Ratio and Proportion
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Solved Examples
Study MaterialSolved Examples – Ratio and Proportion
Solved examples help students understand the practical application of Ratio and Proportion concepts in competitive examinations. These examples are designed from basic to advanced level and cover important ratio and proportion problems frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement aptitude tests.
Topics Covered in Solved Examples
- Basic Ratio Problems
- Simplification of Ratios
- Comparison of Ratios
- Duplicate and Compound Ratios
- Direct and Inverse Proportion
- Distribution Problems
- Partnership Problems
- Age Ratio Problems
- Mixture Ratio Problems
- Advanced Proportion Applications
Example 1: Basic Ratio Problem
Question: Find the ratio of 24 to 36 in simplest form.
Solution:
Ratio:
= 24 : 36
HCF of 24 and 36 = 12
Divide both terms by 12:
= 2 : 3
Therefore:
Required ratio = 2 : 3
Example 2: Ratio of Quantities with Different Units
Question: Find the ratio of 3 kg to 750 g.
Solution:
Convert into same units:
3 kg = 3000 g
Ratio:
= 3000 : 750
= 4 : 1
Therefore:
Required ratio = 4 : 1
Example 3: Comparison of Ratios
Question: Compare 5 : 8 and 7 : 10.
Solution:
Cross multiplication:
5 × 10 = 50
8 × 7 = 56
Since:
50 < 56
Therefore:
5 : 8 < 7 : 10
Example 4: Duplicate Ratio
Question: Find the duplicate ratio of 4 : 7.
Solution:
Duplicate ratio:
= 4² : 7²
= 16 : 49
Therefore:
Duplicate ratio = 16 : 49
Example 5: Compound Ratio
Question: Find the compound ratio of 2 : 5 and 3 : 7.
Solution:
Compound ratio:
= (2 × 3) : (5 × 7)
= 6 : 35
Therefore:
Compound ratio = 6 : 35
Example 6: Basic Proportion Problem
Question: Find x if 3 : 5 :: x : 20.
Solution:
Using proportion property:
3 × 20 = 5 × x
60 = 5x
x = 12
Therefore:
x = 12
Example 7: Distribution Problem
Question: Divide 2400 in the ratio 3 : 5.
Solution:
Total ratio parts:
= 3 + 5
= 8
First share:
= (3/8) × 2400
= 900
Second share:
= (5/8) × 2400
= 1500
Therefore:
Required shares = 900 and 1500
Example 8: Direct Proportion Problem
Question: If 8 workers complete a work in 15 days, how many days will 12 workers take?
Solution:
Workers and days are inversely proportional.
8 × 15 = 12 × x
120 = 12x
x = 10
Therefore:
Required days = 10
Example 9: Inverse Proportion Problem
Question: A car travels 300 km in 5 hours. How much time will it take to travel 420 km at the same speed?
Solution:
Distance and time are directly proportional.
300 : 420 = 5 : x
300x = 2100
x = 7
Therefore:
Required time = 7 hours
Example 10: Partnership Problem
Question: A and B invest 20000 and 30000 respectively in a business. Find their profit-sharing ratio.
Solution:
Profit-sharing ratio:
= 20000 : 30000
= 2 : 3
Therefore:
Profit-sharing ratio = 2 : 3
Example 11: Partnership with Time
Question: A invests 10000 for 12 months and B invests 15000 for 8 months. Find their profit-sharing ratio.
Solution:
A's share:
= 10000 × 12
= 120000
B's share:
= 15000 × 8
= 120000
Ratio:
= 1 : 1
Therefore:
Profit-sharing ratio = 1 : 1
Example 12: Age Ratio Problem
Question: The ages of A and B are in the ratio 4 : 5. If their total age is 54 years, find their ages.
Solution:
Total ratio parts:
= 4 + 5
= 9
One part:
= 54/9
= 6
A's age:
= 4 × 6
= 24
B's age:
= 5 × 6
= 30
Therefore:
Required ages = 24 years and 30 years
Example 13: Ratio Change Problem
Question: The ratio of boys and girls in a class is 3 : 2. If there are 18 boys, find the number of girls.
Solution:
3 parts = 18 boys
1 part = 6
Girls:
= 2 × 6
= 12
Therefore:
Number of girls = 12
Example 14: Mixture Ratio Problem
Question: Milk and water are mixed in the ratio 5 : 2. If total mixture is 35 litres, find the quantity of water.
Solution:
Total ratio parts:
= 5 + 2
= 7
Water quantity:
= (2/7) × 35
= 10 litres
Therefore:
Quantity of water = 10 litres
Example 15: Sub-Duplicate Ratio
Question: Find the sub-duplicate ratio of 49 : 81.
Solution:
Sub-duplicate ratio:
= √49 : √81
= 7 : 9
Therefore:
Sub-duplicate ratio = 7 : 9
Example 16: Triplicate Ratio
Question: Find the triplicate ratio of 2 : 5.
Solution:
Triplicate ratio:
= 2³ : 5³
= 8 : 125
Therefore:
Triplicate ratio = 8 : 125
Example 17: Compound Proportion Problem
Question: If 12 men can complete a work in 15 days, how many men are needed to complete it in 9 days?
Solution:
Men and days are inversely proportional.
12 × 15 = x × 9
180 = 9x
x = 20
Therefore:
Required men = 20
Example 18: Advanced Distribution Problem
Question: Divide 720 among A, B, and C in the ratio 2 : 3 : 4.
Solution:
Total ratio parts:
= 2 + 3 + 4
= 9
One part:
= 720/9
= 80
A's share:
= 2 × 80
= 160
B's share:
= 3 × 80
= 240
C's share:
= 4 × 80
= 320
Therefore:
Required shares = 160, 240, and 320
Important Exam Tips
- Always convert quantities into same units before forming ratios.
- Simplify ratios completely.
- Use cross multiplication in proportion problems.
- Practice distribution-based questions regularly.
- Remember direct and inverse proportion concepts carefully.
- Use fraction methods for quick calculations.
- Verify calculations carefully in partnership problems.
Practicing solved examples regularly improves conceptual clarity, logical thinking, and calculation speed in solving Ratio and Proportion questions in competitive examinations.