Ratio and Proportion Revision Notes for VCE SSCE General Mathematics

Ratio and Proportion

What are ratios?

Ratios provide a way to numerically compare two or more quantities. They show the relative sizes of different amounts and are fundamental tools in mathematics for expressing relationships between values.

A ratio can be written in two main ways:

Using colon notation: $a : b$ (read as "a to b")
As a fraction: $\frac{a}{b}$

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Order Matters!

The order of numbers in a ratio matters! The ratio $a : b$ is not the same as $b : a$ . Always ensure you write ratios in the correct order as specified in the question.

This is one of the most common mistakes students make with ratios.

Expressing quantities as ratios

Two quantities

When comparing two quantities, we simply write them in the order specified, separated by a colon. The key is to maintain the correct order as given in the question.

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Example 17: Expressing Two Quantities as a Ratio

In a Year 10 class of 26 students, there are 14 girls and 12 boys. Express the number of girls to boys as a ratio.

Solution:

Since we have 14 girls and 12 boys, the ratio of girls to boys is $14 : 12$ .

This could also be written as the fraction $\frac{14}{12}$ .

More than two quantities

Ratios can also compare more than two quantities. We simply extend the notation to include all values in the correct order.

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Example 18: Expressing Multiple Quantities as a Ratio

A survey of the same group of 26 students showed that 10 students walked to school, 11 came by public transport, and 5 were driven by their parents. Express as a ratio the number of students who walked to school, to the number who came by public transport, to the number who were driven to school.

Solution:

The order specified is: walked, public transport, driven.

Therefore, the ratio is $10 : 11 : 5$ .

Remember that the order must match what the question asks for.

Simplifying ratios

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What is Simplest Form?

A ratio is in simplest form when it uses the smallest possible whole numbers. This means the highest common factor (HCF) of all numbers in the ratio is 1.

Ratios in simplest form are easier to interpret and work with in further calculations.

We can simplify ratios by:

Dividing all terms by their common factor
Multiplying all terms to eliminate decimals or fractions

Simplifying whole number ratios

For whole number ratios, we divide all terms by their common factor. Look for the largest number that divides evenly into all terms.

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Worked Example: Simplifying Whole Number Ratios

Simplify $15 : 20$

Solution:

Both 15 and 20 can be divided by 5.

$15 : 20 = \frac{15}{5} : \frac{20}{5} = 3 : 4$

The simplified ratio is 3 : 4.

Simplifying decimal ratios

For ratios containing decimals, we multiply all terms by a power of 10 to convert them to whole numbers first, then simplify if possible.

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Worked Example: Simplifying Decimal Ratios

Simplify $0.4 : 1.7$

Solution:

Multiply both terms by 10 to eliminate decimals:

$0.4 : 1.7 = 0.4 \times 10 : 1.7 \times 10 = 4 : 17$

Since 4 and 17 share no common factors, this is already in simplest form.

The answer is 4 : 17.

Simplifying fractional ratios

When ratios contain fractions, we have two methods available. Both methods aim to eliminate the fractions first, then simplify.

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Example: Simplifying Fractional Ratios

Simplify the ratio $\frac{3}{4} : \frac{5}{3}$

Method 1 - Eliminate fractions step by step:

First, multiply both fractions by 4:

$\frac{3}{4} \times 4 : \frac{5}{3} \times 4 = 3 : \frac{20}{3}$

Then multiply both terms by 3:

$3 \times 3 : \frac{20}{3} \times 3 = 9 : 20$

Method 2 - Use the LCM:

Find the lowest common multiple (LCM) of the denominators 4 and 3, which is 12.

Multiply both fractions by 12:

$\frac{3}{4} \times 12 : \frac{5}{3} \times 12 = 9 : 20$

Both methods give the same answer: 9 : 20.

Note: Method 2 is usually quicker when you can identify the LCM easily.

Simplifying ratios with different units

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Critical Rule: Same Units Required

Before simplifying a ratio, both quantities must be in the same units. This is a crucial step that students often miss.

Always convert to the same unit before attempting any simplification!

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Example: Simplifying Ratios with Different Units

Express 15 cm to 3 m as a ratio in its simplest form.

Solution:

Step 1: Write the ratio: $15 \text{ cm} : 3 \text{ m}$

Step 2: Convert 3 m to cm by multiplying by 100:

$15 \text{ cm} : 3 \times 100 \text{ cm} = 15 \text{ cm} : 300 \text{ cm}$

Step 3: Now both parts are in cm, we can simplify by dividing by 15:

$15 : 300 = \frac{15}{15} : \frac{300}{15} = 1 : 20$

Step 4: Write the answer: 1 : 20

Notice that the units cancel out in the final simplified ratio.

Equivalent ratios

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Understanding Equivalent Ratios

Ratios are equivalent when they represent the same relationship between quantities, even though the actual numbers are different. Understanding equivalent ratios is essential for solving proportion problems.

Think of equivalent ratios like equivalent fractions - they look different but represent the same value.

For example, consider the ratio $12 : 8$ :

If we multiply both terms by 2: $12 : 8 = 24 : 16$
If we divide both terms by 4: $12 : 8 = 3 : 2$

All three ratios ( $12 : 8$ , $24 : 16$ , and $3 : 2$ ) are equivalent because they represent the same proportional relationship. The ratio $3 : 2$ is the simplest form.

Finding missing values in ratios

When we know that two ratios are equivalent, we can find missing values using algebraic methods. This is a practical application of proportional reasoning and builds on your algebra skills.

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Example: Finding Missing Values in Ratios

Find the missing value for the equivalent ratios $3 : 7 = \square : 28$

Solution:

Step 1: Let the unknown value be $x$ and write the ratios as fractions:

\begin{aligned} 3 : 7 &= x : 28\\ \frac{3}{7} &= \frac{x}{28} \end{aligned}

Step 2: Solve for $x$ by multiplying both sides by 28:

\begin{aligned} \frac{3}{7} \times 28 &= \frac{x}{28} \times 28\\ x &= 12 \end{aligned}

Therefore: $3 : 7 = 12 : 28$

We can verify this is correct: if we simplify $12 : 28$ by dividing both terms by 4, we get $3 : 7$ ✓

Alternative method using CAS:

You can also use the solve function on a CAS calculator:

$\text{solve}\left(\frac{3}{7} = \frac{x}{28}, x\right)$

This gives x = 12

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Key Points to Remember:

Ratios compare quantities numerically - they show relative sizes between two or more values
Order is crucial - the ratio $3 : 5$ is completely different from $5 : 3$ , so always write ratios in the order specified
Convert to same units first - before simplifying any ratio involving measurements, ensure both quantities use the same units
Simplest form uses smallest whole numbers - a ratio is fully simplified when the HCF of all terms equals 1
Equivalent ratios represent the same relationship - you can multiply or divide all terms by the same number without changing the fundamental relationship
Two methods for fractions - eliminate fractions step-by-step OR use the LCM of denominators
Algebraic thinking - finding missing values in equivalent ratios uses the same skills as solving equations

Ratio and Proportion (VCE SSCE General Mathematics): Revision Notes