BTT Mathematics / Primary Mathematics Learning Hub / Guide 5
Ratio compares quantities multiplicatively, rate compares quantities measured in different units, and percentage expresses a quantity relative to a base of one hundred. All three become easier when the learner can answer two questions: what quantities are being compared, and what amount does one unit of comparison represent?
A child may calculate accurately and still attach the calculation to the wrong base. Twenty per cent of eighty is sixteen, but an increase from eighty to ninety-six is also sixteen and therefore twenty per cent of the original eighty. If the base changes, the percentage relationship changes even when the absolute difference remains sixteen.
This guide begins with ratios as equal units, then develops unitary reasoning, rates, percentages, percentage change and reverse problems. The current MOE Primary Mathematics syllabus organises Primary Mathematics through Number and Algebra, Measurement and Geometry, and Statistics. Ratio and percentage are upper-primary ideas; use the later sections only when the learner has the required fraction, decimal and multiplication background.
Understand ratio · Use one-unit reasoning · Understand rate · Understand percentage · Reverse the relationship · 24 questions · Worked answers
1. A ratio compares quantities by equal units
Suppose a box contains twelve red counters and eight blue counters. The ratio of red to blue is 12:8. Both terms can be divided by four, giving the equivalent ratio 3:2. This does not mean there are three red counters and two blue counters in the actual box. It means the quantities have the same multiplicative relationship as three equal units to two equal units.
If one unit represents four counters, three units represent twelve red counters and two units represent eight blue counters. The unit is a model amount. It can represent four counters, ten dollars, five centimetres or any other quantity required by the problem.
Ratios preserve multiplicative structure. Adding the same number to both terms usually changes the ratio. The ratio 3:2 becomes 4:3 after adding one to each term, and 4:3 is not equivalent to 3:2. Multiplying or dividing both terms by the same positive factor preserves the relationship.
Part-to-part and part-to-whole are different comparisons
With twelve red and eight blue counters, red:blue is 12:8 = 3:2. Red:total is 12:20 = 3:5. The first comparison asks how red relates to blue. The second asks how red relates to the whole collection.
A learner who writes 3:5 when asked for red:blue may understand the numbers but have changed the second quantity being compared. Ask the child to name the two quantities before simplifying. A ratio answer should preserve both the order and the labels.
The fraction of the collection that is red is 12/20 = 3/5. This equals the ratio red:total expressed as a fraction, but it is not the same object as the part-to-part ratio red:blue = 3:2. Keeping these relationships separate makes later percentage work easier.
Order matters
If the ratio of boys to girls is 4:5, then the ratio of girls to boys is 5:4. These ratios describe the same two groups from opposite directions, but they are not written identically. The order in the words determines the order in the notation.
When a child repeatedly reverses a ratio, write the labels above the terms: boys:girls = 4:5. Then carry the labels through the working. This small record is often more useful than repeating “read carefully,” because it shows exactly what must remain attached to each number.
Equivalent ratios are scaled copies
Start with 3:4. Multiply both terms by seven to obtain 21:28. The ratio is unchanged because both quantities were enlarged by the same factor. Divide 21:28 by seven and the original 3:4 returns.
A table can make this visible:
3 : 4
6 : 8
9 : 12
21 : 28
Every row describes the same relationship. Moving between rows is multiplication or division by one common factor. If the left term is multiplied by seven while the right term is multiplied by five, the relationship has changed.
A ratio with a total creates a unit count
Two quantities are in the ratio 5:7 and total ninety-six. The model contains twelve equal units altogether. One unit is 96 ÷ 12 = 8. The quantities are forty and fifty-six.
The crucial step is not division by twelve as a memorised rule. It is recognising that the total contains five units plus seven units. If the given ninety-six represented a difference instead, the unit count would be the difference between seven units and five units: two units, not twelve.
A ratio with a difference uses the gap in units
Two quantities are in the ratio 3:7 and differ by thirty-two. The larger amount has four more equal units than the smaller. Four units therefore represent thirty-two, so one unit is eight. The quantities are twenty-four and fifty-six.
Check both conditions. Their difference is thirty-two, and 24:56 simplifies to 3:7. The sum eighty is not the given number; it is a consequence found only after the quantities have been solved.
2. One-unit reasoning turns a ratio into quantities
Unitary reasoning asks what one unit is worth, then builds the quantity required. It is a powerful bridge between ratio, fractions, rates and percentage.
Example A: Find a missing quantity from a ratio
The ratio of blue pens to black pens is 4:5. There are twenty-eight blue pens. Four units represent twenty-eight, so one unit is seven. Five units represent thirty-five black pens.
Blue = 28, black = 35. The total is sixty-three. Check 28:35 by dividing both terms by seven; the ratio returns to 4:5.
Writing 28 ÷ 4 × 5 is a compact calculation, but the labels explain it. Divide by four because twenty-eight represents four units; multiply by five because the required quantity contains five units.
Example B: Share a quantity in a ratio
Share forty-five dollars in the ratio 2:3. There are five units altogether. One unit is 45 ÷ 5 = 9 dollars. The shares are eighteen dollars and twenty-seven dollars.
Check the total: 18 + 27 = 45. Check the ratio: 18:27 simplifies to 2:3. Both checks matter. A pair of fifteen and thirty would total correctly but has ratio 1:2, not 2:3.
Example C: A changed quantity creates a new ratio
A box begins with twelve red and twenty blue counters, a ratio of 3:5. Six red counters are added. The new counts are eighteen red and twenty blue. The new ratio is 18:20 = 9:10.
Do not add six to the ratio term three. The ratio term is not the actual counter count. It is a scaled description. Changes happen to the real quantities first; then a new ratio is calculated from the new state.
This distinction becomes particularly important in before-and-after questions. Record the actual quantities at each stage rather than trying to edit the simplified ratio directly unless the unit scale is explicitly known.
Example D: Use a ratio to compare a fraction of the whole
If red:blue = 3:2, the whole collection contains five equal units. Red therefore represents 3/5 of the whole and blue represents 2/5. A part-to-part ratio can be converted into fractions of the whole by including all the units in the denominator.
This conversion is valid because the same five units exhaust the collection. If a third colour is present but omitted from the ratio, red:blue alone does not determine red as a fraction of the entire collection.
3. A rate compares quantities measured in different units
A rate might compare dollars with notebooks, kilometres with hours, litres with minutes or words with pages. The two quantities use different units, so the answer should retain both until a useful unit rate is formed.
Example E: Price per item
Five identical notebooks cost fifteen dollars. The rate is $15 for 5 notebooks. Divide both quantities by five to obtain the unit rate: $3 per notebook.
If eight notebooks are bought at the same rate, the cost is 8 × $3 = $24. The assumption “same rate” matters. If a bulk discount or fixed delivery fee is introduced, the simple proportional relationship may no longer apply.
Example F: Average speed as a rate
A vehicle travels 180 kilometres in three hours. Its average speed over the whole journey is 180 ÷ 3 = 60 kilometres per hour.
This does not claim the vehicle travelled at exactly sixty kilometres per hour at every instant. It says that total distance divided by total time is sixty kilometres for each hour on average.
To travel another two hours at the same constant rate of sixty kilometres per hour would add 120 kilometres. But if the question gives a different speed for the next part, calculate that part separately before combining the distances.
Example G: Compare rates by making one quantity common
Shop A sells six pens for twelve dollars. Shop B sells eight of the same pens for twenty dollars. Shop A’s unit rate is two dollars per pen; Shop B’s is two dollars fifty cents per pen. Under the stated conditions, Shop A has the lower price per pen.
Comparing twelve dollars with twenty dollars alone would ignore that different quantities of pens are bought. Comparing six with eight alone would ignore price. A rate comparison needs a common basis such as one pen, one kilogram or one hour.
Do not infer that the lower unit price is always the better real-world purchase. A buyer may not need the larger quantity, and other conditions may matter. The mathematics answers the stated comparison; it does not replace the decision criteria that were not included.
4. Percentage names an amount relative to one hundred
Twenty-five per cent means twenty-five out of every hundred equal parts: 25/100 = 1/4 = 0.25. Percentage connects directly to fractions and decimals when the same reference whole is used.
Example H: Find a percentage of a quantity
Find 25% of eighty. Since 25% = 1/4, divide eighty by four to obtain 20. Another route is 80 × 25/100 = 20.
The fraction route may be faster for familiar percentages such as 50%, 25%, 20%, 10% and 5%. The hundredths route is more general. The important part is that eighty is the base quantity to which the twenty-five per cent applies.
Example I: Find a less convenient percentage
Find 15% of 240. Ten per cent is twenty-four. Five per cent is half of that, twelve. Add them: fifteen per cent is thirty-six.
Equivalently, 240 × 0.15 = 36. The first route decomposes the percentage; the second converts it to a decimal multiplier. Both preserve the same relationship.
A quick size check helps. Fifteen per cent is less than one fifth, so the answer should be less than 48. Thirty-six is plausible; three hundred sixty is not.
Example J: Find what percentage one quantity is of another
A learner scores seventy-two marks out of ninety. The fraction of the total obtained is 72/90 = 4/5. Since 4/5 = 80/100, the score is 80%.
The base is ninety, the possible total under this calculation. Dividing ninety by seventy-two would reverse the relationship. Write “part ÷ whole” before calculating when the two numbers can easily be confused.
Example K: Percentage discount
An item costs seventy-five dollars and is discounted by 20%. Twenty per cent of seventy-five is fifteen dollars. Subtract the discount from the original price: 75 − 15 = $60.
A shorter multiplier method uses the percentage remaining. If twenty per cent is removed, eighty per cent remains. 75 × 0.8 = 60. The two methods describe the same change from different directions.
Be careful when an additional condition such as tax, service charge or a second discount appears. Apply each percentage to the base stated or implied at that stage. Two successive 20% discounts do not equal one 40% discount of the original price because the second discount is taken from a reduced amount.
Example L: Percentage increase
A quantity increases from 320 to 352. The increase is thirty-two. To express that increase as a percentage of the original amount, calculate 32 ÷ 320 = 0.1 = 10%.
The original 320 is the base. Dividing by the new amount 352 would answer a different question: the increase as a percentage of the new amount.
To increase 320 by 10% directly, find 10% of 320, which is thirty-two, then add it. Or multiply by 110% = 1.10. A percentage increase changes the quantity by adding a fraction of the original base.
5. Reverse problems: when the base is unknown
Example M: A percentage is known, the whole is unknown
Eighteen students represent 25% of a group. Since 25% is one quarter, eighteen is one of four equal quarters. The whole group contains 18 × 4 = 72 students.
A common error is to calculate 25% of eighteen, which would find a smaller amount. But eighteen is already the part. The question asks for the larger base from which that part came.
Example N: Work backwards through a discount
After a 20% discount, an item costs sixty-four dollars. The sale price is 80% of the original price. If eighty per cent equals sixty-four, ten per cent equals eight and one hundred per cent equals eighty dollars.
Dividing sixty-four by 0.8 gives the same result. Adding twenty per cent of sixty-four would be incorrect because the discount was defined relative to the original price, not the sale price.
Example O: Work backwards from an amount remaining
A child has 260 dollars left after spending 35% of the original amount. Sixty-five per cent remains. If 65% equals 260, then 1% equals four dollars and 100% equals 400 dollars.
Check forward: 35% of 400 is 140, and 400 − 140 = 260. The check reconstructs the stated final amount from the inferred base.
Example P: Percentage change is not symmetric
A price falls from one hundred dollars to eighty dollars, a decrease of twenty dollars or 20% of the original one hundred. To return from eighty to one hundred requires an increase of twenty dollars, but twenty is 25% of eighty.
Equal absolute changes can represent different percentage changes when the bases differ. This is why “down 20%, then up 20%” does not return to the starting amount. The second twenty per cent is applied to a smaller base.
6. Practice: 24 questions
Keep the worked answers covered. Label the two quantities in every ratio or rate. For percentage questions, circle or write the base before calculating. The three groups increase in structural demand; they are not official ability bands.
Questions 1–8: Ratios and one unit
1. Simplify the ratio 12:8.
2. Share $45 in the ratio 2:3.
3. The ratio of boys to girls is 4:5. There are 28 boys. How many girls are there?
4. A basket contains 6 green apples and 9 red apples. What is the ratio of green apples to all apples in simplest form?
5. Complete the equivalent ratio 3:4 = 21:__.
6. Two quantities are in the ratio 5:7 and total 96. Find both quantities.
7. Two quantities are in the ratio 3:7 and differ by 32. Find both quantities.
8. Twelve red and twenty blue counters are in a box. Six red counters are added. Find the new ratio red:blue in simplest form.
Questions 9–16: Rates and percentages
9. Five notebooks cost $15 at a constant rate. What is the cost per notebook?
10. At the same rate as question 9, what do eight notebooks cost?
11. A vehicle travels 180 km in 3 hours at a constant average rate. What is the average speed?
12. Find 25% of 80.
13. Find 40% of 150.
14. Find 15% of 240.
15. 72 is what percentage of 90?
16. An item costs $75 and is discounted by 20%. What is the sale price?
Questions 17–24: Reverse and compare
17. Increase 320 by 10%.
18. Eighteen students represent 25% of a group. How many students are in the whole group?
19. After a 20% discount, an item costs $64. What was the original price?
20. A child has $260 left after spending 35% of the original amount. How much did the child have at first?
21. A drink is mixed in the ratio syrup:water = 2:3. The total volume is 250 ml. Find the volume of each part.
22. Shop A sells 6 identical pens for $12. Shop B sells 8 identical pens for $20. Which shop has the lower price per pen, and by how much?
23. A price falls from $100 to $80 and later rises from $80 to $100. Find the percentage decrease and the percentage increase.
24. Class A has 18 correct submissions out of 24. Class B has 22 correct submissions out of 25. Which class has the higher percentage, and by how many percentage points?
7. Worked answers
Answers 1–8
1. 3:2. Divide both terms by four. Twelve red units and eight blue units become three equal ratio units to two. Because the same factor is used on both terms, the comparison is preserved.
2. $18 and $27. The ratio contains five units altogether. One unit is $45 ÷ 5 = $9. Two units give $18 and three units give $27. Check both total and ratio.
3. 35 girls. Four units represent twenty-eight boys, so one unit is seven. Five units represent thirty-five girls. Check 28:35 = 4:5 after dividing by seven.
4. 2:5. The total number of apples is fifteen. Green:total is 6:15, which simplifies by dividing both terms by three. Green:red would be 6:9 = 2:3, a different comparison.
5. 28. The first term was multiplied by seven, from three to twenty-one. Multiply the second term by the same factor: 4 × 7 = 28.
6. 40 and 56. Five plus seven gives twelve units. One unit is 96 ÷ 12 = 8. The quantities are 5 × 8 = 40 and 7 × 8 = 56.
7. 24 and 56. The gap from three units to seven units is four units. Four units equal thirty-two, so one unit equals eight. Multiply by three and seven respectively.
8. 9:10. The actual red count changes from twelve to eighteen while blue remains twenty. The new ratio is 18:20, which simplifies by dividing by two. Adding six to the simplified term three would not model the actual change.
Answers 9–16
9. $3 per notebook. Divide both the cost and the number of notebooks by five. Fifteen dollars for five notebooks becomes three dollars for one notebook.
10. $24. Eight notebooks at three dollars each cost 8 × 3 = 24 dollars. This uses the stated constant rate.
11. 60 km/h. Divide total distance by total time: 180 ÷ 3 = 60. This is the average distance travelled per hour over the stated period.
12. 20. Twenty-five per cent is one quarter. One quarter of eighty is twenty. Check that four groups of twenty make eighty.
13. 60. Forty per cent is forty hundredths or two fifths. One fifth of 150 is thirty, so two fifths is sixty.
14. 36. Ten per cent of 240 is twenty-four and five per cent is twelve. Add them to obtain thirty-six. It is less than one fifth of 240, which would be forty-eight.
15. 80%. The part is seventy-two and the whole is ninety. 72/90 simplifies to 4/5, which is 80/100. Therefore the percentage is eighty.
16. $60. Twenty per cent of seventy-five is fifteen. Subtract from the original price: 75 − 15 = 60. Equivalently, eighty per cent remains.
Answers 17–24
17. 352. Ten per cent of 320 is thirty-two. Add the increase: 320 + 32 = 352. The new quantity represents 110% of the original.
18. 72 students. Twenty-five per cent is one quarter. If one quarter is eighteen, four quarters contain 18 × 4 = 72 students.
19. $80. After a twenty per cent discount, eighty per cent remains. If 80% is sixty-four, 10% is eight and 100% is eighty. Check that twenty per cent of eighty is sixteen and the discounted price is sixty-four.
20. $400. Spending thirty-five per cent leaves sixty-five per cent. If 65% equals 260, then 1% equals four and 100% equals 400. Check that 35% of 400 is 140, leaving 260.
21. 100 ml syrup and 150 ml water. The ratio contains five units. One unit is 250 ÷ 5 = 50 ml. Two units give 100 ml and three units give 150 ml.
22. Shop A, by $0.50 per pen. Shop A costs $12 ÷ 6 = $2 per pen. Shop B costs $20 ÷ 8 = $2.50 per pen. The unit-rate difference is fifty cents.
23. 20% decrease and 25% increase. The twenty-dollar fall is 20/100 of the original hundred. The twenty-dollar rise is 20/80 of the later base, which is one quarter or twenty-five per cent. The same dollar change uses a different base.
24. Class B, by 13 percentage points. Class A: 18/24 = 3/4 = 75%. Class B: 22/25 = 88%. The difference between the percentages is 88% − 75% = 13 percentage points.
8. Diagnose the base before increasing the difficulty
If a learner can simplify 12:8 but cannot share forty-five dollars in 2:3, the issue may be moving from a comparison to actual quantities. Ask how many equal units make the whole, and what one unit represents.
If ratio work is secure but percentage questions fail, check whether the learner is naming the base. Write “25% of 80” and circle eighty before calculating. For a reverse problem, write “80% = 64” before finding one per cent or one unit. This makes the direction visible.
If a child treats a changed ratio by adding directly to a simplified ratio term, return to actual quantities. Ratios are descriptions of a state. A change happens to the quantities in that state, after which the new comparison is calculated.
Separate multiplicative comparison from additive difference
“A is seven more than B” and “A is seven times B” are different relationships even though the number seven appears in both. The first concerns a fixed difference. The second concerns repeated equal copies.
Ask the learner to draw one bar for B. For “seven more,” add a short segment labelled seven. For “seven times,” draw seven equal copies of the entire B bar. The picture should express the grammar of the comparison.
Use percentage change only after the base is explicit
Before calculating a percentage increase or decrease, write the difference and the original base. For a change from 80 to 96, the increase is sixteen and the original is eighty, so the change is 16/80 = 20%.
For the reverse change, from 96 back to 80, the decrease is still sixteen but the base is ninety-six. The percentage decrease is therefore not twenty per cent. This pair of questions is useful because it shows that percentage belongs to a relationship, not to the difference alone.
Return the answer to the units
Ratios may be written without physical units after equivalent quantities have been normalised, but rate answers need their paired units: dollars per notebook, kilometres per hour, millilitres per serving. Percentage answers need a reference statement: what percentage of what?
A final numerical answer that cannot be read back into the original quantities is unfinished. The labels help detect whether a correct calculation has answered a different question.
Continue through the Primary Mathematics series
For fraction and decimal foundations, use Fractions, Decimals and the Same Whole. For equal-group foundations, use Equal Groups, Division and Remainders. For story representation, use Word Problems, Bar Models and Checking.
Continue the second collection with Measurement, Units, Perimeter, Area and Volume, Geometry, Angles, Symmetry and Coordinates, and Data, Graphs and Average.
Return to the BTT Primary Mathematics Learning Hub. For the wider conceptual route, see Ratio, Rate and Proportion.
Original learning guide. Curriculum reference checked 6 September 2026. Examples and questions are educational illustrations, not official examination questions or predictions.