👁 Preview — Study, Practice and Revise are open; mock tests and the rest of the syllabus unlock on subscription. Unlock all · ₹4,999
← Back to Arithmetic
Study mode

Ratio and Proportion

Study Practice Revise 🔒 Test

Introduction to Ratio and Proportion

In everyday life, we often compare quantities to understand their relationship. For example, if a recipe calls for 2 cups of flour and 1 cup of sugar, we say the ratio of flour to sugar is 2:1. This comparison is what we call a ratio. When two such ratios are equal, they form a proportion. Understanding ratio and proportion helps solve many quantitative problems quickly and accurately, especially in competitive exams.

Definition and Basics of Ratio

A ratio is a way to compare two quantities of the same kind by division. It tells us how many times one quantity contains another. For example, if there are 3 apples and 2 oranges, the ratio of apples to oranges is 3:2.

Mathematically, the ratio of two quantities \(a\) and \(b\) is written as:

\[ \text{Ratio} = \frac{a}{b} \]

where \(a\) and \(b\) are quantities of the same kind.

Ratios can be expressed in three ways:

  • Using a colon: \(a:b\)
  • As a fraction: \(\frac{a}{b}\)
  • Using the word "to": "a to b"

Ratios can be of two types:

  • Part-to-Part Ratio: Compares one part of a whole to another part. Example: Ratio of boys to girls in a class.
  • Part-to-Whole Ratio: Compares one part to the entire quantity. Example: Ratio of boys to total students.
3 parts 2 parts

In the above bar, the green part represents 3 units and the yellow part represents 2 units, illustrating the ratio 3:2 visually.

Proportion and its Properties

A proportion is an equation stating that two ratios are equal. For example, if \(\frac{a}{b} = \frac{c}{d}\), then \(a:b\) is in proportion to \(c:d\).

This can be written as:

\[ \frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c \]

This is called the fundamental property of proportion, which means the product of the means equals the product of the extremes.

Here, in the proportion \(a : b = c : d\), the terms \(b\) and \(c\) are called the means, and \(a\) and \(d\) are called the extremes.

graph TD    A[Start with proportion] --> B{Are ratios equal?}    B -- Yes --> C[Cross multiply: a x d and b x c]    C --> D{Are products equal?}    D -- Yes --> E[Proportion holds true]    D -- No --> F[Proportion does not hold]

Continued proportion refers to a sequence of three or more quantities where the first is to the second as the second is to the third, and so on. For example, \(a : b = b : c\) means \(b\) is the mean proportional between \(a\) and \(c\).

Worked Examples

Example 1: Simplify the Ratio of 150 cm to 2.5 m Easy
Simplify the ratio of 150 cm to 2.5 m.

Step 1: Convert both quantities to the same unit. Since 1 m = 100 cm, 2.5 m = 2.5 x 100 = 250 cm.

Step 2: Write the ratio in the same units: 150 cm : 250 cm.

Step 3: Simplify the ratio by dividing both terms by their greatest common divisor (GCD). GCD of 150 and 250 is 50.

\(\frac{150}{50} : \frac{250}{50} = 3 : 5\)

Answer: The simplified ratio is 3:5.

Example 2: Find the value of \(x\) if \(\frac{3}{5} = \frac{x}{20}\) Easy
Given the proportion \(\frac{3}{5} = \frac{x}{20}\), find the value of \(x\).

Step 1: Use the fundamental property of proportion: cross multiply.

\(3 \times 20 = 5 \times x\)

Step 2: Calculate the left side.

\(60 = 5x\)

Step 3: Solve for \(x\).

\(x = \frac{60}{5} = 12\)

Answer: \(x = 12\)

Example 3: A Mixture Contains Milk and Water in the Ratio 7:3. How Much Water Should Be Added to Make the Ratio 7:5? Medium
A mixture contains milk and water in the ratio 7:3. How much water should be added to make the ratio 7:5?

Step 1: Assume the total quantity of the mixture is 10 liters (7 liters milk + 3 liters water).

Step 2: Let the amount of water to be added be \(x\) liters.

Step 3: After adding water, milk remains 7 liters, water becomes \(3 + x\) liters.

Step 4: New ratio is given as 7:5, so:

\(\frac{7}{3 + x} = \frac{7}{5}\)

Step 5: Cross multiply:

\(7 \times 5 = 7 \times (3 + x)\)

\(35 = 7 \times 3 + 7x\)

\(35 = 21 + 7x\)

Step 6: Solve for \(x\):

\(7x = 35 - 21 = 14\)

\(x = \frac{14}{7} = 2\)

Answer: 2 liters of water should be added.

Example 4: If 5 kg of Sugar Costs INR 250, What is the Cost of 8 kg? Medium
The cost of 5 kg sugar is INR 250. Find the cost of 8 kg sugar.

Step 1: This is a direct proportion problem: cost is directly proportional to weight.

Step 2: Set up the proportion:

\(\frac{5}{250} = \frac{8}{x}\)

Step 3: Cross multiply:

\(5 \times x = 8 \times 250\)

\(5x = 2000\)

Step 4: Solve for \(x\):

\(x = \frac{2000}{5} = 400\)

Answer: The cost of 8 kg sugar is INR 400.

Example 5: Two Quantities are in the Ratio 4:5. If Their Sum is 180, Find the Quantities. Easy
Two quantities are in the ratio 4:5. Their sum is 180. Find the quantities.

Step 1: Let the quantities be \(4x\) and \(5x\).

Step 2: Their sum is given as 180:

\(4x + 5x = 180\)

\(9x = 180\)

Step 3: Solve for \(x\):

\(x = \frac{180}{9} = 20\)

Step 4: Find the quantities:

First quantity = \(4 \times 20 = 80\)

Second quantity = \(5 \times 20 = 100\)

Answer: The quantities are 80 and 100.

Tips & Tricks

Tip: Always convert units to the same system before comparing ratios.

When to use: When quantities in ratio are given in different units, like cm and m.

Tip: Use cross multiplication to quickly solve proportions with one unknown.

When to use: When a proportion equation has one unknown variable.

Tip: In mixture problems, set up ratios before and after addition or subtraction to find unknown quantities.

When to use: When solving mixture and alligation problems.

Tip: Simplify ratios fully before using them in calculations to avoid errors.

When to use: Before applying ratios in problem solving.

Tip: Remember that in proportion \(a:b = c:d\), the product of means equals product of extremes.

When to use: To verify or solve proportion problems.

Common Mistakes to Avoid

❌ Comparing quantities without converting to the same unit
✓ Always convert all quantities to the same metric unit before forming ratios
Why: Different units lead to incorrect ratio values and wrong answers.
❌ Incorrect cross multiplication order
✓ Multiply means and extremes correctly: \(a \times d = b \times c\)
Why: Swapping terms leads to wrong answers and confusion.
❌ Not simplifying ratios before solving problems
✓ Simplify ratios to their lowest terms first
Why: Simplification reduces calculation errors and complexity.
❌ Confusing part-to-part ratio with part-to-whole ratio
✓ Identify clearly whether ratio compares parts or part to whole
Why: Misinterpretation changes problem approach and solution.
❌ Ignoring the difference between direct and inverse proportion
✓ Understand and apply correct proportion type based on problem context
Why: Applying wrong proportion type leads to incorrect results.

Fundamental Property of Proportion

\[\frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c\]

Two ratios are in proportion if their cross products are equal

a,b,c,d = Quantities forming two ratios

Ratio

\[\text{Ratio} = \frac{a}{b}\]

Represents the relative size of two quantities a and b of the same kind

a,b = Quantities being compared

Compound Ratio

\[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\]

Product of two ratios to form a compound ratio

a,b,c,d = Quantities in ratios
Curated videos per subtopic
Top YouTube explainers, AI-ranked for your exam and language. Unlocks with subscription.
Unlock

Try Practice next.

Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.

Go to practice →
Ask a doubt
Ratio and Proportion · 10 free messages
Ask me anything about this subtopic. You have 10 free messages this session — chat history isn't saved in preview.