UNIT-3

Ratio, Proportion, Percentage, Profit and Loss:

Points to be covered:

• Ratio- definition, continued Ratio, Inverse Ratio

• Proportion- continued proportion, Direct proportion, inverse proportion, variation, joint variation

• Percentage- meaning and computation of percentage.

• Profit and Loss- terms and formula, discount, problem involving cost price, Selling price, marked price

Ratio

• A ratio is comparative measure of two quantities of the same kind in the same unit of measurement.

• If a and b are two quantities of same kind, the ratio of a and b is expressed as a:b.

• In the ratio a:b, ‘a’ is known as antecedent and ‘b’ is known as consequent.

• It is also express how many times one quantity is grater or lesser than other.

• A ratio is pure number.

Examples

1) If two students scored marks 60 and 70 in

mathematics, the ratio of their marks are 6:7.

2) If the investments for two persons X and Y in

a business are 40000 and 30000 respectively,

then ratio of their investment is 4:3.

Notes:

1. A ratio is always expressed in the simplest form. For example, the ratio 6:10 is expressed in the simplest form of 3:5.

2. The order of term in the ratio is very important. For example, the ratio 4:5 is not same as 5:4.

3. The terms of a ratio is multiplied or divided by a non zero number. For example, the ratio 2:3 is same as 4:6 or 6:9.

4. There is no ratio between the marks of a student and the salary of his teacher. The ratio exist if the qualities are of the same kind.

5. In a ratio, the qualities to be compared must be in the same unit. For example, if the weight of two quantities are given as 500 gms and 3 kg. the ratio between two quantities is 500: 3000 or 1:6.

Continued Ratio

• A ratio is said to be in continue if consequent

of one ratio is antecedent of the other.

• For example, a: b and b: c is in continued

form.

Inverse Ratio

• For the ratio a: b, the inverse ratio is defined

by b:a.

• In other words, two ratios are inverse of each

other if their product is 1.

ILLUSTRATIONS

1. If A : B = 4: 5 and B: C = 7: 9, then find A:B:C.

Answer:

A : B = 4 : 5 and (multiply by 7)

B:C = 7 : 9 (multiply by 5), then

A : B = 4*7 : 5*7

B:C = 7*5 : 9*5

A : B = 28 : 35

B:C = 35 : 45

A:B:C = 28:35:45

2. The angles of triangle are in the ratio 3:4:5.

what are the angles of triangle?

Answer:

Let the angles be 3x, 4x and 5x.

Then 3x + 4x + 5x = 180

12x = 180

X = 180 / 12

X = 15

The angles of triangle are 3*15, 4*15 and 5*15.

i.e. 45, 60 and 75.

3. Three persons A, B and C invest Rs. 15000, 10000 and 5000 respectively in a business. The total profit earned by them is Rs. 36000. what is A’s profit?

Answer:

Their investment are in the ratio 15000 : 10000 : 5000

i.e.15 : 10 : 5 i.e.3 : 2 : 1.

And total profit is 36000.

So A’s profit = 3 * (36000/6)

= 3* 6000

= 18000.

4. The monthly incomes of two persons are in the ratio 7:9. Their monthly expenses are in the ration 4:5 and their savings are Rs. 100 and Rs. 200 per month. What are their incomes?

Answer:

Let the incomes be 7x and 9x.

Also their expenses be 4y and 5y.

So, 7x – 4y = 100 and 9x – 5y = 200.

Solving these two equations, we get

x = 300.

So their incomes are 7*300 = 2100 and 9*300 = 2700.

5. Two numbers are in the ratio 2:3. if 5 is subtracted from each number the ratio becomes 3:5. Find the numbers.

Answer:

Let the numbers be 2x and 3x.

5(2x – 5) = 3 (3x – 5)

10x – 25 = 9x – 15

x = 10.

Two numbers are 2*10 = 20 and 3*10 = 30

2 5 3

3 5 5

x

x

Proportion

• Four quantities a, b, c and d are said to be

proportion if a /b = c/ d.

• a, b, c and d are called first, second, third and

fourth proportion respectively.

• A proportion is also considered as the equality

of two ratios.

• From above ratio, we can say that ad = bc.

• Product ‘a*d’ is called product of extremes.

• Product ‘b*c’ is called product of means.

Continued Proportion

• For four quantities a, b, c and d are said to be in

continued proportion if

• i.e. b2 = ac, c2 = bd.

a b c

b c d

Mean Proportional

• If a, b, c and d are in continued proportion then

‘b’ is called the mean proportional between a

and c.

• C is called the third proportional.

Illustration

1. Find the 4th proportional to 3, 5 and 15.

Answer:

Here a = 3; b = 5 and c = 15.

3*d = 15 *5

d = 75/3

d = 25

Therefore, fourth proportion is 25

3 15

5

a c

b d

d

2. Find mean proportional between 9 and 16.

Answer:

Here a = 9 and c = 16.

And to find the value of ‘b’.

So, from continuous proportion,

2

2

2

*

9*16

144

12

a b

b c

b a c

b

b

b

EXAMPLES

1) If a:b = 5:9 and b:c = 4:7, find a:b:c.

2) Divide Rs. 1162 among A,B and C in the ratio 35:28:20.

3) If 2A = 3B = 4C then find A:B:C.

4) Find :(a) the 4th proportional to 4, 9, 12.

(b) the third proportional to 16 and 36.

(c) the mean proportional between 0.08 and 0.16.

5) A bag contains 50 paisa, 25 paisa and 10 paisa coins in the ratio 5:9:4, amounting to Rs. 206. Find the number of coins of each type.

6) In a school 10% of boys are same in number as 1/4th of the girls. What is the ratio of boys to girls in that school.

6) 60 Kg. of an alloy A is mixed with 100 Kg. of alloy B. If alloy A has lead and tin in the ratio 3:2 and alloy B has tin and copper in the ratio 1:4, then find the amount of tin in the new alloy.

7) The ages of A and B are in the ratio 3:1. Fifteen years hence the ratio will be 2:1. Find their present ages.

8) The average age of three boys is 25 years and their ages are in the proportion 3:5:7. Find the age of youngest boy.

9) Three containers have their volumes in the ratio 3:4:5. They are full of mixtures of milk and water. The mixtures contain milk and water in the ratio (4:1), (3:1) and (5;2) resp. The content of all these three containers are poured into a fourth container. Find the ratio of milk and water in the 4th

container.

10)A sum of Rs. 1300 is divided amongst P,Q and R such that

(P’s share/ Q’s share) = (Q’s share/ R’s share) = (R’s share/ S’s share) = 2/3. Find P’s share.

Percentage

• A percent is a ratio whose second term is 100.

Percent means parts per hundred.

• The word comes from the Latin phrase per

centum, which means per hundred.

• In mathematics, we use the symbol % for

percent.

Percentage:

1) 9% of what number is 36?

2) If 30% of a number is 48, then what is 70% of that number.

3) If 1/5 of a number is 25 less then 1/4 of a same number . what will be 3/2 of number.

4) If a number is increased by 12% and then decreased by18% , then find net percentage change in the number.

5) Difference of two numbers is 1660. If 7.5% of one number is 12.5% of the other number, find the numbers.

(6) (i) 2 is what percent of 50?

(ii) ½ is what percent of 1/3?

(iii)What percent of 7 is 84?

(iv)What percent of 2 metric tones is 40 quintals?

(v)What percent of 6.5 litres is 130 ml.?

(7) 65% of a number is 21 less than four-fifth of the number . Find the number.

(8) If 50% of (x-y) = 30% of (x+y), then what percent of x is y?

(9) Mr. Jones gave 40% of the money he had , to his wife. He also gave 20% of the remaining amount to each of his three sons. Half of the amount now left was spent on miscellaneous item and the remaining amount of Rs.12,000 was deposited in the bank. How much money did Mr. Jones have initially?

(10) Difference of two numbers is 1660. If 7.5%

of one number is 12.5% of the other number,

find the two numbers.

(11)When a producer allows 36% commission

on the retail price of his product, he earns a

profit of 8.8%. What would be his profit

percent if the commission is reduced by 24%.

Profit and Loss

Cost Price: The price, at which an article is purchased, is called its cost price, abbreviated as C.P.

Selling Price: The price, at which an article is sold, is called its selling prices, abbreviated as S.P.

Profit or Gain: If S.P. is greater than C.P., the seller is said to have a profit or gain.

Loss: If S.P. is less than C.P., the seller is said to have incurred a loss.

IMPORTANT FORMULAE

• Gain = (S.P.) - (C.P.)

• Loss = (C.P.) - (S.P.)

• Loss or gain is always reckoned on C.P.

• Gain Percentage: (Gain %)

Gain % = (Gain x 100) / C.P.

• Loss Percentage: (Loss %)

Loss % = (Loss x 100) / C.P.

• Selling Price: (S.P.)

SP = [(100 + Gain %)/ 100] x C.P

• Selling Price: (S.P.)

SP = [ (100 - Loss %) / 100] * C.P.

• Cost Price: (C.P.)

C.P. = [100/ (100 + Gain %)] x S.P.

• Cost Price: (C.P.)

C.P. = [100 /.(100 - Loss %)] x S.P

• If an article is sold at a gain of say 35%, then

S.P. = 135% of C.P.

• If an article is sold at a loss of say, 35% then

S.P. = 65% of C.P.

• When a person sells two similar items, one at a

gain of say x%, and the other at a loss of x%,

then the seller always incurs a loss given by:

Loss % =(Common Loss and Gain % / 10)2

= (x / 10)2

• If a trader professes to sell his goods at cost

price, but uses false weights, then

Gain % = [ *100]%error

truevalue error

Profit and Loss:

(1) If 1/5 of a number is 25 less then 1/4 of a same number . what will be 3/2 of number.

(2) A man buys an article for Rs. 27.50 and sells it for Rs. 28.60. Find gain percent.

(3) The ratio of the cost price and the selling price is 4:5. Find the profit percent.

(4) A fair price shopkeeper takes 10% profit on his goods. He lost 20% goods during theft. Find his loss percent.

(5) The C.P. of 21 articles is equal to S.P. of 18 articles. Find the gain or loss percent.

(6) Pure ghee costs Rs.100 per kg. After adulterating it with vegetable oil costing Rs. 50 per kg., a shopkeeper sells the mixture at the rate of Rs. 96 per kg., there by making a profit of 20%. In what ratio does he mix the two?

(7) An article is sold at a certain price. By selling it at 2/3 of that price one losses 10%. Find the gain percent at original price.

(8) The ages of A and B are in the ratio 3:1. Fifteen years hence the ratio will be 2:1. Find their present ages.

(9) The average age of three boys is 25 years and their ages are in the proportion 3:5:7. Find the age of youngest boy.

(10)A dealer sold ¾ of his articles at a gain of 20% and the remaining at a cost price. Find the gain earned by him in the whole transaction.

References:

• https://books.google.co.in/books?id=TjpEBAAAQBAJ&pg=SL1-

PA4&lpg=SL1-

PA4&dq=ratio+definition,+continued+Ratio,+Inverse+Ratio&sour

ce=bl&ots=JmvQbzd5m0&sig=idcOTCKJHHtfswjoD2g5vJquCLs

&hl=en&sa=X&ei=PCjPVJ2cFMK1mwXZ1YDQBw&ved=0CD

wQ6AEwBQ#v=onepage&q=ratio%20definition%2C%20continue

d%20Ratio%2C%20Inverse%20Ratio&f=false

• http://www.mathgoodies.com/lessons/vol4/meaning_percent.html

• http://www.indiabix.com/aptitude/profit-and-loss/formulas

• Quantitative Aptitude by R. S. Agraval