How to Solve Ratio and Proportion Questions in CA Foundation

Ratio and proportion questions form a significant portion of CA Foundation Quantitative Aptitude. These concepts are foundational for advanced mathematics and frequently appear in exam papers. Let's master this topic systematically.

Understanding Ratios

A ratio compares two quantities. Written as a:b, it shows the relationship between them.

Example: If A has Rs. 300 and B has Rs. 200, the ratio A:B = 300:200 = 3:2. This means for every 3 units A has, B has 2 units.

Key Properties:

Types of Ratios

Equivalent Ratios

Ratios with the same value are equivalent. Example: 3:2 = 6:4 = 9:6 = 12:8

To check if two ratios are equal: a:b = c:d if ad = bc

Example: Are 4:6 and 8:12 equal? Check: 4 × 12 = 48 and 6 × 8 = 48. Yes, they're equal.

Compound Ratios

When you multiply ratios, you get a compound ratio.

If A:B = 3:2 and B:C = 4:5, find A:C.

Solution: Make B the same in both ratios.

A:B = 3:2 = 6:4

B:C = 4:5

Therefore, A:B:C = 6:4:5

So A:C = 6:5

Understanding Proportion

A proportion states that two ratios are equal. Written as a:b = c:d or a:b::c:d

This reads as: "a is to b as c is to d"

Example: 2:3 = 4:6 is a proportion because both ratios equal 2/3.

Types of Proportions

Direct Proportion

If x increases, y increases proportionally. Written as x ∝ y or y = kx

Example: If 5 workers complete 100 units of work, how many units will 10 workers complete?

5 workers → 100 units

10 workers → x units

5/100 = 10/x (this is incorrect—should be proportional)

Correct: x/100 = 10/5

x = 100 × (10/5) = 200 units

Inverse Proportion

If x increases, y decreases proportionally. Written as x ∝ 1/y or xy = k

Example: 10 workers complete a job in 20 days. How many days will 4 workers take?

10 workers × 20 days = constant work

4 workers × x days = same constant work

10 × 20 = 4 × x

200 = 4x

x = 50 days

4 workers take longer because fewer people work.

Problem-Solving Framework

Step 1: Identify the ratio type

Determine if quantities are in direct or inverse proportion.

Step 2: Set up the equation

Use the appropriate formula based on type identified.

Step 3: Solve for the unknown

Cross-multiply or simplify as needed.

Step 4: Verify the answer

Check if the answer is reasonable and satisfies the original condition.

Practical Exam Problems

Problem 1: Partnership Division

Three partners invest Rs. 50,000, Rs. 60,000, and Rs. 40,000. They agree to share profits in the ratio of investments. If profit is Rs. 15,000, find each partner's share.

Solution:

Investment ratio = 50:60:40 = 5:6:4

Total parts = 5 + 6 + 4 = 15

Profit per part = 15,000 / 15 = Rs. 1,000

Partner A's share = 5 × 1,000 = Rs. 5,000

Partner B's share = 6 × 1,000 = Rs. 6,000

Partner C's share = 4 × 1,000 = Rs. 4,000

Problem 2: Time and Work

If 6 machines produce 240 articles in 8 hours, how many articles will 9 machines produce in 12 hours?

Solution:

Articles ∝ Number of machines × Time

240 articles = 6 machines × 8 hours × k

240 = 48k

k = 5

For 9 machines, 12 hours:

Articles = 9 × 12 × 5 = 540 articles

Common Mistakes to Avoid

Don't mix direct and inverse proportions. Always identify which applies first.

Avoid simplifying ratios incorrectly. Always express in lowest terms: 12:8 becomes 3:2, not 6:4.

Don't forget units. If comparing quantities with different units, convert first.

Never cross-multiply incorrectly in proportions. Always ensure a:b = c:d means a/b = c/d.

Exam Tips

Practice at least 50 ratio-proportion problems before the exam. Repetition builds speed and confidence.

Solve mental math shortcuts. Example: If ratio is 3:4 and total is 70, parts are 30 and 40 instantly.

Use approximation for multiple choice questions. Eliminate obviously wrong options first.

Time yourself. These problems should take 2-3 minutes maximum per question.

Use CA Saarthi's Quantitative Aptitude practice bank to solve hundreds of ratio and proportion problems with instant feedback. Master shortcuts and exam strategies that help you solve these questions in under two minutes!