Ratios and percentages often appear together in multi-step questions. Start by identifying each share and the quantity that counts as the whole before calculating.
A multi-step ratio or percentage question usually combines ideas you already know. The difficulty is deciding what the whole quantity is at each stage. A percentage may refer to one person’s share rather than the original total. A second discount may apply to a reduced price rather than the first price.
This is a prerequisite review of ratios and percentages for students now in Grade 9. The ratio and percentage sections revisit skills you may have studied previously, then combine them in original multi-step questions with worked reasoning and answers. Use the page to strengthen foundations alongside your current lessons. It does not describe the complete current Grade 9 curriculum or replace work on compound proportions, rates and other topics in your course. All prices and classroom situations are invented for the exercises.
Ratios and percentages: label each whole quantity
Write what each ratio part represents and what quantity counts as 100%. Those two labels prevent many mistakes. In a red-to-blue ratio of 2:5, the total is seven equal parts. Red is 2/7 of the total, while red compared with blue is 2/5. The fraction depends on the comparison requested.
For percentages, “15% of the original price” and “15% of the discounted price” have different bases. The general relationship is part/base = percentage/100, as explained in OpenStax’s proportion and percentage guide. Before calculating, replace “base” with the actual whole used in your question.
Warm-up: make the units agree
Simplify the ratio 750 mL : 1.5 L. Convert 1.5 L to 1,500 mL, giving 750:1,500. Divide both terms by 750 to obtain 1:2.
The first quantity is half the second, but it is one third of their combined total. The total is three ratio parts, not two. This small distinction becomes important in sharing and mixture questions. A ratio compares quantities using division, so incompatible units must be dealt with before simplifying.
Ratios and percentages: share money, then calculate a percentage
Three school clubs share KSh 3,600 in the ratio 2:3:4. The second club sets aside 25% of its share for stationery. Find each club’s share and the amount the second club has left.
- Add the ratio parts: 2 + 3 + 4 = 9.
- Find one part: 3,600 ÷ 9 = KSh 400.
- Calculate the shares: 2 × 400 = KSh 800; 3 × 400 = KSh 1,200; 4 × 400 = KSh 1,600.
- Use the second club’s share as the percentage base: 25/100 × 1,200 = KSh 300.
- Subtract its allocation: 1,200 − 300 = KSh 900 remaining.
Check the original shares: 800 + 1,200 + 1,600 = 3,600. Do not take 25% of KSh 3,600 for the stationery calculation because the question says 25% of the second club’s share. Underlining that phrase helps identify the correct base.
Worked problem 2: two discounts in succession
A bag has an example marked price of KSh 2,500. A first discount removes 20%. A second discount removes 10% of the reduced price. Find the final price and the overall percentage discount.
After the first discount, 80% remains. The price becomes 0.80 × 2,500 = KSh 2,000. After the second discount, 90% of that amount remains: 0.90 × 2,000 = KSh 1,800.
The total reduction is 2,500 − 1,800 = KSh 700. Relative to the original marked price, the discount is 700/2,500 × 100 = 28%. Adding 20% and 10% would give 30%, but the discounts use different bases.
A compact check is 0.80 × 0.90 = 0.72. The customer pays 72% of the original price, leaving a 28% reduction. Multipliers are especially useful when a question contains several successive changes, because each multiplication carries the updated base forward.
Worked problem 3: recover the original amount
After a 15% discount, an exercise item costs KSh 2,040. What was its original price?
The sale price represents 85% of the original amount. Let the original price be p. Then 0.85p = 2,040, so p = 2,040 ÷ 0.85 = KSh 2,400.
Check forward: 15% of 2,400 is 360, and 2,400 − 360 = 2,040. Adding 15% of 2,040 does not reverse the discount, because that uses the reduced amount as the base. The correct reverse step divides by the remaining proportion.
You can also use a unit-percentage method: 85% corresponds to 2,040, so 1% corresponds to 24 and 100% corresponds to 2,400. Choose the method you can explain clearly and check reliably.
Worked problem 4: a mixture whose ratio changes
An imaginary drink mixture contains concentrate and water in the ratio 2:7. There are 4.5 L altogether. Another 1.5 L of water is added. Find the original amounts, the new ratio and the final percentage of concentrate by volume. Assume the volumes add.
The original mixture has 9 parts, so one part is 4.5 ÷ 9 = 0.5 L. Concentrate is 2 × 0.5 = 1.0 L and water is 7 × 0.5 = 3.5 L.
Adding only water leaves the concentrate amount unchanged. Water becomes 3.5 + 1.5 = 5.0 L. The new concentrate-to-water ratio is 1.0:5.0, or 1:5. The final mixture volume is 6.0 L, so concentrate forms 1/6 × 100 = 16⅔% of the total.
The ratio 1:5 does not mean concentrate is 20% of the mixture. One divided by five compares concentrate with water. For its percentage of the complete mixture, divide by all six parts.
Worked problem 5: percentages of different groups
A fictional group of 96 learners contains boys and girls in the ratio 5:7. A reading activity includes 30% of the boys and 25% of the girls. How many learners participate, and what percentage of the entire group is that?
There are 12 ratio parts, so one part represents 96 ÷ 12 = 8 learners. The group contains 5 × 8 = 40 boys and 7 × 8 = 56 girls.
Participating boys: 0.30 × 40 = 12. Participating girls: 0.25 × 56 = 14. Total participants: 12 + 14 = 26. Their percentage of the full group is 26/96 × 100 ≈ 27.1%, to one decimal place.
The answer is not the ordinary average of 30% and 25%, because the two groups contain different numbers of learners. Calculate the actual subgroup counts before combining them. Also check that your answer lies between 25% and 30%, which is sensible for a mixture of these two participation rates.
Understand which percentage change is being requested
An exercise price rises from KSh 800 to KSh 920. The increase is KSh 120, and the percentage increase is 120/800 × 100 = 15%. If it later falls from KSh 920 back to KSh 800, the decrease is still KSh 120, but the percentage decrease is 120/920 × 100 ≈ 13.0%.
The starting amount changes between the two calculations. The OpenStax percentage-change summary uses the original amount for the relevant change as the denominator. Write a short label beside your denominator to avoid reversing the comparison.
Percentage points are different again. If an activity’s attendance rate rises from 70% to 84%, that is an increase of 14 percentage points. The relative percentage increase in the rate is 14/70 × 100 = 20%. State which measure you are giving.
Diagnose a percentage error with one sentence
Before correcting arithmetic, write a sentence identifying the whole: “The 25% applies to the second club’s KSh 1,200 share.” That sentence exposes a wrong denominator before a calculator hides it behind a plausible number. For the drink mixture, the matching sentence is “The concentrate percentage uses the complete 6.0 L mixture.”
When checking a reverse percentage, state what the given amount represents: “KSh 2,040 is 85% of the original price.” If your proposed original price is lower than the discounted price, the answer fails a reasonableness check. If it is higher, that is necessary but not sufficient: apply the discount forward to test it exactly. This habit makes the same foundations useful in unfamiliar multi-step problems.
Your practice set
- Share 45 counters between two groups in the ratio 4:5. How many counters does each receive?
- An example price of KSh 1,200 increases by 8%, then decreases by 10%. Find the final price and overall percentage change.
- An item costs KSh 1,728 after a 4% discount. Find its original price.
- A 6.4 kg mixture contains two grains in the ratio 5:3. Find the mass of each grain and the percentage represented by the first.
Solutions and checks
1. There are nine parts. Each part is 45/9 = 5 counters, giving 20 and 25 counters. Their sum is 45 and their ratio simplifies to 4:5.
2. Multiply 1,200 × 1.08 × 0.90 = KSh 1,166.40. The decrease from the original is KSh 33.60, which is 33.60/1,200 × 100 = 2.8%.
3. The remaining proportion is 0.96. Original price = 1,728/0.96 = KSh 1,800. Four percent of 1,800 is 72, giving the stated sale price.
4. Eight parts make 6.4 kg, so each is 0.8 kg. The masses are 4.0 kg and 2.4 kg. The first grain is 5/8 × 100 = 62.5% of the mixture.
One connected activity: share, shop and draw to scale
This fictional classroom display task connects the sharing and shopping methods above with same-unit scale conversion. It is prerequisite practice for junior learners, not a claim to cover the complete current Grade 9 syllabus. No purchase or classroom survey actually took place.
Worked activity brief
Three clubs share KSh 6,000 in the ratio 2:3:5. The middle club buys a display panel whose marked price is KSh 1,500, discounted by 20%, with all costs included. The panel is 2.4 m long and 1.2 m high. Prepare a scale drawing of the panel at 1:20. Find all club shares, the middle club’s amount left, and both drawing dimensions. Explain how the three stages connect.
One complete sample response
- There are 2 + 3 + 5 = 10 parts. One part is 6,000/10 = KSh 600. Shares are KSh 1,200, KSh 1,800 and KSh 3,000, adding back to KSh 6,000.
- The middle club pays 1,500 × 0.80 = KSh 1,200. The 20% reduction is based on the marked price, not on the club’s share. It has 1,800 − 1,200 = KSh 600 left.
- At 1:20, drawing length:actual length uses the same unit. Convert 2.4 m to 240 cm and 1.2 m to 120 cm. Drawing dimensions are 240/20 = 12 cm and 120/20 = 6 cm. On paper, draw and label a 12 cm by 6 cm rectangle.
- Check in reverse: 12 × 20 = 240 cm = 2.4 m, and 6 × 20 = 120 cm = 1.2 m. The aspect ratio remains 2:1. The available share determines whether the purchase fits the budget; the physical panel dimensions determine the drawing, independently of its price.
The same-unit scale step is the method used in the prepared bearings/scale lesson’s “convert a scale before drawing” case, but no bearing calculation is needed here. Do not measure this webpage as a 1:20 plan: display resizing changes physical centimetres.
Try a fresh joined task
Two groups share KSh 7,200 in the ratio 3:5. The first group buys a board marked KSh 3,000 after a 10% discount, all costs included. Its dimensions are 1.8 m by 0.9 m. Find both shares, the first group’s money left and the dimensions of a 1:15 drawing. Explain any zero balance rather than calling it a loss.
Sample answer and review guide
Eight parts give KSh 900 per part, so shares are KSh 2,700 and KSh 4,500. The board costs 3,000 × 0.90 = KSh 2,700, leaving KSh 0: the share exactly covers the stated purchase. Convert to 180 cm by 90 cm, then divide both by 15 to obtain a 12 cm by 6 cm drawing. Check that the shares sum to KSh 7,200, the discounted price is 90% of the marked price and multiplying the drawing dimensions by 15 restores the board dimensions.
Review criteria: all shares use the total number of ratio parts; the discount uses its named base; the spending is subtracted from the correct share; scale uses matching length units; and both reverse checks are shown. Multipliers, unit percentages or proportional reasoning are equally valid when they preserve these relationships. A wrong final number with a correct earlier method should be corrected at its first error.
Frequently asked questions about ratios and percentages
Does a ratio need units?
A ratio of comparable quantities is usually written without units after both quantities have been expressed consistently. A rate such as kilometres per hour compares different kinds of quantities and retains the relevant units.
How do I simplify a three-part ratio?
Divide every term by the same common factor. For example, 6:9:12 becomes 2:3:4 after division by three.
Can a percentage exceed 100%?
Yes. A quantity can exceed its comparison base. An amount of 150 is 150% of 100, although a part of a fixed whole cannot exceed that whole.
Why do successive changes use multiplication?
Each percentage acts on the amount reached at that stage. A multiplier such as 1.08 or 0.90 updates the amount before the next operation.
Does a 10% rise followed by a 10% fall cancel out?
No. The multiplier is 1.10 × 0.90 = 0.99, leaving 99% of the original amount, an overall decrease of 1%.
What if my answer gives part of a learner?
Recheck the numbers, ratio and stated interpretation. Expected values may be fractional in some contexts, but an actual headcount needs a whole-number interpretation supported by the question.
When should I use percentage points?
Use them for the arithmetic difference between two percentages. Use relative percentage change when the question asks how large the change is compared with the starting rate.
How can I check a reverse-percentage answer?
Apply the original percentage change to your recovered starting amount. It should return exactly to the amount stated in the question, subject to any specified rounding.
Build on the skill with topical revision
After solving the examples, choose one and change its total while keeping the same ratio. Predict which steps stay the same. The existing Grade 9 topical questions and answers page is a relevant next destination on this site; its material currently requires login. If you need access, check the membership information for the current terms before joining.
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