Secondary 1 · Mathematics · Number
Ratios and rates
Compare quantities using ratios, simplify ratios and solve everyday problems involving sharing in a given ratio and simple rates such as cost per kilogram.
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These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
Learning goals
Learners will
- Write and simplify ratios comparing two quantities.
- Share an amount in a given ratio.
- Calculate and compare simple rates such as cost per kilogram and kilometres per hour.
Success looks like
- I can simplify a ratio such as 12 : 18 to 2 : 3.
- I can share $45 in the ratio 2 : 3.
- I can work out a unit rate and use it to decide which deal is better value.
The big idea
A ratio compares quantities of the same kind. If a fruit bowl has 4 apples and 6 bananas, the ratio of apples to bananas is 4 : 6, read “4 to 6”. Order matters: the ratio of bananas to apples is 6 : 4.
Ratios are simplified like fractions: divide both parts by their highest common factor. 4 : 6 becomes 2 : 3 (dividing by 2). This tells us that for every 2 apples there are 3 bananas. Ratios such as 2 : 3, 4 : 6 and 10 : 15 are equivalent.
Before simplifying, both quantities must be in the same units. To compare 50 cm with 2 m, change 2 m to 200 cm first: 50 : 200 = 1 : 4.
To share an amount in a ratio, think in parts:
- Add the parts of the ratio to find the total number of parts.
- Divide the amount by the total number of parts to find the value of one part.
- Multiply to find each share, then check the shares add to the original amount.
A rate compares two different kinds of quantity, such as distance and time (km/h) or cost and mass ($/kg). A unit rate tells you the amount for one unit, which makes comparisons easy. If 3 kg of potatoes cost $7.50, the unit rate is $7.50 ÷ 3 = $2.50 per kilogram.
Shoppers use unit rates to find the best buy: work out the price per kilogram (or per 100 g, or per litre) for each option, then choose the lowest.
Worked examples
Simplifying a ratio
Simplify the ratio 12 : 18.
- Find the highest common factor of 12 and 18. It is 6.
- Divide both parts by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3.
Answer: 12 : 18 = 2 : 3
Sharing in a ratio
Sam and Priya share $45 in the ratio 2 : 3. How much does each person get?
- Total parts: 2 + 3 = 5.
- One part: $45 ÷ 5 = $9.
- Sam gets 2 parts: 2 × $9 = $18. Priya gets 3 parts: 3 × $9 = $27.
- Check: $18 + $27 = $45.
Answer: Sam gets $18 and Priya gets $27.
Comparing unit rates
A 3 kg bag of potatoes costs $7.50 and a 5 kg bag costs $11.50. Which is better value?
- Small bag: $7.50 ÷ 3 = $2.50 per kg.
- Large bag: $11.50 ÷ 5 = $2.30 per kg.
- $2.30 is less than $2.50, so the large bag costs less per kilogram.
Answer: The 5 kg bag is better value at $2.30 per kg.
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Simplify the ratio 15 : 25.
3 : 5 — The highest common factor is 5. 15 ÷ 5 = 3 and 25 ÷ 5 = 5, so 15 : 25 = 3 : 5.
A class has 12 girls and 8 boys. What is the ratio of boys to girls in simplest form?
2 : 3 — Boys come first: 8 : 12. Dividing both by 4 gives 2 : 3.
Which ratio is equivalent to 4 : 6?
2 : 3 — Dividing both parts of 4 : 6 by 2 gives 2 : 3.
A car travels 240 km in 3 hours. What is its average speed?
80 km/h — Speed is distance ÷ time: 240 ÷ 3 = 80 km/h.
$60 is shared in the ratio 1 : 3. How much is the larger share?
$45 — There are 1 + 3 = 4 parts. One part is $60 ÷ 4 = $15, so the larger share is 3 × $15 = $45.
Simplify the ratio 18 : 24.
3 : 4 — The highest common factor of 18 and 24 is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4.
4 kg of apples cost $14. What is the cost per kilogram?
$3.50 per kg — $14 ÷ 4 = $3.50 per kilogram.
A recipe uses flour and sugar in the ratio 5 : 2. How many grams of sugar are needed for 300 g of flour?
120 g — 5 parts = 300 g, so 1 part = 60 g. Sugar is 2 parts: 2 × 60 = 120 g.
Watch out for
- Writing the ratio in the wrong order. Underline the words in the question (“boys to girls”) and write the numbers in that same order.
- Dividing the amount by one part of the ratio instead of the total. For $45 in the ratio 2 : 3, the learner may calculate $45 ÷ 2. Remind them to add the parts first.
- Simplifying a ratio with mixed units, such as 50 cm : 2 m = 25 : 1. Always convert to the same units first (50 : 200 = 1 : 4).
Tips for parents
- Use real recipes: if a pancake recipe needs 2 cups of flour to 1 cup of milk, ask how much milk is needed for 5 cups of flour.
- At the shops, have the learner compare the unit prices on shelf labels and check one by calculating it themselves.
- Use coloured counters to model sharing in a ratio: lay out groups of “2 red, 3 blue” until all the counters are used.
Go further
- Mix a drink or paint in different ratios (such as 1 : 4 and 2 : 5 of cordial to water) and predict which will be stronger before tasting or looking.
- Keep a log of a walk or bike ride: record the distance and time, and calculate the average speed in km/h.
- Investigate a map or model scale such as 1 : 100 and work out the real lengths of objects in a plan.
About 45 minutes
You'll need
- Paper and pencil
- Two colours of counters or blocks
- A calculator
- Grocery receipts or supermarket shelf labels (optional)
Key words
- Ratio
- A comparison of two or more quantities of the same kind, written with a colon, such as 2 : 3.
- Simplest form
- A ratio whose numbers have no common factor other than 1, such as 2 : 3 rather than 12 : 18.
- Equivalent ratios
- Ratios that compare quantities in the same way, such as 1 : 2, 2 : 4 and 5 : 10.
- Rate
- A comparison of two quantities with different units, such as kilometres per hour or dollars per kilogram.
- Unit rate
- A rate written for one unit of the second quantity, such as $2.50 per 1 kilogram.
Still tricky?
For parents
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