Ratios Between Two Quantities | JHS Mathematics

Introduction

A ratio compares two quantities of the same kind, showing how much of one thing there is compared to another. Ratios are written with a colon (:) between the numbers.

\[ \text{Ratio of a to b is written as } a : b \]

Understanding Ratios

Ratios help us compare quantities in different situations:

Part-to-part: Comparing parts of a whole (e.g., 3 boys to 5 girls)
Part-to-whole: Comparing a part to the whole (e.g., 3 boys to 8 total students)

Example: In a class of 10 boys and 15 girls, the ratio of boys to girls is 10:15

Determining Ratios

Finding the Ratio

To determine the ratio between two quantities:

Identify the two quantities being compared
Write them in order with a colon between
Simplify the ratio if possible

Example 1: Simple Ratio

There are 4 red balls and 6 blue balls. What is the ratio of red to blue balls?

\[ \text{Red : Blue} = 4 : 6 \] \[ \text{Simplified} = 2 : 3 \text{ (divided both by 2)} \]

Example 2: Different Units

Convert to same units first: 50cm to 2m

\[ 2m = 200cm \] \[ \text{Ratio} = 50 : 200 \] \[ \text{Simplified} = 1 : 4 \]

Expressing Ratios

Different Ways to Write Ratios

Ratios can be expressed in several forms:

Form	Example
Using colon	3 : 4
As a fraction	\(\frac{3}{4}\)
Using "to"	3 to 4
In words	"The ratio is three to four"

Order Matters in Ratios

The order in which quantities are written in a ratio is important:

\[ \text{The ratio 2:3 is different from 3:2} \]

Example: For every 2 apples there are 3 oranges (2:3) is not the same as for every 3 apples there are 2 oranges (3:2)

Ratio Language

Describing Ratios

We can describe ratios using different words and phrases:

Comparison Phrases

"For every 2 boys, there are 3 girls"
"The ratio of teachers to students is 1:20"
"There are 4 red marbles for each blue marble"

Simplified Forms

Always simplify ratios when possible:

\[ 6:9 \text{ becomes } 2:3 \] \[ 15:20 \text{ becomes } 3:4 \]

Practical Examples

Understanding ratio language is useful in real-world situations:

Recipe: "Mix water and juice concentrate in a 3:1 ratio" means 3 parts water to 1 part concentrate

Map Scale: "Scale of 1:50,000" means 1cm on map represents 50,000cm in real life

Classroom: "Student-teacher ratio of 25:1" means 25 students per teacher

Practice Exercise

Question 1

There are 12 boys and 18 girls in a class. What is the ratio of boys to girls in simplest form?

\[ \text{Boys : Girls} = 12 : 18 = 2 : 3 \]

Question 2

Express the ratio 15:25 in simplest form.

\[ 15 : 25 = 3 : 5 \text{ (divided both by 5)} \]

Question 3

In a basket, there are 5 apples and 7 oranges. Write the ratio of apples to total fruits.

\[ \text{Total fruits} = 5 + 7 = 12 \] \[ \text{Apples : Total} = 5 : 12 \]

Question 4

Convert 200m to 1km as a simplified ratio.

\[ 1km = 1000m \] \[ 200 : 1000 = 1 : 5 \]

Question 5

Describe the ratio 3:7 using the phrase "for every..."

For every 3 of the first quantity, there are 7 of the second quantity.

Question 6

A recipe calls for flour and sugar in a 5:2 ratio. If you use 10 cups of flour, how much sugar is needed?

\[ \text{Flour : Sugar} = 5 : 2 = 10 : x \] \[ 5 \times 2 = 10 \text{ so } 2 \times 2 = 4 \] \[ \text{Sugar needed} = 4 \text{ cups} \]

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Test your skills with ratios between quantities