Ratio, Proportion and Percentages • Topic 1 of 3

Ratios and Equivalent Ratios

What is a Ratio?

A ratio is a comparison of two quantities of the same kind, showing how many times one quantity contains the other. It is written as \(a : b\) or \(\frac{a}{b}\), where \(a\) and \(b\) are the two quantities.

Example: If there are 3 boys and 5 girls in a class, the ratio of boys to girls is \(3 : 5\).

Simplifying Ratios:

To simplify a ratio, divide both terms by their greatest common factor (GCF).

Steps to Simplify a Ratio:

  1. Find the GCF of both numbers
  2. Divide both numbers by the GCF
  3. Write the simplified ratio in the form \(a : b\)

Equivalent Ratios:

Ratios that represent the same comparison are called equivalent ratios. They can be found by multiplying or dividing both terms by the same non-zero number.

Examples: \(3 : 5 = 6 : 10 = 9 : 15 = 12 : 20\)

Comparing Ratios:

Convert ratios to fractions and compare, or use cross multiplication.

Ratios and Equivalent RatiosRatio of blue to red circles: 3 : 2Blue = 3: Red = 2Equivalent Ratios — Multiply/Divide both parts by same number3:2→×2→6:4→×3→9:6→= 3:2RatioFractionDecimalPercentage3:23/50.660%1:41/50.220%2:32/50.440%
Solved Examples
1
Worked Example

Simplify the ratio \(24 : 36\).

Solution
  • GCF of 24 and 36 is 12
  • \(24 \div 12 = 2\), \(36 \div 12 = 3\)
  • Simplified ratio = \(2 : 3\)

-

Answer: $2 : 3$ **Example 2:** Find three equivalent ratios of $4 : 7$. *Solution:* - Multiply both terms by 2: $4 \times 2 = 8$, $7 \times 2 = 14$ → $8 : 14$ - Multiply by 3: $12 : 21$ - Multiply by 4: $16 : 28$ - **Answer:** $8 : 14$, $12 : 21$, $16 : 28$ (any three) **Example 3:** The ratio of two numbers is $5 : 8$. If the sum of the numbers is 104, find the numbers. *Solution:* - Let the numbers be $5x$ and $8x$ - Sum = $5x + 8x = 13x = 104$ - $x = 104 \div 13 = 8$ - First number = $5 \times 8 = 40$ - Second number = $8 \times 8 = 64$ - **Answer:** 40 and 64

Key Points

  • A ratio compares two quantities of the same kind: \(a : b = \frac{a}{b}\)
  • To simplify a ratio, divide both terms by their GCF
  • Equivalent ratios are obtained by multiplying or dividing both terms by the same number
  • Order matters in ratios: \(a : b\) is different from \(b : a\)
  • Ratios have no units (they compare same kinds of quantities)
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.A ratio compares two quantities of the:
Explanation: Same kind.
Q2.The ratio $4:6$ in simplest form is:
Explanation: Divide by $2$.
Q3.Equivalent ratios have the same:
Explanation: Same value.
Q4.The ratio $10:5$ simplifies to:
Explanation: $2:1$.
Practice more MCQs → 📝 Assignment · 35 marks