Simplifying ratios
🎯 In this topic you will
- Simplify and compare ratios
🧠 Key Words
- adapt
- common factor
- highest common factor
- proportion
- ratio
- simplest form
- simplify
Show Definitions
- adapt: To change or adjust a method to suit a different situation or problem.
- common factor: A number that divides exactly into two or more numbers.
- highest common factor: The largest number that divides exactly into two or more numbers.
- proportion: A statement that two ratios are equal, or a part compared to the whole.
- ratio: A way of comparing two quantities by division.
- simplest form: A fraction that has been fully reduced so numerator and denominator share no common factors other than 1.
- simplify: To reduce a fraction, ratio, or expression to its simplest equivalent form.
In this necklace there are two red beads and six yellow beads.
You can write the ratio of red beads to yellow beads as $2 : 6$. You say “The ratio of red beads to yellow beads is two to six.”
For every one red bead there are three yellow beads, so you can simplify the ratio $2 : 6$ to $1 : 3$.
You write a ratio in its simplest form by dividing the numbers in the ratio by the highest common factor. Here, the highest common factor of 2 and 6 is 2, so you divide both the numbers by 2.
red : yellow
$2 : 6$
$\div 2 \ \ \ \ \ \div 2$
$1 : 3$
A ratio written in its simplest form must not contain any decimals or fractions.
📘 Exercise
1. For each of these necklaces, write down the ratio of green beads to blue beads, in its simplest form.
👀 Show answer
a) 1 : 2 (given)
b) 1 : 6
c) 3 : 2
d) 1 : 3
2. Write the following ratios in their simplest form:
- a) 6 : 12
- b) 3 : 18
- c) 20 : 4
- d) 24 : 6
👀 Show answer
a) 1 : 2
b) 1 : 6
c) 5 : 1
d) 4 : 1
3. Simplify each ratio:
- a) 2 : 8
- b) 2 : 12
- c) 3 : 6
- d) 3 hours : 15 hours
- e) 4 kg : 8 kg
- f) 4 cm : 12 cm
- g) 25 : 5
- h) 60 : 5
- i) 36 : 6
- j) 14 days : 7 days
- k) 24 g : 8 g
- l) 54 mL : 9 mL
👀 Show answer
a) 1 : 4
b) 1 : 6
c) 1 : 2
d) 1 : 5
e) 1 : 2
f) 1 : 3
g) 5 : 1
h) 12 : 1
i) 6 : 1
j) 2 : 1
k) 3 : 1
l) 6 : 1
4. Complete the sequence 32, 28, □, 20, 16, □, 8, 4. Find the ratio of the first missing number to the second missing number.
👀 Show answer
Sequence decreases by 4. Missing numbers are 24 and 12. Ratio = 24 : 12 = 2 : 1.
5. The Eiffel Tower is 324 m tall. A model is 3 m tall. Find the ratio of real height to model height.
👀 Show answer
324 : 3 = 108 : 1
6. Simplify these ratios:
- a) 8 : 12
- b) 4 : 18
- c) 32 : 12
👀 Show answer
a) 2 : 3
b) 2 : 9
c) 8 : 3
7. Simplify these ratios:
- a) 6 weeks : 21 weeks
- b) 8 mL : 10 mL
- c) 8 tonnes : 14 tonnes
- d) 24°C : 10°C
- e) 28 km : 12 km
- f) 21 litres : 9 litres
👀 Show answer
a) 2 : 7
b) 4 : 5
c) 4 : 7
d) 12 : 5
e) 7 : 3
f) 7 : 3
8. Helena says “The ratio of margarine to flour is 4 : 1.” (200 g flour, 50 g margarine, 50 g lard)
- a) Explain Helena’s mistake.
- b) Write the correct ratio statement.
👀 Show answer
a) She reversed the ratio. Margarine : flour = 50 : 200 = 1 : 4, not 4 : 1.
b) Correct: flour : margarine = 200 : 50 = 4 : 1.
9. A recipe uses 45 mL vinegar and 120 mL oil. Write the ratio of vinegar to oil in simplest form.
👀 Show answer
45 : 120 → ÷15 = 3 : 8
🧠 Think like a Mathematician
Task: Compare two mixtures of paint and decide which produces the darker blue shade. Then reflect on the method you used and whether it works in general.
Scenario:
- Bryn mixes tins of white and blue paint in the ratio $1:2$.
- Alun mixes tins of white and blue paint in the ratio $2:3$.
Questions:
👀 show answer
- a) - Bryn’s ratio 1:2 means 1 white and 2 blue, total = 3 parts. Fraction of blue = $\tfrac{2}{3} \approx 66.7\%$. - Alun’s ratio 2:3 means 2 white and 3 blue, total = 5 parts. Fraction of blue = $\tfrac{3}{5} = 60\%$. ✔ Bryn’s paint is darker (higher proportion of blue).
- b) Possible methods: - Calculate the fraction of blue in each mixture. - Compare ratios directly by making common totals (e.g., 15 parts: Bryn = 10 blue, Alun = 9 blue). Preferred method: fractions give the clearest comparison.
- c) Yes, the fraction method works for any mixing ratio problem because it reduces the ratio to a proportion of the total.
❓ Exercise
11. Melania makes a drink by mixing orange juice with water in the ratio 2 : 5. Boris makes a drink by mixing orange juice with water in the ratio 1 : 3. Who has the drink with the higher proportion of orange juice? Explain how you worked out your answer.
🔎 Reasoning Tip
The drink with the higher proportion of orange juice is the drink that has more orange juice.
👀 Show answer
Step 1 – Melania: Ratio = 2 parts orange : 5 parts water → total parts = 2 + 5 = 7. Proportion of orange juice = 2/7 ≈ 0.286 (28.6%).
Step 2 – Boris: Ratio = 1 part orange : 3 parts water → total parts = 1 + 3 = 4. Proportion of orange juice = 1/4 = 0.25 (25%).
Step 3 – Compare: 28.6% (Melania) > 25% (Boris).
✅ Final Answer: Melania’s drink has the higher proportion of orange juice.
12a. At the Borrowdale golf club there are 15 women and 60 men. Write the ratio of women : men in its simplest form.
12b. At the Avondale golf club there are 12 women and 56 men. Write the ratio of women : men in its simplest form.
12c. Which golf club has the greatest proportion of men members? Explain.
👀 Show answer
13. Use the information from Q12. In the following year:
- Borrowdale: +7 women and +6 men
- Avondale: +12 women and +10 men
Which club now has the greatest proportion of men? Explain.
👀 Show answer
14. Is the ratio 20x : 15x in its simplest form?
Zara says: "I think it is possible to simplify this ratio."
Sofia says: "I’m not sure. What would your answer be?"
14a. Explain whether Zara is correct or incorrect.
14b. Discuss your answer with a partner. Do you agree with each other?
👀 Show answer
20x : 15x → cancel x → 20 : 15 → ÷5 → 4 : 3.
✅ Zara is correct. The ratio simplifies to 4 : 3.
🔗 Learning Bridge
You’ve just learned how to simplify and compare ratios by dividing both terms by their highest common factor and, if needed, converting ratios into fractions for comparison. Next, you’ll extend these skills to ratios with different units or decimals.
- Same units first: if a ratio has mixed units (kg vs g, minutes vs hours), convert into the same unit before simplifying.
- Clear decimals: multiply both terms by powers of 10 to turn decimals into whole numbers.
- HCF rule still applies: once in whole-number form, divide through by the highest common factor.
- Check by scaling back: the simplified ratio must stay equivalent to the original measurement.
Quick examples
- Flour : Butter =
0.5 kg : 250 g→ convert →500 g : 250 g→ divide by 250 →2 : 1 - Time ratio =
36 s : 1 min→ convert →36 s : 60 s→ divide by 12 →3 : 5
Key idea: when units or decimals appear, first make the numbers “look alike,” then simplify as usual.
A ratio is a way of comparing two or more quantities.
In this pastry recipe, the ratio of flour to butter is $0.5\ \text{kg} : 250\ \text{g}$.
Pastry recipe
0.5 kg flour
250 g butter
water to mix
Before you simplify a ratio, you must write all quantities in the same units.
$0.5\ \text{kg} : 250\ \text{g}$ is the same as $500\ \text{g} : 250\ \text{g}$, which you write as $500 : 250$.
You can now simplify this ratio by dividing both numbers by the highest common factor. In this case the highest common factor is 250.
Divide both numbers by 250 to simplify the ratio to $2 : 1$.
$500 : 250 \ \div 250 = 2 : 1$
If you cannot work out the highest common factor of the numbers in a ratio, you can simplify the ratio in stages. Divide the numbers in the ratio by common factors until you cannot divide any more.
In the example above you could start by:
- dividing by 10
- then dividing by 5
- then dividing by 5 again
- giving you the same answer of 2 : 1.
$500 : 250 \ \div 10 = 50 : 25$
$50 : 25 \ \div 5 = 10 : 5$
$10 : 5 \ \div 5 = 2 : 1$
🔎 Reasoning Tip
When the units are the same, you do not need to write the units with the numbers.
📘 Exercise
1. Simplify these ratios:
- a) 2 : 10
- b) 3 : 18
- c) 5 : 25
- d) 30 : 5
- e) 36 : 12
- f) 180 : 20
- g) 4 : 6
- h) 9 : 15
- i) 10 : 35
- j) 75 : 10
- k) 72 : 20
- l) 140 : 112
👀 Show answer
a) 1 : 5
b) 1 : 6
c) 1 : 5
d) 6 : 1
e) 3 : 1
f) 9 : 1
g) 2 : 3
h) 3 : 5
i) 2 : 7
j) 15 : 2
k) 18 : 5
l) 5 : 4
2. Simplify these ratios:
- a) 5 : 10 : 15
- b) 8 : 10 : 12
- c) 20 : 15 : 25
- d) 18 : 15 : 3
- e) 27 : 9 : 45
- f) 72 : 16 : 32
👀 Show answer
a) 1 : 2 : 3
b) 4 : 5 : 6
c) 4 : 3 : 5
d) 6 : 5 : 1
e) 3 : 1 : 5
f) 9 : 2 : 4
3. Ben’s classwork: He simplified 6 : 12 : 3 incorrectly. a) Explain the mistake he made. b) Work out the correct answer.
👀 Show answer
Ben’s mistake: He only divided some terms by 6, not all of them consistently. Correct working: 6 : 12 : 3 → ÷3 = 2 : 4 : 1. ✅ Final Answer: 2 : 4 : 1
🧠 Think like a Mathematician
Task: Compare two methods for simplifying the ratio $4\ \text{mm} : 6\ \text{cm}$. Decide which is more efficient and why.
Discussion Statements:
Questions:
👀 show answer
- a) Sofia’s method is better because it keeps both numbers as whole numbers in millimetres: $4\ \text{mm} : 60\ \text{mm} = 4:60 = 1:15$. Arun’s method works too ($0.4\ \text{cm} : 6\ \text{cm} = 0.4:6 = 1:15$) but involves decimals.
- b) Most people prefer Sofia’s method, as working with whole numbers is simpler and reduces mistakes. However, both methods give the same simplified ratio of $1:15$.
📘 Exercise
5. Simplify these ratios:
- a) 500 m : 1 km
- b) 36 seconds : 1 minute
- c) 800 ml : 2.4 l
- d) 1.6 kg : 800 g
- e) 3 cm : 6 mm
- f) 2 days : 18 hours
- g) 2 hours : 48 minutes
- h) 8 months : 1 year
🔎 Reasoning Tip
Remember that both quantities must be in the same units before you simplify.
👀 Show answer
a) 500 m : 1000 m = 1 : 2
b) 36 : 60 = 3 : 5
c) 800 : 2400 = 1 : 3
d) 1600 g : 800 g = 2 : 1
e) 30 mm : 6 mm = 5 : 1
f) 48 h : 18 h = 8 : 3
g) 120 min : 48 min = 5 : 2
h) 8 : 12 = 2 : 3
6. Zara says: “The ratio of oranges to sugar is 2 : 1.” The recipe is 750 g oranges and 1.5 kg sugar.
Is Zara correct? Explain your answer.
👀 Show answer
Oranges = 750 g, Sugar = 1500 g. Ratio oranges : sugar = 750 : 1500 = 1 : 2. ✅ Zara is incorrect. The correct ratio is 1 : 2, not 2 : 1.
7. Simplify these ratios:
- a) 600 m : 1 km : 20 m
- b) 75 cm : 1 m : 1.5 m
- c) 300 ml : 2.1 l : 900 ml
- d) 3.2 kg : 1600 g : 0.8 kg
- e) $1.08 : 90 cents : $9
- f) 4 cm : 8 mm : 0.2 m
👀 Show answers Q7
a) 600 : 1000 : 20 → 600 : 1000 : 20 = 30 : 50 : 1
b) 75 : 100 : 150 = 3 : 4 : 6
c) 300 : 2100 : 900 = 1 : 7 : 3
d) 3.2 kg : 1.6 kg : 0.8 kg = 3.2 : 1.6 : 0.8 = 4 : 2 : 1
e) 1.08 : 0.90 : 9 = 108 : 90 : 900 = 6 : 5 : 50
f) 4 cm : 0.8 cm : 20 cm = 4 : 0.8 : 20 = 5 : 1 : 25
8. Marcus and Sofia are mixing paint. They mix 250 ml of white paint with 750 ml of red paint and 1.2 litres of yellow paint. Marcus says the ratio of white : red : yellow = 1 : 3 : 5. Sofia says the ratio is 25 : 75 : 12. Is either of them correct? Explain your answer.
👀 Show answer
White = 250 ml, Red = 750 ml, Yellow = 1200 ml. Ratio = 250 : 750 : 1200 → ÷250 → 1 : 3 : 4.8. Marcus is not correct because he simplified wrongly to 1 : 3 : 5. Sofia is not correct either because 25 : 75 : 12 doesn’t match. ✅ The correct simplified ratio is 1 : 3 : 4.8 (or 5 : 15 : 24 in whole numbers).
9. Preety’s question: Five cups hold 1.2 litres and three mugs hold 900 ml. Which holds more liquid, one cup or one mug?
Preety shows: 5 cups = 1200 ml → 1 cup = 240 ml 3 mugs = 900 ml → 1 mug = 300 ml So a mug holds 60 ml more than a cup.
Use Preety’s method to answer:
- a) Four bags of sugar weigh 1.3 kg, three bags weigh 960 g. Which has greater mass, one bag of sugar or one bag of flour?
- b) Eight pens have a total length of 1.2 m, five pencils have a total length of 90 cm. Which is longer, a pen or a pencil?
👀 Show answer
Preety’s original: Mug (300 ml) > Cup (240 ml).
9a. Sugar: 4 bags = 1300 g → 1 bag = 325 g. Flour: 3 bags = 960 g → 1 bag = 320 g. ✅ One bag of sugar is heavier (325 g vs 320 g).
9b. Pens: 8 = 120 cm → 1 pen = 15 cm. Pencils: 5 = 90 cm → 1 pencil = 18 cm. ✅ One pencil is longer (18 cm vs 15 cm).
🧠 Think like a Mathematician
Task: Investigate Jed’s method for simplifying ratios by multiplying through by powers of 10, and consider its advantages and limitations.
Scenario: This is part of Jed’s homework. To simplify ratios like $1.5:2$ or $0.8:3.6$, Jed’s first step is to multiply both numbers by 10.
Questions:
👀 show answer
- a) Multiplying both numbers by 10 removes the decimals, making the ratio easier to simplify. Example: $1.5:2 \;\; \to\;\; 15:20 \;\; \to\;\; 3:4$.
- b) Advantages: Removes decimals, keeps the ratio in whole numbers, simplifies calculation. Disadvantages: Might involve large numbers if the decimals are very small (e.g. $0.003:0.015$ would become 3:15 after multiplying by 1000).
- c)$0.03:0.15$. Multiply both by 100 → $3:15$. Simplify by dividing by 3 → $1:5$.
- d) Jed’s method is reliable: multiply through by a power of 10 to remove decimals, then simplify the ratio. It always works, but choosing the smallest possible power of 10 keeps the numbers manageable.
📘 Exercise
11. Use Jed’s method (clear decimals first) to simplify these ratios:
- a) 0.5 : 2
- b) 1.5 : 3
- c) 1.2 : 2.4
- d) 3.6 : 0.6
- e) 7.5 : 1.5
- f) 2.4 : 4
- g) 1.8 : 6.3
- h) 2.1 : 0.7 : 1.4
👀 Show answer
a) ×10 → 5:20 → 1:4
b) ×10 → 15:30 → 1:2
c) ×10 → 12:24 → 1:2
d) ×10 → 36:6 → ÷6 → 6:1
e) ×10 → 75:15 → ÷15 → 5:1
f) ×10 → 24:40 → ÷8 → 3:5
g) ×10 → 18:63 → ÷9 → 2:7
h) ×10 → 21:7:14 → ÷7 → 3:1:2
12. Oditi runs three times a week. Times recorded:
- Monday: 1 hour 40 minutes
- Wednesday: 50 minutes
- Friday: 2½ hours
12a. Oditi says the ratio Monday : Wednesday : Friday is 1 : 2 : 3. Without doing calculations, explain why she’s wrong.
12b. Oditi’s mum converts to hours as 1.4 : 0.5 : 2.5, then to 14 : 5 : 25 and finally 14 : 1 : 5. Explain the mistakes.
12c. Work out the correct ratio of Oditi’s times. Show your working.
👀 Show answer
12a. In the order Monday, Wednesday, Friday, Wednesday is the shortest run, not the middle value of 1:2:3. Also 1:2:3 would make Monday the smallest (same as Wednesday), which it isn’t.
12b. She treated 1 hour 40 minutes as 1.4 hours. That is incorrect: 0.4 of an hour is 24 minutes, not 40. You must convert all times to the same unit (e.g., minutes) before forming the ratio.
12c – Correct working:
Monday = 1 h 40 min = 100 min; Wednesday = 50 min; Friday = 2.5 h = 150 min.
Ratio = 100 : 50 : 150 → ÷50 → 2 : 1 : 3.
⚠️ Be careful! Ratios
- Keep the order exactly as stated. “green : blue” ≠ “blue : green”.
- Match units before simplifying. Convert first (e.g., 2 m : 50 cm → 200 cm : 50 cm → 4 : 1).
- Clear decimals/fractions before reducing. Multiply both parts by a power of 10 or a common denominator (e.g., 1.5 : 2 → 15 : 20 → 3 : 4).
- Divide by the HCF—both (all) terms. One term only is invalid: 8 : 12 ÷4 → 2 : 3 (not 2 : 12!).
- Ratios in simplest form are whole numbers only. No units, no decimals, no fractions.
- Comparing mixes? Use proportions of the whole. For \(a:b\), the second part’s share is \( \frac{b}{a+b} \).
- Drop units once matched. Write 500 g : 250 g as 500 : 250, then simplify.
- Scaling up/down keeps the ratio equivalent. \(2:3=4:6=10:15\).