Ratios, Proportions and Rates
Build the four core methods—simplify, share, unitary, and rates—with clear Lower Secondary examples and practice.
🎯 Key idea: Ratios, proportions, and rates are all one skill — multiplying or dividing every part by the same number. Master that and the whole topic opens up.
You already use ratios every day: recipes, maps, sale prices, and travel times all depend on them.
🔢 1. Simplifying and Equivalent Ratios
A ratio compares quantities part to part. When a cafe mixes coffee and milk in the ratio $3:1$, it means:
for every $3$ parts of coffee, there is $1$ part of milk.
The size of a part can change, but the relationship stays fixed. That is why ratios survive scaling.
Worked example: Simplify $12:18$.
Step 1
Find the highest common factor of $12$ and $18$.
The HCF is $6$.
Step 2
Divide both parts by $6$.
$12 [[div]] 6 = 3$
$18 [[div]] 6 = 4$
Answer: $12:18 = 3:4$.
Check yourself: A ratio is fully simplified when no whole number bigger than $1$ divides every part evenly.
Equivalent ratios work the other direction — multiply all parts by the same number and the ratio is unchanged:
$$3:5 = 6:10 = 9:15 = 30:50$$
To find a missing term, first ask: what was one part multiplied by?
Worked example: Find the missing value in $3:5 = ?:20$.
Step 1
Ask what the second part was multiplied by.
$20 [[div]] 5 = 4$
Step 2
Multiply the first part by the same number.
$3 [[times]] 4 = 12$
Answer: $3:5 = 12:20$.
⚖️ 2. Sharing in a Ratio — the Parts Method
When sharing an amount in a ratio, the golden rule is: add the parts first, then divide. Never just divide by $2$ or $3$.
Worked example: Share USD 60 in the ratio $2:3$.
Step 1
Add the parts to find the total.
$2 + 3 = 5$
Step 2
Find the value of one part.
$60 [[div]] 5 = 12$
Step 3
Multiply each share by one part.
$2 [[times]] 12 = 24$
$3 [[times]] 12 = 36$
Step 4
Check the shares add up to the original total.
$24 + 36 = 60$ ✓
Answer: USD 24 and USD 36.
The parts method:
For any amount $T$ shared in ratio $p:q$:
$$[[text{one part}]] = A [[div]] (p + q)]]$$
Then multiply by each ratio number to get each share.
Use the tool: Try the Ratio Sharer below — enter any total and ratio and watch the parts method run step by step.
⚡ 3. Direct Proportion and Rates
When two quantities grow together (more notebooks means more cost), the fastest safe route goes through one unit — the unitary method.
Worked example: $8$ notebooks cost USD 40. What do $5$ notebooks cost?
Step 1
Find the cost of one notebook.
$40 [[div]] 8 = 5$
Step 2
Multiply by the target quantity.
$5 [[times]] 5 = 25$
Answer: USD 25.
A rate compares two different units: kilometres per hour, litres per minute, or dollars per kilogram.
Worked example (speed): A train covers $180$ km in $2.5$ hours. What is its speed?
Divide distance by time.
$180 [[div]] 2.5 = 72$
Answer: $72$ km/h.
Worked example (best buy): Pack A is $500$ ml for USD 7. Pack B is $750$ ml for USD 12. Which is better value?
Compare the price per $250$ ml for each pack.
Pack A: $500$ ml is $2$ lots of $250$ ml, so USD $7 [[div]] 2 = 3.50$ per $250$ ml.
Pack B: $750$ ml is $3$ lots of $250$ ml, so USD $12 [[div]] 3 = 4.00$ per $250$ ml.
Answer: Pack A is better value — USD 3.50 vs USD 4.00 per $250$ ml.
Unit discipline: Write the units at every line. Most rate errors are really unit errors — minutes mixed with hours, grams with kilograms. Convert first, calculate second.
Stretch — exchange rates: If GBP 1 buys USD 1.30, how many dollars do you get for GBP 50?
Multiply by the rate.
$50 [[times]] 1.30 = 65$
Answer: USD 65.
Multiply when converting into the more expensive currency; divide when going back. If your answer gets smaller when it should grow, you picked the wrong direction.
The five classic mistakes to avoid:
$1$. Dividing the total by $2$ or $3$ instead of by the total parts.
$2$. Adding to both parts and expecting the ratio to stay the same ($3:5$ is NOT $5:7$).
$3$. Comparing pack prices instead of unit prices in best-buy questions.
$4$. Mixing units — minutes vs hours, grams vs kilograms.
$5$. Rounding too early — keep full precision until the final line.
Practice Problems
Try these problems. Work through each one on paper first, then open the answer to check your method.
1. Simplify the ratio $24:36$.
Show answer
Step 1
Find the HCF of $24$ and $36$.
The HCF is $12$.
Step 2
Divide both parts by $12$.
$24 [[div]] 12 = 2$
$36 [[div]] 12 = 3$
Answer: $2:3$
2. Find the missing number: $4:7 = ?:21$.
Show answer
Step 1
Find the scale factor.
$21 [[div]] 7 = 3$
Step 2
Multiply the first part by $3$.
$4 [[times]] 3 = 12$
Answer: $12:21$
3. Share GBP 80 in the ratio $3:5$.
Show answer
Step 1
Add the parts.
$3 + 5 = 8$
Step 2
Find one part.
$80 [[div]] 8 = 10$
Step 3
Multiply each share.
$3 [[times]] 10 = 30$
$5 [[times]] 10 = 50$
Answer: GBP 30 and GBP 50
4. A car travels $252$ km in $3.5$ hours. What is its average speed?
Show answer
Divide distance by time.
$252 [[div]] 3.5 = 72$
Answer: $72$ km/h
5. If $6$ pens cost USD 18, what do $11$ pens cost?
Show answer
Step 1
Find the cost of one pen.
$18 [[div]] 6 = 3$
Step 2
Multiply by $11$.
$3 [[times]] 11 = 33$
Answer: USD 33
Interactive Quiz
Choose the best answer for each question, then submit.