Ratio, Rate, and Proportion (Grade 10 NSC Matric Mathematical Literacy): Revision Notes

Worked Example: Writing ratios in simplest form

Question: Write these ratios in their simplest forms:

1. 5 : 30
2. 14 : 18
3. 18 : 30
4. 7 : 280

Solutions:

1. 5 and 30 are both divisible by 5, so the ratio simplifies to 1 : 6

2. 14 and 18 are both divisible by 2, so the ratio simplifies to 7 : 9

3. 18 and 30 are both divisible by 6, so the ratio simplifies to 3 : 5

4. 7 and 280 are both divisible by 7, so the ratio simplifies to 1 : 40

Worked Example: Writing ratios in unit form

Question:

1. There are 23 nurses in a hospital and 7567 patients. How many patients does each nurse care for?
2. In a Grade 10 class, learners vote for a class badge. 4 learners vote for badge A and 17 vote for badge B. How many learners vote for badge B for each learner voting for badge A?

Solutions:

1. 23 : 7567

Divide both numbers by 23:

$23 \div 23 : 7567 \div 23 = 1 : 329$

So each nurse cares for 329 patients on average.

2. 4 : 17

Divide both numbers by 4:

$4 \div 4 : 17 \div 4 = 1 : 4.25$

So there are 4.25 votes for Badge B for every one vote for Badge A.

Worked Example: Calculating rates

Question:

1. Elias, a star athlete, runs 100 m in 15 seconds. a) What is his speed in metres per second? b) If he could keep running at this speed, how long would it take to cover 1 km?
2. Cheese costs R 56 per kg. Thandi buys 200 g of cheese. How much does she pay?

Solutions:

a) $100 \text{ m} \div 15 \text{ sec} = 6.67 \text{ m/sec}$

b) 1 km = 1000 m $1000 \text{ m} \div 6.67 \text{ m/sec} = 100 \text{ seconds or } 1 \text{ min } 40 \text{ sec}$

There are 1000 g in 1 kg. We can either work out that there are 5 × 200 g in 1000 g and divide the cost by 5, or calculate the cost per 100 g first.

Cost per 100 g = $\text{R 56} \div 10 = \text{R 5.60}$

200 g costs $\text{R 5.60} \times 2 = \text{R 11.20}$

Worked Example: Finding missing numbers in a ratio

Question: Thenji makes a fruit salad for breakfast at a restaurant. She uses pieces of fruit in the following ratio:

banana : apple : paw-paw 1 : 2 : 3

1. If she uses 20 pieces of apple, how many pieces of banana and paw-paw should she use?

2. If she uses 12 pieces of banana, how many pieces of apple and paw-paw should she use?

Solutions:

1. Start with the banana : apple ratio: $1 : 2 = \text{banana pieces} : \text{apple pieces}$ $1 : 2 = 10 : 20$ So we have 10 banana pieces and 20 apple pieces.

   Next, look at the apple : paw-paw ratio: $2 : 3 = \text{apple pieces} : \text{paw-paw pieces}$ $2 : 3 = 20 : 30$ So we have 20 apple pieces and 30 paw-paw pieces.

   Complete ratio: 10 : 20 : 30

2. Start with banana : apple ratio: $1 : 2 = \text{banana pieces} : \text{apple pieces}$ $1 : 2 = 12 : 24$ So we have 12 banana pieces and 24 apple pieces.

   For apple : paw-paw ratio: $2 : 3 = \text{apple} : \text{paw-paw}$ $2 : 3 = 24 \text{ apple pieces} : ? \text{ paw-paw pieces}$

   Using cross-multiplication: $3 \times 24 \div 2 = 36 \text{ pieces}$

   Complete ratio: 12 : 24 : 36

Worked Example: Working with inverse proportion

Question: Learners at a school want to hire a hall for a party. They can hire the hall for one evening for R 3000. The learners who attend the party need to split the cost between them.

1. Draw up a table to show the cost per learner if 30, 50, 100, 200 and 300 learners attend the party.

2. The learners decide they can't hold the party if they need to pay more than R 25 each. What number of learners must go to the party for it to be affordable?

Solutions:

2. $\text{R 3000} \div \text{R 25} = 120 \text{ learners}$ So there must be at least 120 learners going to the party.

This is an example of inverse proportion because as the number of learners increases, the cost per learner decreases.