Ratio, Proportion, and Rate Revision Notes for Grade 12 NSC Matric Mathematical Literacy

Ratio, Proportion, and Rate

What is a ratio?

A ratio compares two or more quantities and shows how much of one thing there is compared to another. Ratios help us understand the relationship between different amounts and are essential for making comparisons in mathematics and everyday life.

Different ways to write ratios

Ratios can be expressed in three different ways:

Using a colon: 3 : 2
Using words: 3 to 2
As a fraction: 3/2

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All three methods represent the same relationship between the quantities. Choose the format that works best for your specific problem or context.

Equivalent ratios

Ratios can be scaled up or down while maintaining the same relationship. This creates equivalent ratios.

The key rule for creating equivalent ratios is to multiply or divide both numbers by the same value. For example:

3 : 2 can become 6 : 4 (multiply both by 2)
3 : 2 can become 9 : 6 (multiply both by 3)

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Always maintain the same order when writing ratios. The ratio 3:2 is different from 2:3 because the order matters for the comparison being made.

Writing ratios in simplest form

To write a ratio in its simplest form, find the largest number that divides into both terms and divide both by this number. This works the same way as simplifying fractions.

For example, to simplify 80 : 120:

Both numbers can be divided by 10: 80 ÷ 10 = 8, 120 ÷ 10 = 12
Both 8 and 12 can be divided by 4: 8 ÷ 4 = 2, 12 ÷ 4 = 3
The simplest form is 2 : 3

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Worked Example: Pancake Recipe

A pancake recipe uses 3 cups of flour and 2 cups of milk, giving a ratio of 3:2.

To make pancakes for more people, multiply both amounts by 4:

3 × 4 = 12 cups of flour
2 × 4 = 8 cups of milk
The ratio remains the same: 12:8 = 3:2

Ratios comparing 3 numbers

Sometimes you need to work with ratios that compare three quantities. The same principles apply - you can scale all three numbers up or down by the same factor.

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Worked Example: Sharing Money

A grandmother wants to share R800 between her three grandchildren in the ratio of their ages: 20:15:5 years.

Solution:

Simplify the ratio: 20:15:5 = 4:3:1 (divide all by 5)
Find total parts: 4 + 3 + 1 = 8 parts
Calculate each share:
- First child: 4/8 × R800 = R400
- Second child: 3/8 × R800 = R300
- Third child: 1/8 × R800 = R100
Check: R400 + R300 + R100 = R800 ✓

Types of ratio problems

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There are several types of ratio problems you might encounter. Understanding these different categories helps you choose the right approach for each situation.

Common types include:

Working out ratios from given quantities
Finding equivalent ratios
Finding unit ratios
Finding total number of parts
Finding total quantities
Finding quantities in specific parts
Finding missing numbers

Unit ratios

A unit ratio has one of the terms equal to 1. This makes comparisons easier by showing the amount per single unit.

To create a unit ratio, divide both terms by the smaller number. For example:

5 : 9 becomes 1 : 1.8 (divide both by 5)

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Worked Example: Hospital Staffing

There are 23 nurses and 7,567 patients in a hospital. How many patients does each nurse care for?

Solution:

Write the ratio: 23 : 7,567
Convert to unit ratio: 1 : 329 (divide both by 23)
Each nurse cares for approximately 329 patients

Proportions

A proportion is an equation stating that two ratios are equal. It has the form: $\frac{a}{b} = \frac{c}{d}$

Cross-multiplication method

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When one number in a proportion is unknown, use cross-multiplication to find it:

If $\frac{a}{b} = \frac{c}{d}$ , then $a × d = b × c$

This is the most reliable method for solving proportion problems.

Using tables for proportion problems

Tables can help organise proportion problems and make them easier to solve by clearly showing the relationships between quantities.

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Worked Example: Flower Bunches

Sipho makes flower bunches with a rose-to-daisy ratio of 1:3. If he has 15 daisies, how many roses does he need?

Solution:

Set up the proportion: $\frac{1}{3} = \frac{?}{15}$
Cross-multiply: $1 × 15 = 3 × ?$
Solve: $15 = 3 × ?$ , so $? = 5$
Sipho needs 5 roses

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Worked Example: Summer Camp

A summer camp has a boy-to-girl ratio of 8:11. If there are 88 boys, what is the total number of children?

Solution:

Set up proportion: $\frac{8}{11} = \frac{88}{g}$ (where g = girls)
Cross-multiply: $8g = 11 × 88 = 968$
Solve: $g = 968 ÷ 8 = 121$ girls
Total children: 88 boys + 121 girls = 209 children

Rate

A rate compares two quantities with different units. This is particularly useful in real-world applications where you need to compare different types of measurements.

Common examples include:

Cost rates (R16.95 per kg)
Speed rates (60 km/h)
Exchange rates

Unit rates

A unit rate shows the amount per one unit. To find a unit rate, divide one quantity by the other.

Example: If 2 kg of flour costs R20, the unit rate is: R20 ÷ 2 kg = R10 per kg

Distance, speed and time relationships

The relationship between distance, speed, and time can be remembered using triangles:

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Key Formulas:

Distance = Speed × Time
Speed = Distance ÷ Time
Time = Distance ÷ Speed

These formulas help solve problems involving motion and travel.

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Key Points to Remember:

Ratios compare quantities and can be written as a
, a to b, or a/b
Always maintain the same order when writing ratios - 3:2 is not the same as 2:3
Equivalent ratios are created by multiplying or dividing both terms by the same number
Use cross-multiplication to solve proportion problems when one value is unknown
Unit ratios and unit rates make comparisons easier by showing amounts per single unit
The Distance-Speed-Time triangle helps remember motion formulas

Ratio, Proportion, and Rate (Grade 12 NSC Matric Mathematical Literacy): Revision Notes