Ratio and Proportion

Ratio And Proportion

In our day to day life we compare a lot of things to make better choices. We compare routes to reach our destination faster, compare prices to save money, compare people’s height and weight.

When we compare only two quantities, we either see the difference (a-b) or we perform division. When we divide two numbers to find out the magnitude of one with respect to another, we call it a ratio. The ratio of two numbers say a,b would be a/b represented as “a:b” and which is read as “a is to b”.

Where,

• a and b can be any two non-zero quantities
• “a” is termed as the first term or antecedent.
• “b” is termed as the second term or consequent.

When two ratios are equal, they are said to be in proportion. Symbols like “::”and “=” are used to show equity among them. 

If a : b = c : d, then  a : b is said to be in proportion with c : d.

Ratio And Proportion Formulas

The ratio and proportion formulas are the key to solve any ratio and proportion problem. The ratio and proportion formulas are given below.

Ratio Formula:

 a:b-> a/b

Proportion Formula: 

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

Properties Of Ratio And proportion

• Properties of Ratio

1. Ratio remains the same when you multiply or divide both the quantities with the same non- zero number. 

• a:b=pa:pb=qa:qb
• a:b=a/p:b/p=a/q:a/q

Where p=q≠0

1. Two ratios can be compared as real number in their fraction form

a:b=p:q ⇔ aq=bp

• a:b>p:q ⇔ aq>bp
• a:b

1. If two ratios are equal

• a:b=p:q ⇔ b/a=q/p (invertendo)
• a:b=p:q ⇔ a/p=b/q (Altertendo)
• a:b=p:q ⇔ (a+p)/p=(b+q)/q (componendo)
• a:b=p:q ⇔ (a-p)/p=(b-q)/q (dividendo)

• Properties Of Proportion

1. Product of extremes = product of means i.e., ad = bc
2. a, b, c, d,…. are in continued proportion means, a:b = b:c = c:d
3. a:b = b:c then b is called mean proportional and b2 = ac
4. The third proportional of two numbers, a and b, is c, such that, a:b = b:c
5. d is fourth proportional to numbers a, b, c if a:b = c:d.
6. The two terms ‘b’ and ‘c’ are called ‘means or mean terms’, whereas the terms ‘a’ and ‘d’ are known as ‘extremes or extreme terms.’

Some Solved Ratio and Proportion Examples

Example of Ratio

Example1: In a class there are 30 boys and 25 girls. What is the ratio between boys and girls?

Solution->

 The  ratio of boys and girls would be the number of boys to the number of girls.

= 30/25

= 6/5

∴ ratio= 6:5

Example2: Divide Rs. 100 between Ron and Bob in ratio 2:3.

Solution ->  

let Ron’s part be 2X. 

Then Bob’s part would be 3X.

Thus, 2X+3X= 100

= 5X=100

X=100/5

X=20

∴ Ron’s part = 2*20= Rs. 40

And, Bob’s part = 3*20= Rs. 60

Example3: Seema collected coins of 1 rupee, 50 paisa and 25 paise in a piggy bank. If there are 495 rupees in the piggy bank, with the ratio of 1:8:16 respectively. How many one Rupee coins are there in the piggy bank?

Solution -> 

Let the no. of one Rupee coins = x

Then, no. of 50 paise coin = 8x,

And no. of 25 paise coin= 16x

According to the question,

x+(8x/2)+(16x/4)=495

x+ 4x+4x=495

x=495

x=55

∴ no. of one rupee coins is 55.

Example Of Proportion

Example1: Find out if the following ratios are in proportion:

1. 5:8 and 6:9
2. 1:4 and 5:20

Solution -> 

Two ratios a:b and c:d are said to be in proportion if and only if a*d=c*b.

1. 5:8 and 6:9

5*9=45

8*6=48

Now,  a*dc*b

Therefore these ratios are not in proportion.

1. 1:4 and 5:20

1*20=20

4*5=20

Now,  a*d=c*b

Therefore these ratios are in proportion.

Example2: If 3:4::24:x is in proportion then find the value of x. 

Solution ->

Given,  

             3:4::24:x is in proportion

Therefore ,

3*x=4*24

3x=96

x=96/3

x=32.

Example3: Find the third proportional to 9 and 6. 

Solution ->

Let the third proportional be c.

Now, b2=a*c.

c= b2/a

=62/9

=4

Thus, the third proportional to 9and 6 is 4.

More Ratio And Proportion Examples 

Example1: A bag contains 35 balls of red and white colour. The ratio of red to white balls is 4:1. If 7 more white balls are added to the bag the ratio changes. To obtain the original ratio how many more red balls are needed to be added to the bag.  

Solution ->

Given that the initial ratio if red: white balls=4:1

Let the no. of white balls be x, then the no. of red balls would be 4x.

According to the question,

x+4x=35

⇒ x=7

Now, initially there were 28 white and 7 red balls.

After the addition of 7 new red balls,

No. of red balls= 14 

 Let the no. of new white balls needed to add be t.

Ratio of initial balls should be equal to the ratio of new balls

∴ 28:7=4:1=28+t:14

⇒4*14=1*(28+t)

56 = 28+t

t= 56-28

t=28

Hence, no. of white balls to be added = 28.

Conclusion

With the above discussion, explanation, and examples it is easy to understand the mathematical concept of ratio and proportion. The ratio is the comparison of two numbers with the same unit while proportion is nothing but an extension over ratio which states that two ratios or fractions are equivalent. We also came across numerical problems and terminologies like antecedent, consequent mean proportional, third proportional, means and extremes. We went through ratio and proportion formula and ratio and proportion examples for best understanding.