CODEBHAVYA • QUANTITATIVE APTITUDE

⚖️ Ratio & Proportion

Compare quantities, build equivalent ratios, solve proportions, divide totals fairly and recognise direct and inverse variation.

🎯 1. Learning Objectives

After completing this topic, you should be able to:

  • Write and simplify ratios using common units.
  • Generate equivalent ratios and compare ratios.
  • Recognise and solve proportions using cross multiplication.
  • Divide a quantity in a given ratio.
  • Solve direct and inverse proportion problems.
  • Combine two linked ratios and apply ratios in programs and data.

📖 2. Ratio Fundamentals

Meaning

A ratio compares two quantities by division.

a : b = a/b,   b ≠ 0

Simplest Form

Divide every term by their greatest common divisor.

12 : 18 = 2 : 3
Use the same units first: 2 m : 50 cm becomes 200 cm : 50 cm = 4 : 1.

🔁 3. Equivalent Ratios

Multiplying or dividing all terms by the same non-zero value produces an equivalent ratio.

a : b = ka : kb
Example: 4 : 7 = 20 : 35 because both terms were multiplied by 5.

🔗 4. Proportion

Four quantities are in proportion when two ratios are equal.

a : b = c : d   ⇔   ad = bc

Fourth Proportional

a : b = c : x ⇒ x = bc/a

Mean Proportional

a : x = x : b ⇒ x = √ab

🧩 5. Dividing a Quantity in a Ratio

If total T is divided in the ratio a : b, the number of equal parts is a+b.

First Share

T × a/(a+b)

Second Share

T × b/(a+b)
Example: ₹360 in 2:3 gives ₹360×2/5=₹144 and ₹360×3/5=₹216.

📈 6. Direct Proportion

Two quantities are directly proportional when increasing one increases the other in the same ratio.

y ∝ x ⇒ y/x = constant ⇒ y₂ = y₁x₂/x₁

Examples: cost and quantity at a fixed price, distance and time at a fixed speed.

↔️ 7. Inverse Proportion

Two quantities are inversely proportional when increasing one decreases the other so their product stays constant.

y ∝ 1/x ⇒ xy = constant ⇒ y₂ = x₁y₁/x₂

Examples: workers and days for fixed work, speed and time for a fixed distance.

🧮 8. Compound and Linked Ratios

Compound Ratio

Multiply corresponding terms.

(a:b) × (c:d) = ac : bd

Linked Ratios

Make the common term equal before joining.

a:b=3:5, b:c=10:7 ⇒ a:b:c=6:10:7

🧠 9. Ratio Interpretation

Statement Interpretation Example for 2:3
First : Second Part-to-part comparison 2 parts to 3 parts
First as fraction of total a/(a+b) 2/5
Second as fraction of total b/(a+b) 3/5
Difference in shares |a−b|/(a+b) of total 1/5 of total
INTERACTIVE EXPLORER

🧪 10. Ratio & Proportion Explorer

Select a calculation, enter values and click Calculate. Changing inputs hides the old result.

Select a calculation or example, then click Calculate.

🧾 11. Solved Examples

Example 1: Simplification

Problem: Simplify 36:48.

Solution: GCD=12. Divide both terms by 12: 3:4.

Example 2: Proportion

Problem: x:24=5:8.

Solution: 8x=24×5, so x=15.

Example 3: Direct Variation

Problem: Five books cost ₹600. Find the cost of eight.

Solution: 600×8/5=₹960.

Example 4: Inverse Variation

Problem: 12 workers need 10 days. How long for 15 workers?

Solution: 12×10=15×d, so d=8 days.

💻 12. Ratios in Programming

Screen and Image Scaling

Aspect ratios preserve image shape while dimensions change.

Load Distribution

Servers can receive work according to capacity ratios.

Data Normalisation

Ratios compare measurements collected on different scales.

Mixtures and Weighted Sampling

Programs generate datasets or batches in required class proportions.

⚠️ 13. Common Mistakes

Different Units

Convert both quantities to the same unit before forming a ratio.

Ratio Is Not a Difference

3:5 compares by division; it does not mean a difference of 2 only.

Part-to-Part vs Part-to-Whole

In 2:3, the first part is 2/5 of the total, not 2/3.

Direct vs Inverse

Decide whether the quotient or the product must stay constant.

✍️ 14. Practice Problems

Solve each problem first, then use Show Solution.

1. Simplify 45:60.

2. Divide ₹700 in the ratio 3:4.

3. Find x if 7:9=21:x.

4. Eight machines make 400 parts. How many do twelve machines make at the same rate?

5. Ten workers finish a job in 18 days. Find the time for 15 workers.

📝 15. Quick Revision

  • a:b means a/b and both quantities must use the same unit.
  • Equivalent ratios multiply or divide every term by the same non-zero number.
  • a:b=c:d implies ad=bc.
  • Divide T in a:b using Ta/(a+b) and Tb/(a+b).
  • Direct proportion keeps y/x constant.
  • Inverse proportion keeps xy constant.
  • Linked ratios require the common term to be made equal.
  • In a:b, the first quantity is a/(a+b) of the total.
INTERACTIVE PRACTICE

⚖️ Ratio & Proportion Practice

Complete 20 questions from easy to hard. Score, every option, correct answers and explanations appear only after final submission.

Start 20-Question Test →