Sec 3 G3 Maths Bukit Timah | Fractions and Ratios

Secondary 3 G3 Mathematics Tutor Bukit Timah | Fractional Ratios, Sharing and Recipe Scaling — Did you know? One changed word in a Mathematics question can require a different method even when all the numbers remain the same. The key is to identify what the answer must represent before starting. Here is a quick reference for students, parents and tutors, followed by detailed examples to practise in short sittings.

The quick answer

Convert both compared quantities into consistent units before simplifying a ratio.
A:B=2:3 means the combined total has five parts, not three.
Multiply every term of a fractional ratio by a common denominator to obtain equivalent whole-number parts.
For sharing, divide by the sum of the ratio parts to find the size of one part.
For A:B and B:C, make the two B quantities equal before forming A:B:C.

The skills discussed here are included in the full-course 2027 SEC G3 Mathematics syllabus K310 under Ratio and Proportion. Schools may cover or revise these prerequisites in different secondary years. This guide does not imply that every Secondary 3 school is currently on the same chapter. Check the SEAB 2027 G3 subject listing for official syllabus context.

Jump to: Understand the ideas · Worked examples · Mistakes · Study plan · FAQs.

Understand the relationship, not only the calculation

A ratio compares two named amounts

The ratio 2:3 means that whenever one group has two parts, the other has three comparable parts. Together they have five parts. The first group is therefore 2/5 of the total, not 2/3 of the total. The expression 2/3 compares the first group to the second group.

A student can memorise simplification and still misunderstand the meaning of a share. Ask the learner to draw five equally sized boxes, label two for A and three for B, and identify what the whole represents. This visual model is an efficient defence against a wrong denominator.

Ratios involving fractions are legitimate

For example, (1/2):(2/3) is not an invalid ratio. Multiply both quantities by six, a common multiple of the denominators. The result is 3:4. Both terms have been scaled by the same positive factor, so the original comparison is unchanged.

The G3 K310 syllabus explicitly includes ratios involving rational numbers. A tutor can alternate fractions and whole numbers so the learner sees equivalent scaling rather than learns a completely different technique every time fractions appear.

Units must agree before the ratio is simplified

Suppose a rope is 1.5 metres long and another is 75 centimetres long. Writing 1.5:75 without considering the units hides a mismatch. Convert 1.5 metres to 150 centimetres, then simplify 150:75 to 2:1.

This matters for money, mass and time too. If one duration is recorded in hours and another in minutes, put both in the same units before reducing. A ratio is only meaningful when the two compared quantities have a coherent interpretation.

Sharing a total begins with the sum of parts

To share $200 in ratio 2:3, count five total parts. One part is $200/5=$40, so A receives $80 and B receives $120. The amounts add to the original total and preserve the given ratio.

Students often divide by the larger ratio number instead of the sum of all parts. To repair this misconception, ask what a single part represents and why the combined amount contains five, rather than three, copies of that part.

Three-part ratios extend the same principle

Sharing 180 counters in ratio 2:3:4 means nine parts altogether, or twenty counters per part. The three groups contain forty, sixty and eighty counters. Their sum gives 180.

A similar method works for more than three groups; the number of named quantities changes but the idea does not. State whether the amount being shared is divisible into whole items. If it is not, the problem’s real-world context may impose additional restrictions.

Combined ratios need a matching middle group

If A:B=2:3 and B:C=4:5, the B quantities in the two ratios do not yet match. Multiply the first comparison by four to obtain 8:12 and the second by three to obtain 12:15. Thus A:B:C=8:12:15.

Directly writing A:B:C=2:3:5 would be an invented comparison. The repeated group is the bridge. A learner should explain why scaling both sides of one ratio keeps that ratio valid before using the combined three-part form.

Recipes and mixtures give ratios meaning

If four servings use 240 grams of flour, six servings require 240×6/4=360 grams when the question assumes all ingredients scale proportionally. A syrup-to-water ratio of 2:5 has seven parts of mixture, so the syrup proportion of the whole is 2/7.

These applications ask different things even though they use the same logic. A recipe question often changes a serving factor; a mixture problem asks for relative amounts of components. Read whether the ratio compares syrup with water or syrup with total mixture.

One check can prevent several types of errors

After sharing a total, add the shares back and reduce their ratio to the original. After a unit conversion, reverse it and see whether the initial value returns. After combining ratios, test that both original pairwise comparisons are preserved.

A small-group tutor can ask the student to choose their own checking method. Becoming able to explain why the check is appropriate often improves performance on unfamiliar word problems more than doing another long set of nearly identical calculations.

A three-question diagnostic before more worksheets

Start with one routine exercise, one near-example that changes an important condition, and one word problem with no chapter name. Ask the student to explain the first operation before calculating. The three attempts show whether the problem is a missing definition, an inappropriate method, or a small arithmetic slip. An effective correction fixes the cause, not just the final number.

Small-group teaching can be particularly useful when each learner has to produce original working and explain a decision. It is less useful if everyone is watching one confident student reproduce a finished solution. Use a short follow-up question after a delay to see whether the lesson changed what the student can do without prompting.

40 worked examples with a check

These are constructed educational exercises, not official examination questions. The ten question types each have four changed numerical versions. Do not attempt all forty in one sitting. Work through a small set, hide the method, compare the independent check and then mix question families so that the chapter title no longer supplies the first step.

Fractional ratios

This cluster focuses on fractional ratios. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 1. Question: Simplify (1/2):(2/3). Method: Scale both terms by the denominator product 6. The ratio becomes 3:4; divide by HCF 1 to get 3:4. Check: Their relative quotient is 3/4, consistent with the original two fractions. A common trap: Do not cancel a number in just one ratio term.

Independent practice: Name the kind of quantity the question asks for before doing any arithmetic. Change one input and predict how the result should respond. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Share two amounts

This cluster focuses on share two amounts. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 2. Question: Share $50 in ratio 2:3. Method: There are 5 parts, each worth $10. The two shares are $20 and $30. Check: The shares add to $50, and their ratio is 2:3. A common trap: The divisor for a total share is the sum of all parts.

Independent practice: Hide the working and begin independently; recognising a printed line is not the same as recalling the rule. Build a second solution using a different representation. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Share three amounts

This cluster focuses on share three amounts. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 3. Question: Split 108 counters into three groups in ratio 2:3:4. Method: The 9 parts are each 12 counters. The three groups have 24, 36, 48. Check: Their total is 108, preserving all three proportions. A common trap: Leaving out one group changes the whole number of parts.

Independent practice: Explain why the main step preserves the original comparison or number property. State the condition that makes the shortcut valid. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Combine two ratios

This cluster focuses on combine two ratios. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 4. Question: If A:B=2:3 and B:C=4:5, find A:B:C. Method: Scale A:B by 4, giving 8:12; scale B:C by 3, giving 12:15. The combined simplest ratio is 8:12:15. Check: Both original pairwise ratios reduce correctly from the combined comparison. A common trap: Shared B parts have to match before appending three groups.

Independent practice: Record one common error and the exact checking habit that would expose it next time. Give the result a real-world interpretation where one is appropriate. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Recipe serving factors

This cluster focuses on recipe serving factors. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 5. Question: A four-serving recipe uses 240 g of flour. How much flour for six servings with direct proportional scaling? Method: Serving factor is 6/4=1.5; multiply flour by it: 240×1.5=360 g. Check: Dividing the new amount by 1.5 returns 240 g. A common trap: Do not add a fixed amount without regard to the serving factor.

Independent practice: Try a new numerical case after a break instead of copying the same completed example. Work backwards from the result to construct a possible question. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Units in a length ratio

This cluster focuses on units in a length ratio. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 6. Question: Simplify 1.5 metres : 75 centimetres. Method: Convert metres to centimetres: 1.5 m=150 cm. Then 150:75=2:1. Check: The first length is exactly twice the second. A common trap: Different units must be reconciled before reduction.

Independent practice: Ask whether a diagram, factor list or a reverse calculation can verify the result. Write a short counterexample to one tempting but false shortcut. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Recover an unknown share

This cluster focuses on recover an unknown share. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 7. Question: A:B=3:5 and B receives $40. Find A’s amount. Method: One part is $40/5=$8, so A receives 3 parts: $24. Check: The actual amounts 24:40 reduce to 3:5. A common trap: B is not the entire total unless the context explicitly says so.

Independent practice: Keep the units or mathematical conditions visible in every final line. Change one input and predict how the result should respond. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Syrup and water in a mixture

This cluster focuses on syrup and water in a mixture. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 8. Question: Syrup:water=2:5, with 150 mL per part. Find both amounts and their total. Method: Syrup=300 mL, water=750 mL and total=1050 mL. Check: The components add to 1050 mL and retain the specified ratio. A common trap: The syrup’s fraction of the whole is 2/7, not 2/5.

Independent practice: Compare a superficially similar question requiring a different method. Build a second solution using a different representation. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

A ratio as a fraction of the whole

This cluster focuses on a ratio as a fraction of the whole. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 9. Question: In A:B=3:4, what fraction of both groups together is A? Method: The total has 7 parts, so A’s share is 3/7. Check: The two group fractions 3/7 and 4/7 add to one. A common trap: The total-part denominator is 7, not 4.

Independent practice: Explain the reason to a classmate without saying ‘because this is the formula’. State the condition that makes the shortcut valid. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

When one group gains members

This cluster focuses on when one group gains members. Before substituting numbers, identify the mathematical object being sought and the rule that makes the operation valid. Later, contrast it with a question that looks similar but asks for something else.

Worked example 10. Question: Actual A:B counts are 30:40. Five people join A. Find the new simplest ratio. Method: The new counts are 35:40. Divide both by HCF 5 to obtain 7:8. Check: The first count gained five, and the second stayed 40. A common trap: Five extra people cannot be added directly to the abstract ratio part 3.

Independent practice: Return after two days with the problem heading hidden and choose the method unaided. Give the result a real-world interpretation where one is appropriate. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Fractional ratios

Worked example 11. Question: Simplify (2/3):(3/4). Method: Scale both terms by the denominator product 12. The ratio becomes 8:9; divide by HCF 1 to get 8:9. Check: Their relative quotient is 8/9, consistent with the original two fractions. A common trap: Do not cancel a number in just one ratio term.

Independent practice: Name the kind of quantity the question asks for before doing any arithmetic. Work backwards from the result to construct a possible question. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Share two amounts

Worked example 12. Question: Share $84 in ratio 3:4. Method: There are 7 parts, each worth $12. The two shares are $36 and $48. Check: The shares add to $84, and their ratio is 3:4. A common trap: The divisor for a total share is the sum of all parts.

Independent practice: Hide the working and begin independently; recognising a printed line is not the same as recalling the rule. Write a short counterexample to one tempting but false shortcut. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Share three amounts

Worked example 13. Question: Split 144 counters into three groups in ratio 3:4:5. Method: The 12 parts are each 12 counters. The three groups have 36, 48, 60. Check: Their total is 144, preserving all three proportions. A common trap: Leaving out one group changes the whole number of parts.

Independent practice: Explain why the main step preserves the original comparison or number property. Change one input and predict how the result should respond. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Combine two ratios

Worked example 14. Question: If A:B=3:4 and B:C=5:6, find A:B:C. Method: Scale A:B by 5, giving 15:20; scale B:C by 4, giving 20:24. The combined simplest ratio is 15:20:24. Check: Both original pairwise ratios reduce correctly from the combined comparison. A common trap: Shared B parts have to match before appending three groups.

Independent practice: Record one common error and the exact checking habit that would expose it next time. Build a second solution using a different representation. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Recipe serving factors

Worked example 15. Question: A four-serving recipe uses 300 g of flour. How much flour for six servings with direct proportional scaling? Method: Serving factor is 6/4=1.5; multiply flour by it: 300×1.5=450 g. Check: Dividing the new amount by 1.5 returns 300 g. A common trap: Do not add a fixed amount without regard to the serving factor.

Independent practice: Try a new numerical case after a break instead of copying the same completed example. State the condition that makes the shortcut valid. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Units in a length ratio

Worked example 16. Question: Simplify 2 metres : 100 centimetres. Method: Convert metres to centimetres: 2 m=200 cm. Then 200:100=2:1. Check: The first length is exactly twice the second. A common trap: Different units must be reconciled before reduction.

Independent practice: Ask whether a diagram, factor list or a reverse calculation can verify the result. Give the result a real-world interpretation where one is appropriate. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Recover an unknown share

Worked example 17. Question: A:B=4:6 and B receives $48. Find A’s amount. Method: One part is $48/6=$8, so A receives 4 parts: $32. Check: The actual amounts 32:48 reduce to 4:6. A common trap: B is not the entire total unless the context explicitly says so.

Independent practice: Keep the units or mathematical conditions visible in every final line. Work backwards from the result to construct a possible question. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Syrup and water in a mixture

Worked example 18. Question: Syrup:water=3:6, with 150 mL per part. Find both amounts and their total. Method: Syrup=450 mL, water=900 mL and total=1350 mL. Check: The components add to 1350 mL and retain the specified ratio. A common trap: The syrup’s fraction of the whole is 3/9, not 3/6.

Independent practice: Compare a superficially similar question requiring a different method. Write a short counterexample to one tempting but false shortcut. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

A ratio as a fraction of the whole

Worked example 19. Question: In A:B=4:5, what fraction of both groups together is A? Method: The total has 9 parts, so A’s share is 4/9. Check: The two group fractions 4/9 and 5/9 add to one. A common trap: The total-part denominator is 9, not 5.

Independent practice: Explain the reason to a classmate without saying ‘because this is the formula’. Change one input and predict how the result should respond. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

When one group gains members

Worked example 20. Question: Actual A:B counts are 40:50. Five people join A. Find the new simplest ratio. Method: The new counts are 45:50. Divide both by HCF 5 to obtain 9:10. Check: The first count gained five, and the second stayed 50. A common trap: Five extra people cannot be added directly to the abstract ratio part 4.

Independent practice: Return after two days with the problem heading hidden and choose the method unaided. Build a second solution using a different representation. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Fractional ratios

Worked example 21. Question: Simplify (3/4):(4/5). Method: Scale both terms by the denominator product 20. The ratio becomes 15:16; divide by HCF 1 to get 15:16. Check: Their relative quotient is 15/16, consistent with the original two fractions. A common trap: Do not cancel a number in just one ratio term.

Independent practice: Name the kind of quantity the question asks for before doing any arithmetic. State the condition that makes the shortcut valid. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Share two amounts

Worked example 22. Question: Share $126 in ratio 4:5. Method: There are 9 parts, each worth $14. The two shares are $56 and $70. Check: The shares add to $126, and their ratio is 4:5. A common trap: The divisor for a total share is the sum of all parts.

Independent practice: Hide the working and begin independently; recognising a printed line is not the same as recalling the rule. Give the result a real-world interpretation where one is appropriate. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Share three amounts

Worked example 23. Question: Split 180 counters into three groups in ratio 4:5:6. Method: The 15 parts are each 12 counters. The three groups have 48, 60, 72. Check: Their total is 180, preserving all three proportions. A common trap: Leaving out one group changes the whole number of parts.

Independent practice: Explain why the main step preserves the original comparison or number property. Work backwards from the result to construct a possible question. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Combine two ratios

Worked example 24. Question: If A:B=4:5 and B:C=6:7, find A:B:C. Method: Scale A:B by 6, giving 24:30; scale B:C by 5, giving 30:35. The combined simplest ratio is 24:30:35. Check: Both original pairwise ratios reduce correctly from the combined comparison. A common trap: Shared B parts have to match before appending three groups.

Independent practice: Record one common error and the exact checking habit that would expose it next time. Write a short counterexample to one tempting but false shortcut. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Recipe serving factors

Worked example 25. Question: A four-serving recipe uses 360 g of flour. How much flour for six servings with direct proportional scaling? Method: Serving factor is 6/4=1.5; multiply flour by it: 360×1.5=540 g. Check: Dividing the new amount by 1.5 returns 360 g. A common trap: Do not add a fixed amount without regard to the serving factor.

Independent practice: Try a new numerical case after a break instead of copying the same completed example. Change one input and predict how the result should respond. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Units in a length ratio

Worked example 26. Question: Simplify 2.5 metres : 125 centimetres. Method: Convert metres to centimetres: 2.5 m=250 cm. Then 250:125=2:1. Check: The first length is exactly twice the second. A common trap: Different units must be reconciled before reduction.

Independent practice: Ask whether a diagram, factor list or a reverse calculation can verify the result. Build a second solution using a different representation. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Recover an unknown share

Worked example 27. Question: A:B=5:7 and B receives $56. Find A’s amount. Method: One part is $56/7=$8, so A receives 5 parts: $40. Check: The actual amounts 40:56 reduce to 5:7. A common trap: B is not the entire total unless the context explicitly says so.

Independent practice: Keep the units or mathematical conditions visible in every final line. State the condition that makes the shortcut valid. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Syrup and water in a mixture

Worked example 28. Question: Syrup:water=4:7, with 150 mL per part. Find both amounts and their total. Method: Syrup=600 mL, water=1050 mL and total=1650 mL. Check: The components add to 1650 mL and retain the specified ratio. A common trap: The syrup’s fraction of the whole is 4/11, not 4/7.

Independent practice: Compare a superficially similar question requiring a different method. Give the result a real-world interpretation where one is appropriate. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

A ratio as a fraction of the whole

Worked example 29. Question: In A:B=5:6, what fraction of both groups together is A? Method: The total has 11 parts, so A’s share is 5/11. Check: The two group fractions 5/11 and 6/11 add to one. A common trap: The total-part denominator is 11, not 6.

Independent practice: Explain the reason to a classmate without saying ‘because this is the formula’. Work backwards from the result to construct a possible question. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

When one group gains members

Worked example 30. Question: Actual A:B counts are 50:60. Five people join A. Find the new simplest ratio. Method: The new counts are 55:60. Divide both by HCF 5 to obtain 11:12. Check: The first count gained five, and the second stayed 60. A common trap: Five extra people cannot be added directly to the abstract ratio part 5.

Independent practice: Return after two days with the problem heading hidden and choose the method unaided. Write a short counterexample to one tempting but false shortcut. After you have attempted this example, write one sentence explaining how the initial information leads to the chosen first step. Retain the student’s original working so the later correction can be compared fairly.

Fractional ratios

Worked example 31. Question: Simplify (4/5):(5/6). Method: Scale both terms by the denominator product 30. The ratio becomes 24:25; divide by HCF 1 to get 24:25. Check: Their relative quotient is 24/25, consistent with the original two fractions. A common trap: Do not cancel a number in just one ratio term.

Share two amounts

Worked example 32. Question: Share $176 in ratio 5:6. Method: There are 11 parts, each worth $16. The two shares are $80 and $96. Check: The shares add to $176, and their ratio is 5:6. A common trap: The divisor for a total share is the sum of all parts.

Share three amounts

Worked example 33. Question: Split 216 counters into three groups in ratio 5:6:7. Method: The 18 parts are each 12 counters. The three groups have 60, 72, 84. Check: Their total is 216, preserving all three proportions. A common trap: Leaving out one group changes the whole number of parts.

Combine two ratios

Worked example 34. Question: If A:B=5:6 and B:C=7:8, find A:B:C. Method: Scale A:B by 7, giving 35:42; scale B:C by 6, giving 42:48. The combined simplest ratio is 35:42:48. Check: Both original pairwise ratios reduce correctly from the combined comparison. A common trap: Shared B parts have to match before appending three groups.

Recipe serving factors

Worked example 35. Question: A four-serving recipe uses 420 g of flour. How much flour for six servings with direct proportional scaling? Method: Serving factor is 6/4=1.5; multiply flour by it: 420×1.5=630 g. Check: Dividing the new amount by 1.5 returns 420 g. A common trap: Do not add a fixed amount without regard to the serving factor.

Units in a length ratio

Worked example 36. Question: Simplify 3 metres : 150 centimetres. Method: Convert metres to centimetres: 3 m=300 cm. Then 300:150=2:1. Check: The first length is exactly twice the second. A common trap: Different units must be reconciled before reduction.

Recover an unknown share

Worked example 37. Question: A:B=6:8 and B receives $64. Find A’s amount. Method: One part is $64/8=$8, so A receives 6 parts: $48. Check: The actual amounts 48:64 reduce to 6:8. A common trap: B is not the entire total unless the context explicitly says so.

Syrup and water in a mixture

Worked example 38. Question: Syrup:water=5:8, with 150 mL per part. Find both amounts and their total. Method: Syrup=750 mL, water=1200 mL and total=1950 mL. Check: The components add to 1950 mL and retain the specified ratio. A common trap: The syrup’s fraction of the whole is 5/13, not 5/8.

A ratio as a fraction of the whole

Worked example 39. Question: In A:B=6:7, what fraction of both groups together is A? Method: The total has 13 parts, so A’s share is 6/13. Check: The two group fractions 6/13 and 7/13 add to one. A common trap: The total-part denominator is 13, not 7.

When one group gains members

Worked example 40. Question: Actual A:B counts are 60:70. Five people join A. Find the new simplest ratio. Method: The new counts are 65:70. Divide both by HCF 5 to obtain 13:14. Check: The first count gained five, and the second stayed 70. A common trap: Five extra people cannot be added directly to the abstract ratio part 6.

Six avoidable mistakes and how to repair them

1. Treating 2:3 as though the first group is 2/3 of the total

If this happens, identify the first mathematical assumption that stopped being true. Rewrite that line, explain the relevant definition, then solve a similar problem with different numbers. A correction is complete when the learner can avoid the same error without the tutor giving the next step. Use a reverse check wherever the mathematical structure permits it.

2. Simplifying a ratio before converting incompatible units

3. Dividing a total by one named part instead of all parts

4. Joining A:B and B:C without making B match

5. Scaling one ingredient but not the others in a direct-proportion recipe

6. Stopping at two shares without checking they add to the total

What a helpful Bukit Timah small-group Maths tutorial looks like

For parents exploring a Secondary 3 G3 Mathematics tutor in Bukit Timah, the important questions are whether the teaching fits the learner’s current school chapter, whether the tutor diagnoses misconceptions precisely, and whether the learner becomes less dependent on hints. A three-pupil model may give room for individual explanation, but actual class format and availability should be confirmed directly. Class size by itself does not guarantee learning.

A lesson can start with a five-minute retrieval check, then a concept explanation using one clear example. The tutor offers an unfamiliar contrast question, observes the learner’s first decision, and provides a correction tied to the specific wrong step. After the pupil solves independently, a later return reveals whether the method was retained. The outcome to watch is controlled reasoning and self-checking, not merely more completed pages.

The unchanged Secondary 1 Mathematics Tutor Clementi small-group guide describes a foundation-level teaching approach; this is not a claim that its classes cover the Secondary 3 G3 syllabus. For current year-level context, consult the Secondary 3 G3 Mathematics tuition Bukit Timah overview.

A practical six-week study sequence

Week 1 — diagnose

Bring a marked school question and a new unseen problem. Identify whether the obstacle is reading the question, selecting a method, or carrying out the calculation. For this article the target is making quantities comparable, calculating total parts and preserving a ratio through scaling. Store one dated original and one later independent attempt so progress is based on visible evidence.

Week 2 — explain the principle

Practise one mathematical relationship through two contrasting examples. Write the student’s own explanation beside a clean worked solution. For this article the target is making quantities comparable, calculating total parts and preserving a ratio through scaling. Store one dated original and one later independent attempt so progress is based on visible evidence.

Week 3 — vary the numbers and story

Use multiple representations of the same principle. Do not allow the worksheet heading to decide the method for the student. For this article the target is making quantities comparable, calculating total parts and preserving a ratio through scaling. Store one dated original and one later independent attempt so progress is based on visible evidence.

Week 4 — build an independent check

Require a factor-product check, ratio-total check, divisibility check or other appropriate verification after every completed solution. For this article the target is making quantities comparable, calculating total parts and preserving a ratio through scaling. Store one dated original and one later independent attempt so progress is based on visible evidence.

Week 5 — practise mixed retrieval

Bring back a selection of earlier problems after several days, interleaved with another chapter. Notice whether the first step is still available. For this article the target is making quantities comparable, calculating total parts and preserving a ratio through scaling. Store one dated original and one later independent attempt so progress is based on visible evidence.

Week 6 — measure transfer

Give unseen questions and compare independent accuracy, reasons for the first step and recurring errors with the Week 1 sample. For this article the target is making quantities comparable, calculating total parts and preserving a ratio through scaling. Store one dated original and one later independent attempt so progress is based on visible evidence.

Questions parents can ask at home

Three brief prompts help without taking over the Mathematics: What is the question asking for? Why is the chosen method appropriate? What would make this answer impossible or unreasonable? If the learner can answer calmly and independently, the understanding is becoming portable. Parents need not calculate every answer to notice whether the explanations are improving.

A lower mark is not by itself a diagnosis, just as a higher mark on repeated worksheet questions is not proof of lasting mastery. Compare the original solution with an unseen question solved several days later. Then choose one small weakness to repair before adding more material.

Frequently asked questions

Does A:B=2:3 mean A is 2/3 of the total?

No. The two groups together have five parts, so A is 2/5 of the total. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

What is (1/2):(2/3) in whole-number terms?

Three to four, after multiplying both parts by six. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

Can a ratio contain decimals?

Yes. Convert to equivalent whole-number parts where useful. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

Why convert units first?

The two numbers must describe comparable quantities before they can be reduced. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

What is one ratio part?

The common amount represented by each unit in the simplified comparison. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

How do I combine two ratios?

Scale them until the shared quantity has the same number of parts in both comparisons. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

What is a good sharing check?

Verify the shares sum to the total and preserve the original ratio. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

Can a recipe always scale exactly?

A textbook direct-proportion question assumes it; real cooking may involve additional practical conditions. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

How do I find A when B is known?

Find the value of one B-part, then multiply it by A’s number of parts. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

What if the total includes discrete objects?

Check whether each share is a valid whole-number count under the question’s conditions. A useful next step is to test this explanation on a new numerical case and ask the learner to justify the first operation without prompts.

The official references are the 2027 SEAB G3 syllabus page and the K310 Mathematics syllabus PDF. Readers should use their own school’s scheme of work to decide the next most useful chapter.

The takeaway

Did you know? Some of the most important progress is not visible in the number of questions completed. It appears when the learner can name what they are looking for, choose a method for a reason and verify the answer independently. Focused Mathematics support in Bukit Timah should help make those habits routine across changing question types.