RATIO &

PROPORTION Std VIII

NATIONAL ANTHEM "Jana Gana Mana Adhinayaka Jaya

He Bharat Bhagya Vidhata Punjab Sindh Gujarat Maratha Dravida Utkala Banga Vindhya Himachal Yamuna Ganga Ucchala Jaladhi Taranga Tubh Shubha Name Jage Tubh Shubha Ashisha Mange Gahe Tubh Jaya Gata Jana Gan Mangaldayak Jay He Bharat Bhagya Vidhata Jaye He ! Jaye He ! Jaye He ! Jaye,Jaye,Jaye,Jaye He "

PLEDGE “India is my country and all

Indians are my brothers and

sisters. I love my country and I am proud

of its rich and varied heritage. I shall always strive to be worthy

of it. I shall give my parents, teachers,

and all elders respect and treat

everyone with courtesy. To my country and my people, I

pledge my devotion. In their

well-being and prosperity alone,

lies my happiness.”

Dear students, Welcome to the world of

Mathematics... In this book we are traveling through the concept of ratio and proportion. Let's continue explore in the

world of ratio through this book...



Contents 1.Ratio..............................................5 2.Properties of ratios ......................8 3.Ratio table.....................................9 4.Equivalent ratio...........................10 5.Comparing ratios..........................11 6.Ratio between two quantities.............................12 7.Exercise........................................20 8.Ratio between three quantities........................21 9.Steps to find ratio........................................23 10.Simplifying ratio........................23 11.Exercise......................................31 12.Proportion..................................32 13.Properties of proportion............................36 14.Exercise......................................42

RATIO We use ratio in our everyday life.

it is used in cooking to compare two

quantities etc.To make tea or coffee , to

compare speedof two vehicle, to check which product is better etc we need the

help of ratio. We can consider the case of

making coffee.We add sugar, coffee powder

and milk in a ratio. If we change this ratio it

tastes bitter. To compare the size of two

objects we use the concept of the ratio. Thus ratio can be defined a term that is used

to compare two or more numbers. It is used

to indicate how big or small a quantity is

when compared to another. In a ratio, two

quantities are compared using division. The

symbol : is used to represent ratio. If if

there are two quantities a and b then ratio

between them is given by equation a : b =

a/b.

Here, “a” is called the first antecedent, and

“b” is called the consequent. Let's see an example to make the

concept more clear. Consider two

square of different sizes . We can see that

the length and breadth are different.

Consider that the length and breadth are

10 cm for small square.

Let the length and breadth of big square

be 20cm. We can see that length of small

square is half than that of big square. Or in

other words we can say that big square is

twice than that of small square. We have

length of small square is equal to 10 cm

and that of big square is 20 cm.

Square is 20 cm. Therefore ratio of big

square to that of small square is 20:10. We

said that division can be used to find ratio. Therefore , ratio of the big square to that of small

square = 20 / 10 = 2/1 so the ratio of big square to that of small

square is 2 :1

Key Points to Remember:

* The ratio should exist exist

between the quantities of of

the same kind. *While comparing two two

things, the units should be be

similar. * There should be be

significant order of terms.

Properties of ratio (a) If we multiply and divide each term of

ratio by the same number (non-zero), it

doesn’t affect the ratio. Eg : 4:9 = 8:18 = 12:27 (b) Ratio data doesn’t have any negative

numerical value. (c) Ratio data values can be added,

subtracted, divided, and multiplied. (d) The second term of a ratio cannot be

zero. (e) Ratio is a pure number and does not have

any unit. (f) The ratio of two unlike quantities is not

defined. For example ,the ratio between 5 km and

5l can't be find. (g) Ratio a: b # b: a unless a=b.

(h) In case both the numbers a and

b are equal in the ratio a:b then a: b

= 1 (i) If a greater than the ratio a:b

then a:b >1. (j) If a less than b in the ratio a:b

then a:b < 1. (k) It is to be ensured that two

quantities have same unit before

comparing them.

Ratio table A ratio table is just a table that is used to describe the relationship between two separate quantities. Ratio tables are helpful for visualizing the relationship between two separate quantities. It is a logically organized list of equivalent ratios that aids in understanding the relationship between ratios and numbers.

Equivalent Ratio Equivalent ratio are similar to equivalent fractions. If the antecedent and the consequent of a given ratio are multiplied or divided by the same number other than 0 each gives equivalent ratio. eg. 2:1 and 20:10 are equivalent ratio since if we cancel common factor 10 in 20:10 we get 2:1.

Comparing ratios Let's discuss about how to compare 2 ratios. We can consider an example for this. Compare 1:2 and 2:3 . step 1 : write the ratios in the form of fraction. i.e; 1/2 and 2/3. step 2 : reduce the fractions. 1/2 and 2/3 are already in the reduced from. step 3 : compare these by finding the least common multiple of the denominators. Here lcm of 2 & 3 is 6. step 4 : make the denominators equal by multiplying the numerator and denominator of the first fraction by 3 which gives ratio as 3/6 and multiply numerator and denominator of second fraction by 2 gives ratio as 4/6. step 5 : it is easy to compare 3/6 and 4/6.Thus 2:3 > 1:2.

Ratio between two

quantities

Consider the following example, The number of books and magazines on a library are 500 and 300 respectively. Find ratio between books and magazine We have to find ratio between books and magazines given number of books = 500 number of magazines= 300 We said that ratio ratio between two qualities is given by, a :b = a / b Therefore, book : magazine =500/300 i.e; book : magazine = 5:3 Later it was found out that number of books were 10000 number of magazine were 6000. Find out the new ratio.



number of books = 10000 number of magazines = 6000 Thus the ratio is 1000/6000 = 10/6 = 5/3 From the above two examples we can conclude that the we get the actual quantity when we multiply ratio by a number.





If two quantities are in the

ratio a : b then there exists a

quantity x such that first is

ax and second is bx



Example 1

Ratio formula a : b = a/b

In a class of 100 students, 33 are girls and

the remaining are boys. Using the ratio

formula, find the ratio of the total number of

boys to the number of girls.

Given, Total number of students = 100 Number of girls = 33 Number of boys = Total number of students - Number of girls = 100 - 33 = 67 Number of boys: Number of girls = 67:33

Example 2 In a handwriting competition, there are 5

boys and 3 girls. What will be the ratio

between girls and boys?

The ratio between girls and boys will be 3 : 5 = 3/5.

Example 3 Two numbers are in the ratio 2 : 4. If the sum of two numbers are 180. Find the actual

numbers.

Given ratio = 2 : 4 so first number = 2 x Second number = 4x given ,sum of two numbers are 180 i.e; 2x + 4x = 180



6 x = 180 x = 180/6 x = 30

therefore , first number is 2x = 2 * 30 = 60 second number is 4x = 4 * 30 = 120

Example 4

Two numbers are in the ratio 3:2. If the

sum of the numbers are 75 then find the

numbers.



Let the first number be 3x and second be 2x. Given the sum of numbers are 75. so, 3x + 2x = 75 5 x = 75 x = 75/5 = 13 first number is 39 and second number is 26.



Example 5 Divide 268 into two parts such that their

ratio is 1:3.

Let the two parts be: x & y x:y=3:1 x/ y = 3/1 y=3x x+y=x+3x=268 4x=268 X = 268/ 4 ​

=67 y=3×67=201

x=67,y=201.

∴

∴

Example 6 If Rs. 60 is divided into two parts in the

ratio 2:3, then find the difference between

those two parts. Let the two parts be 2x and 3x given 2x + 3x = 60 5x = 60 x = 12 So, 2x = 2*12 = 24 3x = 3*12 = 36 Therefore the two parts 24 and 36.

Example 7 In a mixture 60 l , the ratio of milk and water is 2:1. If this ratio is to be 1: 2 then find quantity of water to be added. let the quantity of milk be 2x and water be 1x. Given 2x + 1 x = 60 3x = 60

x = 60/3 = 20l therefore amount of milk is 2 * 20 = 40l and amount of water is 1 *20 = 20l. let the quantity of water to be added to get ratio 1:2 be x. therefore quantity of water in new mixture is 20 + x So,. 1: 2 = 40 :20 + x 1*(20 + x) = 40 * 2 x + 20 = 80 x = 80 - 20 = 60 l Quantity of water to be added further = 60l

EXERCISE 1) A class has 32 students, of which 20 are girls. Write the ratio of girls to boys. 2) The ratio between two quantities is 3: 4. If the first is 810, find the second. 3) Two numbers are in the ratio 11:12 and their sum is 460. Find the numbers. 4)If x:y = 2:3 and y:z = 3:2, what is the ratio of x:z ? 5) What must be added to each term of the ratio 9: 16 to make the ratio 2:3 6) James own 2 cats 3 dogs and a parrot as pets. What is the ratio of number of cats to the total number of pets James owns?

Ratio between three

quantities Consider the following example, Let 48 rupees be divided in the

ratio 1: 2: 1 among three people. Find

the share obtained by each. As in the case of ratio between two

quantities we can say, first person's share = 1x second person's share = 2x third person's share = 1x given total is48 Rs so, 1x + 2x + 1x = 48 4x = 48 x = 48/4 = 12 First person's share = 1 * 12 = 12 second person's share = 2 * 12 = 24 third person's share = 1 * 12 = 12

Now assume that ratio is to 2 :

4 : 2 and sum remains same as 48.

Find Each person's share. First person's share = 2x Second person's share = 4x third person's share = 2x given total is 48 so, 2x + 4x + 2x = 48 8x = 48 x = 48/8 x = 6 therefore, share of first person = 2x = 12 share of second person = 4x = 24 share of third person = 2x = 12 From the above two example

we can conclude the following; If three quantities are in the

ratio a:b: c then there exist an x

such that first is ax, second is bx

and third is cx.

Steps to find ratio 1) Find the total number of parts in

ratio by adding the numbers in the

ratio together . 2) Find the value of each part in the

ratio by dividing the given amount

by the total number of parts. 3) Multiply original ratio by the

value of each part.

Simplifying ratio Let the ratio be a: b: c. step 1 : write down the factors of a, b,c. step 2 : find the greatest common factor of a, b and c. step 3 : divide the three numbers in the ratio

by the number obtained in step 2.

eg. Simplify ratio 60:6:24. Find highest common factor. We know

that highest common factor of 60,6 and

24 is 6. Cancel 6 from each number given i.e; 60/6 , 6/6 , 24/6. Then we get 10,1,4 respectively. Therefore simplified form is 10:1:4.

Example 1 Share rupees 20 in the ratio 2: 5: 3. 1) find total number of parts 2+5+3 = 10 2) divide the amount by the total no.

of parts 20/10 = 2 3) multiply each number in the ratio

by the value of 1 part

Therefore shares are 4,10,6.

Example 2 Given the ratio a:b = 1:2 and b:c = 2:1.

Find a:b: c. a:b = 1:2 ----------- (1) b:c = 2:1 ----------- (2) Since the number b is same on

equation 1 and equation 2 we can say

that, a: b: c = 1:2:1 .

Example 3

Given the ratio between 3

quantities are 10: 20:30. Let this be

equivalent to x:2:3. Find x. given x:2:3 = 10:20:30 = 1:2:3 on equating we get x = 1.

Example 4

Express in simplest way: 5×1/2 : 10×1/2 : 11×1/2 we have to simplify 11/2 : 21/2 : 23/2. We cannot use fractional form in

ratio. So we need to multiply each by

2 therefore ratio is 11:21:23.

Example 5

If a = 2b/5 and b = 5c/2. Find a:b:c. Given a = 2b/5 = 2/5*(5c/2) = c Therefore a: b: c = c : 5c/2 : c = 1: 5/2 : 1 = 2 : 5 : 2

(multiply each by 2 to avoid

fraction).

Example 6 Divide rupees 242 among Anil, Anita,

Ashwin such that Anil get double of

Anita and Ashwin gets 2/3 of Anita.

Calculate the amount received by

each. Given Anil gets double of Anita i.e, A = 2B so, A:B = 2:1 ---------(1) Given Ashwin gets 2/3 of Anita i.e, C = 2/3 *(B) so, B:C = 3:2 ----------(2) multiply by 3 in (1) to make value of

B same in (1) and (2). Now ratio becomes A:B:C = 6:3:2 we know that if three quantities are

in the ratio a:b:c then there exist an x

such that first is ax , second is bx

and the third is cx.

So,6x+3x+2x = 242 11x = 242 x = 242/11 = 22

therefore Anil's share = 6*22 =132 Anitha's share = 3*22 = 66 Aswin's share = 2*22 = 44.

Example 7 Three numbers are in the ratio 1:2:3. The

average of the number is 48. Find the

numbers. let's the numbers be 1x, 2x, 3x. given average is 48. so, 1x + 2x +3x = 48 6x = 48 x=7 therefore numbers are 7, 14, 21.

Example 8 Complete the equivalent ratio 2:3:6 = ___:6:___. It is given that they are equivalent

ratio.Therefore each no. In the ratio is

multiplied by the same no. Here 3*2=6,so

each no. has to be multiplied by 2 . Thus answer is 4:6:12.

Example 9 Three factories manufacture identical

products in the ratio 13 6 15.The third

factory makes 924 more products than the

first factory in one month. How many

products does each factory make that

month?

∶∶

let the number of products be 13 x , 6 x and

15 x. given 13 x + 924 = 15 x

2x = 924 x= 462 Therefore no. of products =13*462,6*462,15*462 = 6006,2772,6930

Example 10

The weights of three containers are in the

ratio 14 2 7. The weight of the second

container is 58 kg.Calculate the weights of

the first and third containers.

∶∶

Let the weight of the containers be 14x, 2x

and 7 x. given 2x = 58 x = 29 therefore weight of the first container = 14*29 = 406 kg Weight of the third container = 7 * 29 = 203kg



Exercise 1) Share $ 48 in ratio 3:1:2. 2) Given the ratio between 3 quantities are

10: 20:30. Let this be equivalent to x:4:6.

Find x. 3) Express in simple form (a) 11/3 : 22/3 : 44/3. (b) 12/5 : 13/5 : 14/15. 4) If a = 3b/8 and b = 8c/7. Find a: b:c. 5) If rupees 135 is divided among three boys in the ratio 2:3:4 , find the share of each boy. 6) Some money is shared between Jennifer,

Michael and Daniel in the ratio 16 3 15. Jennifer’s share exceeds Daniel’s share by

875 LE. How much money do each of them

receive?

∶∶

Proportion It is an equation or statement used to depict that two ratios or fractions are equal. Two ratios are said to be in proportion when

the two ratios are equal.Ratio and

proportions are said to be faces of the same

coin. When two ratios are equal in value,

then they are said to be in proportion. In

simple words, it compares two ratios.

Proportions are denoted by the symbol ‘::’.

Direct proportion The direct proportion describes the

relationship between two quantities, in

which the increase/ decrease in one

quantity, there is an increase / decrease in

the other quantity also. It is represented by

the proportional symbol, . Eg. If four pen cost 8 rupees then 8 pen cost

16 rupees.

∝

Inverse proportion When the value of one quantity

increases concerning the decrease in other

or vice-versa, we call it inversely

proportional. If two quantities are in inverse

variation, then we also say that they are

inversely proportional to each other. eg. Speed is inversely proportional to the

time taken to travel.

‘a is inversely proportional to b is written

as a

∝ 1/b

Proportion formula Let two ratios are a:b & c:d. The two terms

‘b’ and ‘c’ are called ‘means or mean term,’

whereas the terms ‘a’ and ‘d’ are known as

‘extremes or extreme terms.’ a/b = c/d or a : b :: c : d

ie, The product of means = the product of

extremes. This can be written as ad = bc

Proportion formula

a:b :: c:d => a/b = c/d

Properties of proportion (1) If a : b = c : d, then a + c : b + d (2) If a : b = c : d, then a – c : b – d (3) If a : b = c : d, then a – b : b = c – d : d (4) If a : b = c : d, then a + b : b = c+d : df (5) If a : b = c : d, then a : c = b: d (6) If a : b = c : d, then b : a = d : c (7) If a : b = c : d, then

a+b:a–b=c+d:c–d (8) The numbers a,b,c and d are

proportional if the ratio of the first two

quantities is equal to the ratio of the last

two quantities, i.e., a:b::c:d (9) Each quantity in a proportion is called

its term. (10) For every proportion, the product of

the extremes is always equal to the

product of the means, i.e., p:q::r:s if and

only if ps=qr. (11) If (a:b)>(c:d) = (a/b>c/d)

Difference between ratio and

proportion

Example 1 Are the two ratios 1:10 and 7:10 in

proportion? We have, 1:10= 1/10= 0.1 and 7:10= 7/10= 0.7 Since both the ratios are not equal, they are

not in proportion.

Example 2 Are the two ratio 1:2 and 2:4 in proportion? We have, 1/2 = 0.5 and 2/4 = 0.5. Therefore they are in

proportion.

Example 3 If a:b:c = 2:3:4. Find a/b:b/c:c/a. a/b:b/c:c/a = 2/3:3/4:4/2

= 2/3:3/4:2/1 we have lcm ( 3,4,1) = 12. ( lcm of denominators) so multiplying by 12 gives a/b:b/c:c/a = 8:9:24

Example 4 Find x,if x:3 :: 8:6 given the ratios are in proportion So by proportion formula we have, 6x = 24 x = 24/6 =4

Example 5

Find the third proportional to 3 and 6. Let the third proportional be c. Then, b² = ac 6x6=3xc C= 36/3

= 12 Thus, the third proportional to 3 and 6 is 12

Key points to be remembered

Two ratios a: b and b: c is said to be in

continued proportion if a: b = b: c. In this

case, the term c is called the third

proportion of a and b whereas b is called the

mean proportion of between the terms a and

c.When the terms a, b and c are in continued

proportion b² = ac

Example 6 l varies directly as m, and l is equal to 5 when

m = 2/3. Find l when m = 16/3. If l varies directly as m. l m l/m = k….(i) l/m = 5/(2/3) = 15/2 That means k = 15/2 Now, m = 16/3 Substituting m = 16/3 and k = 15/2 in

equation (i), we get; l/(16/3) = 15/2 3l/16 = 15/2 l = (15/2) × (16/3) =5×8 = 40

∝

Exercise (1) Are the two ratio 15:2 and 10:2 in proportion? (2) Find x if 2:3 = x:3 (3) Calculate mean proportional between 3 and 27. (4) Suppose x and y are in inverse proportion. If y = 12 then x = 4, find the value of y when x = 8. (5) The sale price of a phone was $150, which was only 80% of normal price. What was the normal price?