Ration and Proportion

1. INTRODUCTION TO RATIO

A ratio is a comparison between two quantities of the same kind. It tells us how many times one quantity is compared with another.

Ratio = First Quantity ÷ Second Quantity
If two quantities are a and b, their ratio is written as a : b.

For example, if a class has 20 boys and 30 girls, the ratio of boys to girls is:

20 : 30 = 2 : 3

The ratio must generally be expressed in the same unit before simplification.

Example: Find the ratio of 2 metres to 50 centimetres.

2 m = 200 cm
Ratio = 200 : 50 = 4 : 1

2. TERMS OF A RATIO

In the ratio a : b:

a is called the antecedent.
b is called the consequent.
a and b are called the terms of the ratio.

If a : b = m : n, then
a/b = m/n

3. SIMPLIFICATION OF RATIO

A ratio is simplified by dividing both terms by their common factor. The simplest form is obtained when the two terms have no common factor other than 1.

Example: Simplify 48 : 72.

HCF of 48 and 72 = 24
48 : 72 = 48÷24 : 72÷24
= 2 : 3

4. IMPORTANT PROPERTIES OF RATIO

a : b = ka : kb, where k ≠ 0

Therefore, multiplying or dividing both terms of a ratio by the same non-zero number does not change its value.

Example:
3 : 5 = 6 : 10 = 9 : 15 = 30 : 50

Componendo

If a/b = c/d, then (a+b)/b = (c+d)/d

Dividendo

If a/b = c/d, then (a-b)/b = (c-d)/d

Componendo and Dividendo

If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d)

Quick Tip: Always check units first. A ratio of kilograms to grams must be converted into the same unit before calculation.

5. EQUIVALENT RATIOS

Two ratios are equivalent when they have the same value.

a : b = c : d if and only if ad = bc

Example: Are 4 : 6 and 10 : 15 equivalent?

4 × 15 = 60
6 × 10 = 60
Since both products are equal, the ratios are equivalent.

6. DIVISION OF A QUANTITY IN A GIVEN RATIO

Suppose a total amount S is divided in the ratio a : b. Then the total number of parts is:

Total Parts = a + b

First Share = S × a/(a+b)
Second Share = S × b/(a+b)

Example: Divide ₹840 in the ratio 3 : 4.

Total parts = 3 + 4 = 7
First share = 840 × 3/7 = ₹360
Second share = 840 × 4/7 = ₹480

Therefore, the shares are ₹360 and ₹480.

7. THREE-PART RATIO

If a quantity is divided in the ratio a : b : c, then:

Total Parts = a + b + c
Shares = S×a/(a+b+c), S×b/(a+b+c), S×c/(a+b+c)

Example: Divide ₹1200 among A, B and C in the ratio 2 : 3 : 5.

Total parts = 2 + 3 + 5 = 10
A = 1200 × 2/10 = ₹240
B = 1200 × 3/10 = ₹360
C = 1200 × 5/10 = ₹600

8. COMPARISON OF TWO RATIOS

To compare a : b and c : d, compare:

a/b and c/d

or cross multiply:
ad and bc

Example: Which ratio is larger: 7 : 9 or 5 : 6?

7 × 6 = 42
5 × 9 = 45
Since 45 > 42,
5 : 6 is larger.

9. COMPOUND RATIO

The compound ratio of a : b and c : d is:

Compound Ratio = ac : bd

Example: Find the compound ratio of 2 : 3 and 4 : 5.

= (2×4) : (3×5)
= 8 : 15

Exam Shortcut: When a total amount is divided according to a ratio, first calculate the total number of parts. This avoids unnecessary long calculations.

10. WHAT IS PROPORTION?

A proportion is a statement that two ratios are equal.

a : b = c : d
or
a/b = c/d

Here, a and d are called extremes, while b and c are called means.

Product of Extremes = Product of Means
ad = bc

Example: Check whether 8 : 12 = 10 : 15.

8 × 15 = 120
12 × 10 = 120
Therefore, 8 : 12 = 10 : 15.

11. FINDING THE UNKNOWN TERM

Example: Find x if 5 : 8 = x : 24.

5/8 = x/24
8x = 5×24
8x = 120
x = 15

12. DIRECT PROPORTION

Two quantities are in direct proportion when an increase in one quantity produces a proportional increase in the other, and a decrease produces a proportional decrease.

For direct proportion:
x/y = constant

Therefore:
x₁/y₁ = x₂/y₂

Example: If 5 notebooks cost ₹100, what is the cost of 8 notebooks?

5/100 = 8/x
5x = 800
x = ₹160

Common examples of direct proportion

Number of items and total cost, when price per item is fixed.
Distance and time, when speed is constant.
Quantity of goods and price, when unit price is fixed.
Wages and working days, when daily wage is fixed.

13. INVERSE PROPORTION

Two quantities are in inverse proportion when an increase in one quantity produces a proportional decrease in the other.

For inverse proportion:
x × y = constant

Therefore:
x₁y₁ = x₂y₂

Example: 6 workers complete a job in 20 days. How many days will 10 workers take, assuming equal efficiency?

6 × 20 = 10 × x
120 = 10x
x = 12 days

Common examples of inverse proportion

Number of workers and time required for the same work.
Speed and time for a fixed distance.
Number of machines and time required for a fixed production.

Remember: Direct proportion → product of corresponding cross terms is used through equal ratios. Inverse proportion → the product of the two changing quantities remains constant.

14. IMPORTANT FORMULAS AT A GLANCE

Concept	Formula
Basic Ratio	a : b = a/b
Equivalent Ratio	a : b = ka : kb
Proportion	a : b = c : d ⇒ ad = bc
Two-Part Division	Shares = Sa/(a+b), Sb/(a+b)
Three-Part Division	Share = Sa/(a+b+c), etc.
Compound Ratio	(a:b) and (c:d) ⇒ ac:bd
Direct Proportion	x/y = constant
Inverse Proportion	xy = constant
Percentage from Ratio	First part % = a/(a+b) × 100

15. RATIO AND PERCENTAGE

If two quantities are in the ratio a : b, their respective percentages of the total are:

First percentage = a/(a+b) × 100
Second percentage = b/(a+b) × 100

Example: Boys and girls are in the ratio 3 : 2. What percentage of the class are boys?

Boys percentage = 3/(3+2) × 100
= 3/5 × 100
= 60%

16. MIXTURE RATIO

When two quantities are mixed, their ratio can be used to determine the quantity of each component.

Example: Milk and water are mixed in the ratio 5 : 2. If the total mixture is 21 litres, find the amount of milk and water.

Total parts = 5 + 2 = 7
Milk = 21 × 5/7 = 15 L
Water = 21 × 2/7 = 6 L

17. SOLVED PRACTICE EXAMPLES

Q1. Simplify 84 : 126.
HCF = 42
Answer = 2 : 3

Q2. If x : 7 = 12 : 21, find x.
x/7 = 12/21
x = 7×12/21
Answer = 4

Q3. Divide ₹900 in the ratio 4 : 5.
Total parts = 9
First = 900×4/9 = ₹400
Second = 900×5/9 = ₹500
Answer = ₹400 and ₹500

Q4. 12 workers finish a job in 15 days. How many days will 20 workers take?
12×15 = 20×x
x = 180/20
Answer = 9 days

18. PRACTICE QUESTIONS

Simplify 72 : 108.
Find x: 7 : 9 = x : 27.
Divide ₹1500 in the ratio 2 : 3.
Find the compound ratio of 3 : 4 and 5 : 6.
Are 12 : 18 and 20 : 30 equivalent?
If 8 pens cost ₹120, find the cost of 15 pens.
10 workers complete a job in 18 days. How many days will 15 workers take?
The ratio of boys to girls is 4 : 5. If there are 36 boys, find the number of girls.
Divide 840 into three parts in the ratio 2 : 3 : 7.
A mixture contains milk and water in the ratio 7 : 3. Find the amount of each in 50 litres.

19. ANSWERS

1. 2 : 3    2. 21    3. ₹600, ₹900
4. 5 : 8    5. Yes    6. ₹225
7. 12 days    8. 45 girls
9. 140, 210, 490
10. 35 L milk, 15 L water

FINAL REVISION TIP:
Ratio compares quantities. Proportion states equality between ratios. For direct proportion, quantities increase or decrease together. For inverse proportion, one increases while the other decreases. Always identify the type of relationship before applying a formula.