How to Calculate Ratios: Direct & Inverse Proportion

A ratio is a way to compare two like quantities, written as a:b. To simplify, divide both numbers by their GCD. There are two kinds: direct proportion (a/b = c/d) when both values rise together, and inverse proportion (a × b = c × d) when one rises as the other falls. Example: 12:8 simplifies to 3:2 by dividing both by 4.

Defining Ratios and Their Context

A ratio is a way to compare two or more like quantities. It is written as (read as "a to b") or . The concept of ratios has been known since Ancient Greece and was used by Euclid in Elements to describe geometric proportion. Today, ratios are everywhere — recipes, map scales, vehicle speeds, and even currency exchange rates.

For instance, if you have 12 apples and 8 oranges, the ratio is 12 : 8, which simplifies to 3 : 2. This concept is closely related to how to calculate percentages, which uses the ratio of a part to 100.

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Formulas and Types of Ratios

There are two main kinds of ratio problems:

Direct Proportion

When one value increases, the other increases proportionally.

What the symbols mean:

• a, b = the original pair (known)
• c = the new value (known)
• d = the value you're solving for
• Cross-multiplication: a × d = b × c

Inverse Proportion

When one value increases, the other decreases proportionally.

What the symbols mean:

• a, b = the first pair of values
• c, d = the second pair of values
• The product stays constant: a × b = c × d

💡 Tip: Sign of a direct proportion: more → more. Sign of an inverse proportion: more → less. Always ask first: "If this one goes up, does the other one go up or down?"

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How to Calculate Ratios: Worked Examples

Example 1: Simplifying a Ratio ⭐

Simplify the ratio 24 : 36.

Step 1: Find the GCD of 24 and 36. Factors of 24: 1, 2, 3, 4, 12, 24 | Factors of 36: 1, 2, 3, 4, 9, 12, 36 → GCD = 12

Step 2: Divide both numbers by the GCD.

✅ Answer: 24 : 36 = 2 : 3

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Example 2: Direct Proportion ⭐⭐

If 5 kg of rice costs Rp75,000, how much does 8 kg cost?

Step 1: More rice → higher price → direct proportion.

Step 2: Set up the equation and solve by cross-multiplying.

✅ Answer: 8 kg of rice costs Rp120,000

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Example 3: Inverse Proportion ⭐⭐

If 6 workers finish a job in 4 days, how many days will 8 workers need?

Step 1: More workers → fewer days → inverse proportion.

Step 2: Use the formula a × b = c × d.

✅ Answer: 8 workers finish the job in 3 days

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Example 4: A Common Mistake — Choosing the Wrong Type ⭐

This is the most common slip students make: confusing direct and inverse proportion.

Problem: If 3 taps fill a pool in 12 hours, how many hours do 6 taps need?

Wrong approach ❌ (using the direct-proportion formula):

The student forgot to ask: do more taps mean longer or shorter?

Correct approach ✅ (inverse proportion — more taps → faster):

✅ Correct answer: 6 taps fill the pool in 6 hours

💡 Remember: Always ask first — "If the first value goes up, does the second go up or down?" Up → direct. Down → inverse.

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Ratios in Everyday Life

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Direct vs Inverse Proportion: A Quick Way to Tell Them Apart

Before tackling a word problem, always run this check:

1. Identify the two quantities being compared (e.g. number of workers and time).
2. Imagine the first one going up — what happens to the second?
   • It also goes up → direct proportion → use
   • It goes down → inverse proportion → use
3. Solve using the matching equation.

This idea also links closely to how to calculate LCM and GCD, which often comes up when simplifying ratios.

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Practice Problems: Calculating Ratios

⭐ Problem 1 (Easy): The ratio of Andi's age to Budi's age is 3 : 5. If Andi is 18, how old is Budi?

⭐ Problem 2 (Easy): Simplify the ratio 45 : 75 to its lowest terms.

⭐⭐ Problem 3 (Medium): A car travels 120 km in 2 hours. How far does it travel in 5 hours at the same speed?

⭐⭐⭐ Problem 4 (Challenge): If 4 taps fill a pool in 6 hours, how many hours do 3 taps need?

Answer Key:

1. Direct proportion: Budi's age = (5/3) × 18 = 30 years old. Andi : Budi = 18 : 30 = 3 : 5 ✓
2. GCD(45, 75) = 15 → 45÷15 : 75÷15 = 3 : 5. No common factor remains beyond 1.
3. Direct (longer time → farther): (120 ÷ 2) × 5 = 60 × 5 = 300 km.
4. Inverse (fewer taps → more time): 4 × 6 = 3 × x → x = 24 ÷ 3 = 8 hours.

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Summary

1. A ratio compares two like quantities; simplify it by dividing both by their GCD.
2. Direct proportion (): one value rises, the other rises too. Solve by cross-multiplying.
3. Inverse proportion (): one value rises while the other falls. The product stays constant.
4. The most common mistake is picking the wrong type — always check the direction of change before choosing a formula.
5. Ratios show up in map scales, recipes, speeds, and many other everyday situations.

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Learn more about how to calculate percentages — a concept closely tied to ratios. And explore effective study strategies in 5 Best Approaches to Learning Maths.