. The average of odd Ntural numbers between a and b = (a+b)/2. The average of even natural between a and b is (a+ b)/2.

Formula: The average of m things is ; the average of n other is . Then the averse of (m + n) things is

Formula: The average of m things is ; the average of n things out of the m things is . Then the averse of remaining things is is

EXAMPLES

0.1. Find the average of first 20 natural numbers. = 10.5

0.2. The average weight of 20 boys is 50 kg. Each boy puts on a dress whose weight is 1 kg. what is the average weight of the boys now.

Sol: 50 kg.+ 1 . kg = 51 kg.

0.3. The average age of 40 students in a class is 12 years. 10 students whose average age is 10 years joined them. What is the present average age. Sol. Present average = = 11. 6 years.

0.4. The average weight of 35 men is 70 kg. Another men whose weight is 88kg joined. What is the present average weight? Sol: Present average weight = Past average + Km./h 70 kg + = 70 kg + kg = 70. 5 kg. Formula: A certain man travelled a distance of x km. partly x1 km. at v1 speed: x2 km. at v2 speed and so on. The average speed is km./h.

0.5 A car travelled 20% of the total distance at 10 km./h., 30% at 20% km/h. and the rest at 40 km./h. what is the average speed? Sol:. Average speed = km/h =

0.6. A man travelled a distance at x km./h. and came back at y km./h. the average Sol. The average speed = km./h. = km./h.

Formula: The average of first n even numbers is : n+ 1.

0.7.The average of first 6 even numbers is 6+ 1 = 7

Formula: The average of first n natural is n

08. The average of first 5 odd numbers is 5

Formula: The average of even numbers till n is + 1

09. The average of even numbers till 6 is + 1 = 4

Formula : The average of odd numbers till n is

10.The average of odd numbers till 5 is = = 3

Formula: The average of squares of first n natural numbers is

11. The average of squares of first 11 numbers is = = 46

Formula: The average of cubes of first n natural numbers is

EXERCISE

1. Average of 20, 24, 32 x, 16 and 30 is 25 find x = ?(1) 26(2) 28 (3) 27 (4) 28. 5

2. The average of three numbers is 45 and that of the last two is 50. Finds the first numbers?(1) 32(2) 34(3) 35(4) 33

3. Average of 5 numbers is 15 and that of the first four is 12. What is the lat number?(1) 27(2) 24(3) 26(4) 28

4. The average of 8 numbers is 45. If each number is divided by 5. The new average will be :(1) 9(2) 8.7 (3)6(4) 8

5. Average of 100 numbers is 100. If the first is increased by I , the second number by 2, the third numbers by 3 and so on. What will be the new average?(1) 145. 5(2) 152. 0(3) 148.5(4) 150.5

6. My average marks in English, Sanskrit, Hindi, Maths and Since are 72 and in maths, Science, Social, Hindi and Sanskrit are 80. Marks in Social exceed the marks in English by ?(1) 38(2) 39(3) 40(4) 41

7. The average age of the father and his 6 children is 12 years, which is reduced by 5 years if the father be excluded. How old is the father?(1) 36 years (2) 42 years(3) 38 years(4) None

8. The average age of a wife and a husband, who were married 6 years back was 22 years then. Including two sons, who were born during the interval, the average age of the family is 15 years now. If one son is 3 years old now. Find the age of the second son at present?(1) 1 year(2) 2 year(3) 3 year(4) None

9. There are four numbers. Average of the first 3 is 15 and that of the last 3 is 16. If the last number is 19, find the first number?(1) 12(2) 16(3) 20(4) 24

10. Of the 6 numbers, average of the five is 16 and that of the last five is 20. If the last numbers is 20. Find the first numbers?(1) 20(2)0(3) 24(4) 26

11. The average of 11 results is 32. If the average of first 6 results is 34 and that of last 6 results is 33, the sixth result is ?(1)48(2) 49(3) 50(4) None 12. The average age of a husband and a wife, who were married 7 years ago, was 25 years then. The average age including a child which was born during the interval is 22 years now. How old is the child now?(1)3(2) 4(3) 5(4) 2

13. The average age of a husband and wife was 23 years when they were married 5 years ago. The average age of the husband, the wife and a child who was born during the interval, is 20 years now. How old is the child now?(1) 3 years(2) 4 years(3) 5 years(4) None

14. Of the three numbers, the first is 2/3rd of the second, and the 2nd is 4/5th of the third. If the average of three numbers is 35, then value of the first numbers is:(1)20(2) 22(3) 26(4) 24

15. Of the three numbers, the first is twice is second and three times the third. The average of the three numbers is 66, and the three numbers in order are?(1)107, 54, 37(2) 108, 53. 36(3) 107, 54, 36(4) 108, 54, 36

10. SIMPLE INTEREST

If a man borrows some money from any other body, he has to pay some extra money in addition to the sum borrowed to th lender. That extra money is called interest. If the interest is calculated uniformly through out the period, the methos is called Simple Interest methos. The money borrowed is called Priniipal the amount at the end of the 1 period payable including interest is called Amount Generally rate of interest is expressed as a percent per annum.

According to this methos, the simple interest payable at R % p.a for a period of T years, on the principal P is Simple Interest = and A = P + Simple Interest , A = P + =

Formula : If the sums P1, P2, P3 yield same interest in N1, N2 and N3 years at R1, R2, R3 rates of interests then P1 : P2 : P3 = : : and P1 : P2 = N2 R2 : N1 R1

EXAMPLES01. Rs. 11,000 was given for SI partly at 5% for 5 years and remaining at 10% for 3 years to give same interest. What is the 2nd sum given ?Sol: P1 : P2 = N2 R2 : N1 R1 = 3 x 10 = 5 x 5 = 30 : 25 = 6.5 P2 = Formula : If the sums P1 , P2, P3 yield same amounts then P1 : P2 : P3 = : : And P1 : P2 = (100 + N2 R2 ) : (100 + N1 R1)

02. A sum of rs. 18,000 is divided into two parts. One part is given at 10% for 5 years and another part at 5% for 4 years. The amounts received on these two parts are same. What is the 1st part?Sol: P1 : P2 + (100 + N2 R2) : (100 + N1R1) (100 + 5 x 4) : (100 + 10 x 5 ) = 120 : 150 = 4 : 5 1 st part = 18000 = Rs . 8000

Formula: If a sum becomes n times in T years ar R% p.a then 100 ( n-1) = TR or T= if the sum doubles, then n= 2

Formula: If a sum becomes m times in Y1 years, then it takes to become n times in Y2 years then years.

03. A sum becomes 10 times in 18 years. In how many times it will become in 6 years? Sol. => = => 3 = => n- 1 = 3 => n= 4.

The sum becomes 4 times in 6 years.

INSTALMENT PAYMENTS: A sum of Z rupee is taken for n year at r% p. a payable at the end of every year in equal installments, if the installment is x rupees then x is got by the relation nx +xr

04. An amount of Rs. 2300 is due after 4 years at 10 % payable in 4 annual equal installments. What is the installment?Sol. Here z = 2300, n = 4, r= 10 Let the instalment be Rs. X then 4x + = 2300 = 2300; x= 500 Each annual instalment = Rs 500.

Formula: If a sum gets x rupees more when it is given for r% more, then x =

05.A sum given for SI fetches Rs. 50 more as interest if it is given at 10% interest of 8% for 5 years. Find the sum given?

Sol. , x= Rs. 50 r= 10- 8 = 2 ,N = 5 => = Rs. 500

06. A certain sum amount to Rs. 2880 in 4 years and Rs. 3120 in 6 years. Find the sum and rate of interest?Sol: A4= 2880, A6= 3120; I2= A6 A4 = 3120- 2880 = 240; I1 = 120, I4 = 120 X 4 = 480 P= A4 I4 2880 480 = 2400.Hence on Rs. 2400. Interest for 2 years is Rs. 240. R= = = 5; P= 2400, R = 5%

EXERCISE

1. Find the simple interest on Rs. 900 for 2 years at 3% p.a (1) Rs. 50 (2) Rs. 54 (3) Rs. 62 (4) None

2. Find the simple interest on Rs. 18000 for 3 years and 4 months at 4 % p. a (1) Rs.225 (2) Rs. 250 (3) Rs. 270 (4) Rs. 255

3. The simple interest on a certain sum of money for 2 years at 3% p.a Rs. 480. Find the sum (1) Rs.8000 (2) Rs. 7200 (3) Rs. 8400 (4) Rs. 8800

4. The simple interest on a certain sum of money for 4 years at 2% p.m is Rs. 3200 find the sum. (1) Rs.33000(2) Rs.42000 (3) Rs. 37000(4) Rs. 40000

5. In what time will the simple interest on Rs. 5000 at 9% p.a be Rs. 1800. (1) 2 years (2) 3 years (3) 4 years (4) 5 years

6. In what time will the simple interest on Rs. 25 00 at 4% p.a be Rs. 625. (1) 6 years(2) 7 years (3) 5 years (4) None

7. At what rate % p.a will the simple interest on Rs. 7500 for 6 years be Rs. 45 00. (1) 9%(2) 11%(3) 12%(4) 10%

8. The simple interest on Rs. 4, 800 for 3 years and 5 months is Rs. 984. Find the rate percent per annum. (1) 5%(2) 6%(3) 7%(4) None

9. A lent Rs. 12, 000 at simple interest for 3 years at 16% p.a . Find the amount to be received. (1) Rs.17760(2) Rs.18840 (3) Rs.167700(4) None

10. What sum will amount to Rs. 6050 in 3 years at 7% p.a simple interest? (1) Rs.4500(2) Rs.5000 (3) Rs. 5200(4) None

11. In how many years will a sum of money become double itself at 5% p.a simple interest? (1) 18 years (2) 19 years (3) 20 years (4) 21 years

12. In how many years will a sum of money become thrice at 12 % per annum simple interest? (1) 14 years (2) 15 years (3) 16 years (4) None

13. At what rate % p.a will a sum of money become twice in 16 years? 1) 5 %(2) 6 %(3) 7 % (4) None

14. A part of a certain sum of money is invested at 9%p.a and the rest 12% p.a if the interest earned in each case for the same period is equal, the ratio of the sums invested is: (1) 4:3 (2) 5: 3(3) 5: 2(4) None

15. A part of the sum of Rs. 3, 00 is invested at 5% p.a and the rest at 6% p.a. the whole annual interest received was Rs. 162. Find the money lent at 6% p.a (1) Rs. 1150 (2) Rs. 1200 (3) Rs. 1250 (4) Rs. 1300

11. COMPOUND INTEREST

If an Compound Interest method, interest is calculated on the interest again. If the borrower does not pay the interest every year to the lender, the lender considers the unpaid interest as sum given and calculates interest on this accrued interest also. Hence for more than one year period, the compound Interest receivable will be more than the simple interest for the same period at the same rate.

The interest may be calculated quarterly. Half- yearly or annually. Generally, it is annually calculated. If the interest is calculated half- yearly or quarterly the rate of interest should also be reduced to that period.

In the working of problems, consider amount = A, sum given = P: no of times the interest is calculated = n: The rate of interest for the period of calculation of interest = r

Formula:(1) Interest calculated annually amount at the end of n years = A = P (2) Interest calculated half- yearly, No. of times interest is calculated = n, then A = (3) Rate of interest for 1st years r1% for 2nd year r2 % and so on. Then A= ..(4) If the total period of time is n years and y part of a year ( like 2 years 9 months = 2 years or 2 years) then A= P , hence for 2 years 9 months period A= P (5) if a sum Rs. z is due after n years and is payable in n annual equal instalments then x x + + x = z where x is the annual instalment.(6) If recent worth of a sum Rs. Z due after n years at this method is (7) If the interest is calculated other than one year, then the equivalent compound interest for one year is called the effective annual rate.(8) If the difference of compound interest and simple interest for 2 years at R% on P Rs. Is Rs. d then d= (9) For more than 2 years. D= P . here compound interest is calculated annually only.

EXAMPLES

01. Find the Compound Interest on Rs. 2000 at 10% p.a for 1 years. Sol: P= 2000 r1 =10, r2= ; A= P = 200 x X = 2310 CI = A- P = ( 2310- 2000) Rs = Rs. 310.

02. Find the difference of CI and SI at 8% p.a on Rs. 1600 for 2 years.Sol. D = = Rs. = Rs.= Rs. 10. 24

03. If the difference of CI and Si on sum foe 3 years at 5% is Rs. 30. 50. Find the sum?Sol : d = 30. 50 = 30 = , R= 5, n= 3We know d= p = P = P X = 4000 the sum Rs. 4000.

100.

04. A sum becomes Rs. 2000 Rs. 2200 in two consecutive years, find r?Sol. X = 2000, y= 2200, r= 100 = 100= 10 Rate of interest = 10% p.a

05. A sum becomes Rs. 4800 in 4 years and Rs. 3600 in 2 years find the sum?Sol. A4= 4800, A2= 3600; = ; A2 = p ; 3600 = p ; P = 36 00 X = 2700.

becomes Rs. 1452 in 2 years at 10% on method. Find the sum?1452, P = ?, n = 2 , r= 10; A = P P = P x ;X = 12 X 10 X 10 = 1200. Sum givem = Rs. 1200.

Formula: If a sum given on Compound interest method becomes x times in y years then it becomes xn times in ny years.

07. A sum given on CI method, becomes 3 times in 4 years, how many times it becomes in 12 years. Sol. X= 3, y= 4, ny = 12, n= = 3 => xn = 33 = 27. hence the sum becomes 27 times in 12 years.

08. A sum doubles itself in 5 years. How much time it takes to become 16 fold. Sol. x= 2, y= 5 Now 16 = 24; n= 4 to become 16 times, it takes ny years or (4 x 5) years that is 2years.

EXERCISE1. Find the amount of Rs. 2, 400 after 3 years, when the interest is compounded annually at the rate of 20% per annum. (1) Rs. 4, 417.30 (2) Rs. 4, 147,20(3) Rs. 4, 741. 60 (4) Rs, 4, 471.40

2. Find the compound interest on Rs. 800 for 2 years at 12 % per annum compounded annually. (1) Rs.312.50 (2) Rs.2 42, 50(3) Rs. 212.50 (4) Rs, 213. 50

3. Vandana deposited Rs. 8, 000 with a finance company for 3 years at an interest of 15% per annum. What is the compound interest that Vandana gets after three years ? (1) Rs. 4, 167 (2) Rs.5, 167(3) Rs. 4, 767 (4) Rs, 4, 168

4. Find the amount of Rs. 1, 000 in 1 year at 2 % for annum when the interest is compounded half- yearly (1) Rs.14, 040, 10 (2) Rs. 1. 025.10 (3) Rs. 1, 820.10 (4) Rs, 1,020.10

5. Compute the compound interest on Rs. 4, 000 for 1 % years at 10% per annum, compoindhalfyearly. (1) Rs. 730.50 (2) Rs.830.50(3) Rs. 630.50 (4) Rs. 930.50

6. Sanjay a hut at a cost of Rs. 700. If the borrowed this sum at 20% per annum compounded six monthly, find the compound interest that he pays after one a half year. (1) Rs. 431.70 (2) Rs.231.70(3) Rs.251,70 (4) Rs. 631.70

7. Find the amount and the compound interest on Rs. 100000 compounded quarterly for 9 months at the rate of 4% p. a (1) Rs. 2,03.030 (2) Rs.1,03,030(3) Rs. 1, 43, 030 (4) Rs,1,93, 030

8. In what time will be Rs. 800 amount to Rs. 882 at 5% per annum? Interest compounded annually. (1) 8 years (2) 6 years (3) 2 years (4) 9 years

9. The interest on a sum of Rs. 2, 000 is being compounded annually at the rate of 4% per annum. Find the period for which compound interest is Rs. 163. 20 (1) 2 years (2) 5years (3) 3 years (4)7 years

10. In how many years a sum of Rs. 6400 compounded quarterly at the same of 5% p. a will amount to Rs. 6461? (1) 9 months (2) 8 months (3) 6 months (4) 1 months

11. Find compound interest on Rs. 5600 for 1 % years at 10% per annum compounded annually. (1) Rs. 867 (2) Rs. 668 (3) Rs. 868 (4) Rs. 886

12. In how many years will a sum of Rs. 800 at 10% per annum compounded semiannually becomes Rs. 926.10? (1) 5 years (2) 3 years (3) 2 years (4) 2 years

13. A sum of Rs. 1200 becomes Rs. 1323 in 2 years at compound interest compounded annually. Find the rate of interest. (1) 5%(2) 10%(3) 20 %(4) 15%

14. Find the rate of compound interest when Rs. 8, 000 amount to Rs. 9261 in 1 years if interest is compounded half yearly. (1) 5%(2) 10%(3) 25%(4) 20%

15. Find the compound interest on Rs. 1800 for 3 years if the rates of interest for the successive years are 4% 5% and 10%. (1) Rs. 4, 621. 60 (2) Rs. 9, 621, 60 (3) Rs. 2, 612. 60 (4) Rs. 3, 621. 60

12. TIME & WORK

If A can do a piece of work in x days, then he can do 1/xth part of work in one day. Again if B can do the same work in y days, he can do 1/yth part of work in one day. If they two together do the work they can do th part of the work in one day. That is th part of the work is done in one day. Hence to do the full work they take days. Similarly if C also joins them, if can do the work in z days, they three can complete the work in days i. e. days. Their capacities will be in proportion with their one days work. Hence the ratio of capacities of A and B is : or y : x Similar the ratio of capacities of A, B and C is : : or yz: zx : xy .

The wages are paid according to their capacities. The more the capacity, the more than wage. If the people working together get the total wage, amount is shared among the people in the ratio of their capacities.

If the number of men to do a piece of work be changed in the ratio m: n then time taken by them to finish the work in changed in the ratio n: m.

As is twice as good as a work man as B then(i) Time taken by A and B are in the ratio 1: 2(ii) Work done by A and B are in the ratio 2: 1

EXAMPLES

1. A and B can do a work in 10 and 15 days individually. In how many days, they can complete the work working together. Find also the rate of their capacities?

Sol . The No. of days they both finish the work working together = = 6 The ratio of their capacities = y : x = 15 : 10 = 3: 2

2. A. B and C with do a work in 20, 30, 40 days indecently. In howmany days they can completer the work working together?

Sol . The no. of days they three can complete the work = = = = = 9

3. P can do of a work in 10 days and Q can do 1/3rd of the same work in 8 days. What is the ratio of their capacities?

Sol . P of work in 10 days Q : 1/3 work in 8 days Full work in 20 days Full work in 24 days Ratio of their capacities = 24: 20 = 6: 5

4. X and Y can do a work independently in 20 days and 24 days. Y started the work and did for 6 days. The remaining work was done by x in howmany days?

Sol. Let x work for n days, then 6X + n x =1 = 1- ; n = x 20 = 15 days.

5. A and B can do a work independently in 20 and 30 days. They two started the work 2 days before completion of the work. A and B dropped from the work. The reaming work was completed by C in 10 days. In how many days, C alone can do the work?

Sol. The work done by A and B in 2 days = 2 = . this work was done by C in 10 days. Number of days, C alone can do the full work = 10 X 6= 60 6. A work can be done by A and B is 15 days; by B and C in 20 days and by C and 30 days. In how many days can C alone do the work?

Sol. C can complete the work in days = = = 120 days.

7. A is 40% more efficient than B, if B can do a work in 21 days in howmany days A can do the same work?

Sol. A can do the work in 21 X days i.e in 15 days.

8. The efficiencies of two persons are in 3: 2 the no. of days they can finish a work will be in ratio?

Sol. In 2: 3 Since efficiently ratio is the inverse ratio of the on. Of days.

9. A, B and C can do a piece of work in 15, 20, 30 days respectively. B and C started the work and after some days 4 joined them. Thw work finished in 8 days from the beginning. When did A join them ?Sol. B and C worked = = = = 15 x = 15 x = 5. Hence A joined B and C after 3 days.

EXERCISE

1. 30 men can do a piece of work in 24 days. In how many days can 18 men do it ? (1) 40 days (2) 60 days (3) 36 days (4) 54 days

2. 16 men can do a piece of work in 45 days. How many men are required to complete it in 20 days? (1) 18 (2) 48 (3) 36(4) 40

3. 8 men can do job in 27 days. How many more men are required to complete it in 18 days? (1) 3 (2) 12 (3) 6(4) 4

4. 24 men can do a job in 21 days working 8 hours a day. In how many days can 16 men do it working 9 hours a day?(1) 89 (2) 28 (3) 30(4) 48

5. 6 men working 8 hours a day can earn Rs. 2, 400 in 10 days. In how many days can 15 men working 6 hours a day earn Rs. 1, 800?(1)5 (2) 9 (3)4(4)7

6. 20 men can complete 2/5 of the work in 20 days working 8 hours a day. How many more men will be required to finish the remaining work in 16 days working 10 hours a day?(1)10 (2) 12 (3) 13(4) 74

7. A and B can do a piece of work in 60 days and 40 days respectively. In howmany days can they finish it together?(1) 41 (2) 24 (3) 44(4) 45

8. A and B can do a job in 16 days and A alone in 24 days. In how many days can B alone do it? (1) 56 (2) 41 (3) 48(4) 85

9. A, B and C can do a piece of work in 18 days, 24 days and 36 days respectively. In how many days can they finish it if they works together? (1) 12(2) 56(3) 74(4) 8

10. A and B can do a job in 40 days, B and C in 24 days and C and A in 60 days. In how many days can all of them together finish it? (1) 45 (2) 12 (3) 10(4) 15

11. A and B can do a job in 30 days and 40 days respectively. They work together for 12 days and then A leaves. In how many days can B finish the remaining work? (1) 12 (2) 14 (3) 13(4) 16

12. A and B can do job in 30 days and 40 days respectively. A works for 16 days and then B joins him. in how many days more will remaining work be finished? (1) 5 (2) 6 (3) 8(4)4

13. A and B can do a job in 30 days and 40 days respectively. They begin the work together and after some days A leaves B finishes the remaining work in 5 days. After how many days does A leave? (1) 20 days (2) 30 days(3) 15 days(4) None

14. A can do job in 30 days. He works for 6 days. B completes the remaining work in 20 days. In how many days can B alone complete the whole work? (1) 45 (2) 25 (3)35(4) 55

15. A and B can do a piece of work in 36 days and 60 days respectively. They finish a job for Rs. 2, 800. What is the share of A?(1) Rs. 1850(2) Rs. 1547(3) Rs. 4562(4) Rs. 1750

13. PIPES & CISTERNS

If a tap fill sin the tank with any liquid, the tap is called filling tap or inlet. If the tap empties the tank the tap is called emptying tap or outlet. Some times inlets and outlets will be working at the same time. Then the tank is filled in slowly. If the outlets empty more liquid than filling in liquid, it is not possible for the tank to be full for any long time. By the by if the tank catain liquid initially the tank will also be emptied gradually.

If two filling taps fill the tank in xh and yh then the tank will be full h. If the at the same time if an empty is also opened, then the part of the tank filled in 1 hours is + - where z is time in hour the emptying pipe empties the tank, hence the tank will be full in h.

EXAMPLES1. Two pipes fill in a tank in 10 min. and 15 min, separately. If both pipes are opened how much time does it take to fill the tank.

Sol. Total time = min = min = 6 min.

2. One tap can fill a tank in 10 h. and another tap can empty the tank in 15 . how much time does it take to fill the tank if both pipes are opened?

Sol. min = h. = 30 h.

3. Two taps fill in tank in 20 h. and 10 h . Two taps are opened first. After how much time the second tap should be closed off so that the tap full in 18 h.?

Sol. Let they work for xh. Then 18 x + x+ = 1 18 + 2x = 20; 2x = 2; x= 1 After 1h. the second tap should be closed.

4. A tank is already full. Two taps fill the tank in 10 h. and 15h. respectively. Another tap can empty the tank in 5h. if the three pipes are opened, how much time does it take to empty the tank.

Sol . Let they work for xh. then x = ; x = X= X 30 or 22 It takes 22 h, to empty the tank.

5. A tap fill s in a tank in 12 min. at the rate of 6 litre/ min. Another tap empties the tank in 36 min. how much remains in the tank after 10 min?

Sol. Total capacity of tank = 6 X 12 litres= 72 litres.The part filled by the taps in 1 min. = = = = Amount of water two in the tank after 10 minutes = 10 x x 72 = 40 litres.

EXERCISE

1. A cistern can be filled separately by two pipes in 12 and 16 min. respectively. If both fill pipes are opened together, when will the cistern be filled? (1) 6 min (2) 7 min (3) 6 min (4) None of these

2. A tap can fill a tank in 25 min. and another can empty it in 50 min. Find whether the tank will be filled up or emptied and in how many min? (1) Filled in 50 m(2) emptied in 25 m(3) Filled in 25 m(4) never be filled

3. A fill pipe can fill of cistern is 12 min. In how many min. it can fill of cistern? (1) 7 min (2) 8 min (3) 9 min (4) 10 min

4. An emptying pipe empty 5/6 of cistern in 20 min. in 9 min, what part of cistern will be emptied ? (1) 8/3 (2) 6/17 (3) 9/3 (4) 3/8

5. Two pipes A and B can fill a cistern in 10 and 15 min. respectively, but an outlet C can empty it in 5 min. the pipes A and B are kept open for 4 min., and then the outlet C is also opened. In what time is the cistern emptied? (1) 10 (2) 15 (3) 20 (4) 25

6. Two pipes A and B can fill a cistern in 3 and 6 min respectively, while an empty pipe C can empty the cistern in 4 min. All the three pipes are opened together and after 2 min. pipe C is closed. Find when the tank will be full? (1) 2 (2) 3 (3) 4 (4) 5

7. A cistern normally takes 6 hours to be filled by a tap but because of a leak, 2 hours more. In how many hours will the leak empty a full cistern ? (1) 23 (2) 24 (3) 25 (4) 26

8. Two fill pipes A and B separately fill a cistern in 25 and 20 min. respectively. Both are opened together, but at the end of 5 min. A is turned off. When will the cistern be full? (1) 16 min (2) 12min (3) 20 min (4) None of these

9. Two pipe can fill a tank in 8 and 12 hours respectively whereas an escape pipe can empty it in 6 hours. If the three pipes are open at 1pm, 2pm and 3pm respectively at what time will the tank be filled?(1)7 A.M (2) 9 A.M (3) 7 P.M (4) 9 P.M

10. A water tank of 1000 litres capacity is connected to a tap which can fill it at the rate of 20 litres per minute and water is let out at the same time at the rate of 5 liters per minute. After an hours the outlet is shut off. Find how long will it take now for the tank to become full? (1) 3 min (2) 4 min(3) 5 min (4) None of these

11. A pipe can fill a cistern in 8 hours and another in 10 hours. Both are opend at the same time. If the second pipe is closed 2 hours before the cistern is filled up, find in what time, the cistern will be filled up? (1) 5 hours (2) 4 hours (3) 6 hours (4) None of these

12. Two pipe, P and Q can fill a cistern in 12 and 15 minutes respectively. Both are opend together, but at the end of 3 minutes the first is turned off. How much longer will the cistern take to fill? (1) 9 minutes (2) 8 minutes (3) 7 minutes (4) 8 minutes

13. Two pipes P and Q world fill a cistern in 18 and 24 minutes respectively. Both pipes being opened find when the first pipe must be turned off so that the cistern may be just filled in 12 minutes? (1) after 12 minutes (2) after 9 minutes (3) after 8 minutes (4) after 10 minutes

14. A cistern which could be filled in 9 hours takes one hour more to be filled owing a leak in its bottom. If the cistern is full in what time will the leak empty it? (1) 45 hours (2) 60 hours (3) 75 hours (4) 90 hours

15. Two pipe A and B can fill a cistern in 20 and 30 minutes respectively, and a third pipe C can empty it in 40 minutes. How long will it take to fill the cistern if all the three are opened at the same time? (1) 19 min. (2) 15 min. (3) 17 min. (4) 7 min.

14. TIME & DISTANCE

Distance = SPEED x TIME

Distance in direct proportion with speed and time while speed and time are in indirect proportion

SPEED = TIME =

If the distance, velocity (speed) and the taken are represented by d1 v1 t respectively, then

1. Fro distance constant , v1 t1 = v2 t2 or v1 : t1 = v2 : t22. For time constant = 3. For speed constant, =

NOTE: If the speed is given in km/ph multiply the number by 5/18 to get speed in m/s. Similarly if 1 m/s means 18/5 km/h

RELATIVE SPEED : If two things are in motion and moving in a straight line either in opposite direction or same direction, to find their time of meeting we have to find their relative speed./ In the first case, the relative speed is equal to the sum of their speeds and in the second case, the relative speed is equal to the difference of their speeds

Initially if two persons are at A and B and move in a straight line, then the time taken for them to meet = distance between them / their relative speed.

EXAMPLES01. A man travelled 250 km. at the rate of 20 km/ph Find the time taken ?Sol:

02. Two men from the same place at the same time with velocities of 50 of km/h and 40 km/h. (1) in same direction (2) and in opposite direction. What will be the distance between them After 4 hours

Sol (1): If they move in same direction, the relative speed = difference of their velocities = (50 km 40 km)/h = 10 = 10 km/h; time = 4 hours Distance between them = t x vR = 4 x 1- km = 40 km

(2) If they move in opposite direction, the relative speed = sum of their velocities (50 + 40) km/h = 90km/h

03. Two persons start from two places 200 km apart at the same with velocities 50 km/h and 30 km/h mov in opposite direction. Where they meet ?

Sol: d = 200 , v1 = 50, v2 = 30, vR = v1 + v2 = 50 + 30 = 80 The distance covered by them will be indirectly proportion to their distance of the persons of meeting form A = 200 x km = 200 x = 125 km

04. A thief runs away from B with a velocity of 20 km/h. A police from a station 60 km away started chasing him after 1 h with a velocity of 40 km/h, when can the police catch the thief?

Sol: Distance travelled by the thief in 1 h = 20 km Total distance of them after 1 h = (60 + 20)km = 80km Their relative speed = (40 20) km/h = 20 km/h Time taken to catch the thief = =

05. A student to his school from the house at 5 km/h by 10 minutes. If he goes at 6 km/h. He would be 5 minutes early . What is the distance of the school from his school ?

Sol: Let the distance be d km then d = Difference of time in hours x

06. The velocities of two persons ar ein 3 : 4 and the times of their travel are 5 : 6 What is the ratio of the distances covered by them ?

Sol: d1 : d2 = v1 t1 : v2t2 = 3 x 5 : 4 x 6 = 5 : 8

EXERCISE1. A starts form a place at 7 a.m with a speed of 10 kmph./ B starts from the same place at 9am with a speed of 15 kmph. When will B meet A ? (1) 1.10pm(2) 1 pm(3) 1.15pm(4) 12.55 pm

2. A car can cover a distance of 100 mts, in 5 sec. Find its speed in mps ? (1) 25 mps(2) 24 mps(3)20 mps (4) 22 mps

3. The speed of a train is 72 kmph. What is the distance in meters covered by the train in 3 minutes ? (1) 3300 mts(2) 3258 mts(3) 3568 mts(4) 3600 mts

4. A train covers half of the distance at 24 kmph and the remaing at 40 kmph. Find the average speed of the whole journey? (1) 30 kmph(2) 25 kmph(3) 15 kmph(4) 60 kmph

5. A train covers a distance in 40 min. if it runs at a 45 kmph. The speed of which the train must run to reduce the time of journey to 30 min will be ? (1) 45 kmph(2) 60 kmph(3) 25 kmph(4) 15 kmph

6. A scooter covers 120 kmps in 4 hours. What is the distance in meters covered by the scooter in one second ? (1) 3mts (2) 3mts (3) 2mts (4) 2mts7. A car covers four successive 3 km stretches at speeds of 10 kmph, 20 kmph, 30 kmph and 60 kmph respectively . Its average speed over this distance is : (1) 30 kmph(2) 18 kmph(3) 24 kmph(4) 20 kmph

8. The driver of a car driving at the speed of 38 km per hour locates a bus 40 meters ahead of him After 20 seconds the bus is 60 meters behnd. Find the speed of the bus. (1) 40 kmph(2) 20 kmph(3) 19 kmph(4) 38 kmph

9. A monkey climbing up a greased pole ascends 12 meters and slips down 3 meters in alternate minutes. If the pole is 63 meters high, how long will it take him to reach the top ? (1) 12mts (2) 11mts (3) 12mts (4) 11mts

10. A thief has got away on a cycle and is cycling at 25 km per hour, is pushed by a motorist 12 minutes after his get-away. At what speed must the motorist travel in order to overtake the cyclist just as he reaches the railway station, which is 20 km away? (1) 22kmph(2) 33kmph (3) 22kmph (4) 33kmph

11. A boy goes to the school with the speed of 3 kmph and retuns with the speed of 2 kmph. If the takes 5 hours in all, then the distance (in kms) between the village and the school is : (1) 6kms(2) 8 kms(3) 7kms(4) 9 kms

12. a man goes uphill with an average aped of 24 kmph and comes down with an average speed of 30 kmph. The distance travelled in both the cases being the same, the average speed for the entire journey is : (1) 24.6 kmph(2) 28.8 kmph(3) 26.8 kmph(4) 28.6 kmph

13. A student walks from his house at 5 kmph and reaches his school 10 minutes late. If his speed had been 6 kmph he would have reached 15 minutes early. The distance of his school from his house is: (1) 11.5 km(2) 13.5 km(3) 12.5km(4) 13km

14. Two trains 121 metres and 99 metres in length respectively are running in opposite direction, one at the rate of 40 km and the other the rate of 32km an hour. In what time will they be completely clear of each other from the moment they meet? (1) 10.5 sec(2) 10 sec(3) 11.5 sec(4) 11 sec

15. Two men together start on a journey in the same direction. They travel 9 and 15 kms. Respectively daily. After travelling for 6 days the man travelling at 9 km per hour doubles his speed and both of them finish the distance in the same time. Find the time taken by them to reach their destination : (1) 24 days(2) 18 days(3) 20 days(4) 16 days

15. TRAINS, BOATS & STREAMS

TRAINS

A train has a travel its own length to cross a man, a telegraph pole or a car standing beside the tract. To across a bridge, a platform or a standing train, the train has to travel its own length as well as the length of the bridge etc.

EXAMPLES

1. Find the length of the train if it takes 15 sec. travelling at 54 kmph, to cross a man standing beside the tract.

Sol: Length = = 54 x

2. A train takes 20 sec. to a cross a pole beside the tract and 30 sec. to cross a platform of length 100m. Find the length and velocity of the train?

Sol: Since velocity is constant , l = 200 Length of train = 200m.Velocity of the train =

THEORY: When two trains travel in opposite directions, their relative velocity = the sum of the velocities if they in same directions, the relative velocity = the difference of their velocity

The time taken by a man in 1st train to cross the 2nd train is equal to the time taken by the 2 nd train to cross the man with the relative velocity o the trains

03. Two trains travel in opposite direction with velocities 36 km/h, and 54 km/h and cross each other in 10 sec. The length of 1st train is 100 km. What is the length of 2nd train ?

Sol. t = ; 10 = ;10 x Length of 2nd train = 150m

04. Two trains cross each other in 36 ec while going it same direction and in 18 sec . while coming in opposite direction. Find the ratio of the velocities of the trains ?

Sol: Since the distance to be covered in both cases is same vR tl = vR t2

05. The lengths of two are 200m, and 300m, It takes 50 secs for the trains to cross each other the velocity of one faster train 54 km/h. What is the Velocity of slower train.

Sol: V2 = 54 - 36 = 18 Velocity of 2nd train = 18 km/h

EXERCISE1. A train a distance of 80 km at a speed 40 kmph for the first 60 km and the remaining distance at the speed of 20 kmph. What is the average speed of the train on the journey? (1) 40 kmph(2) 32 kmph(3) 34 kmph(4) 42 kmph

2. Two trains are moving in opposite directions at the speed of 50 and 70 km per hour. Their lengths are 150m and 100m. The time required for their crossing each other is (1) 8.5 sec(2) 7.5 sec(3) 6.5 sec(4) 4.5 sec

3. A train 150 metres long running at a speed of 60 kmph takes 30 seconds to cross a bridge. What is the length of the bridge? (1) 250m(2) 260m(3) 350m(4) 360m

4. A railway 140 metres long travelling at 90 kmph will overtake another train of length 160 metres moving in the same direction at 72 kmph in what time ? (1) 5 minutes(2) 8 minutes(3) 2 minutes(4) 1 minutes

5. If a train running at 72 km per hour crosses a tree standing by the side of the track in 7 seconds the length of the train is ? (1) 140m(2) 120m(3) 160m(4) 180m