How G3 G2 G1 Works | Fractions, Ratios and Percentages: Which Number Is the Whole?

Quick answer for students and parents. Most difficult fractions, ratios and percentage word problems are not difficult because of multiplication. They become confusing when the student loses track of which quantity is the whole, which is a part, and which amount is being used as the base. Before calculating, write the relationship in words: “this is two parts out of five”, “this is 20% of the original price”, or “this is one-third of what remains”. Only then choose a fraction or formula.

Students studying Mathematics at G1, G2 and G3 under Singapore’s Full Subject-Based Banding (Full SBB) may meet these concepts at different levels of demand. The problems below teach general mathematical structure; they are original teaching examples, not official 2027 SEC questions or a universal G-level syllabus comparison. Check the correct subject and topic in the SEAB 2027 syllabuses.

A fast diagnostic: when your child sees a percentage, ask “Percent of what?” When they see a fraction, ask “A fraction of which whole?” When they see a ratio, ask “Which two quantities are being compared, and is the total one of them?” Many mistakes become visible before the first calculation.

1. Why three familiar representations still cause confusion

Fractions, ratios and percentages all represent relationships, but their reference quantities need careful attention. In a recipe, two cups of rice to three cups of water is a part-to-part ratio of 2:3. Rice is two-fifths of the combined five cups only if the quantities are treated as directly addable volume measures for that comparison. Writing 2/3 for the rice share of the combined total confuses the water amount with the whole.

A percentage is a way of expressing a proportion per hundred. It needs a base. “20%” is incomplete for a real decision unless we know 20% of what. Twenty percent of $80 is $16; twenty percent of $50 is $10. The same percentage produces different amounts because the whole changes.

A fraction can also be based on a changing whole. If a student uses one-third of the money remaining, the base is not necessarily the original amount. The word remaining changes the denominator’s meaning even when the fraction stays one-third.

Strong teaching makes these relationships explicit before introducing a long calculation sequence. This follows eduKate’s immutable Secondary 1 Mathematics Tutor Clementi reference: locate the first unstable concept, bridge concrete and symbolic representations, and check independent reasoning.

2. Four questions to ask before choosing an operation

1. What is the whole? Is it the entire class, original price, combined mixture, initial distance or the remaining amount?
2. Which parts are being compared? Is this a ratio of boys to girls, boys to all students, or two stages of a process?
3. Has the whole changed? Was something added, removed, discounted or used before the next calculation?
4. What is the question asking? Does it need a part, a total, a ratio, a percentage, a final price or an original price?

Answer those questions in plain language first. The calculation should follow the meaning, not determine it.

3. Worked ratio problem: part-to-part is not part-to-whole

Question: A school club has boys and girls in the ratio 2:3. There are 30 students altogether. How many boys and girls are there?

The ratio contains 2 + 3 = 5 equal parts. One part represents 30 ÷ 5 = 6 students. Boys: 2 × 6 = 12. Girls: 3 × 6 = 18. Check 12 + 18 = 30 and 12:18 simplifies to 2:3.

Now ask three different questions about the same club:

• Ratio of boys to girls: 12:18 = 2:3.
• Fraction of the club who are boys: 12/30 = 2/5.
• Percentage of the club who are boys: (12/30) × 100% = 40%.

All three are correct, but they answer different relationships. The 2/3 ratio compares boys with girls; the 2/5 fraction compares boys with the entire group. A child who substitutes 2/3 into a question asking for the fraction of the whole has chosen the wrong denominator.

Transfer: If the club’s boy-to-girl ratio stays 2:3 but the total rises to 45, one part equals 9, so boys = 18 and girls = 27. The actual counts change, but the share of boys remains 2/5 = 40%. The student should be able to explain what changed and what stayed constant.

4. Worked ratio problem: the given amount is only one part

Question: A jar contains red and blue counters in the ratio 2:3. There are 18 blue counters. How many red counters?

Here 18 corresponds to three ratio parts, not the combined five parts. One part is 18 ÷ 3 = 6, so red counters = 2 × 6 = 12. The total is 30.

A common mistake is dividing 18 by 5, because the student remembers that ratios should be added before dividing. That rule applies when the given quantity is the combined total; here the given quantity is just the blue part.

Key diagnostic: ask the learner to point to the quantity represented by 18 on a bar model. If they place it across all five parts, the error is in representing the whole. Repeating more multiplication drills does not repair this misunderstanding. Draw two red blocks and three blue blocks, label the blue group 18, and only then calculate.

5. Worked percentage problem: 20% off is not 20% of the final price

Question: A bag costs $80 before a 20% discount. What price does the customer pay?

The original price is the base: 20% × $80 = $16 discount. The final price is $80 − $16 = $64. Equivalently, the customer pays 80% of the original price: 0.8 × $80 = $64.

Reverse question: A bag costs $64 after a 20% discount. What was the original price? If $64 represents 80% of the original price, then original price = $64 ÷ 0.8 = $80. Multiplying $64 by 1.2 gives $76.80, which is not the original amount because 20% of the final price is not equal to 20% of the original price.

This is a common confusion in reverse-percentage questions. The operations may appear symmetrical, but they use different bases. Ask the student to label the original price as 100%, the discount as 20% of that original and the final price as 80% of it.

6. Why a 20% increase followed by a 20% decrease does not cancel

Start with $100. Increase it by 20%: $100 × 1.2 = $120. Then decrease the result by 20%: $120 × 0.8 = $96. The overall result is $4 below the starting value.

Why? The first 20% was based on $100, so it added $20. The second 20% was based on $120, so it removed $24. Equal percentage labels do not mean equal dollar amounts when the base changes.

To return from $120 to $100 requires a decrease of $20 on a $120 base: 20 ÷ 120 × 100% = approximately 16.67%. This is a useful demonstration that percentage increase and decrease are not simply inverse operations with the same percentage when the base changes.

Ask the child to try a different starting value, such as $50. Increase by 20% to $60, then decrease 20% to $48. The absolute difference changes, but the final amount remains 96% of the original. The multiplicative relationship is stable.

7. Fractions of a remainder: the whole changes midway

Question: A student has $60. They spend one-third of the money on a book. Then they spend one-quarter of the remaining money on lunch. How much is left?

First expense = 1/3 × $60 = $20. Money remaining = $60 − $20 = $40. The second expense is one-quarter of the remaining $40, so lunch costs 1/4 × $40 = $10. Final amount left = $40 − $10 = $30.

A frequent wrong answer calculates 1/4 of the original $60 as $15. That overlooks the word remaining. The arithmetic 60 ÷ 4 = 15 is correct for a different whole but wrong for the question’s changed base.

Represent the process as a timeline: $60 original → spend $20 → $40 remaining → spend $10 → $30 final. The second fraction attaches to the $40 stage, not the initial $60. A simple diagram or labelled line can prevent the loss of meaning.

8. Ratios can be equivalent even when actual quantities differ

A ratio of 2:3 is equivalent to 4:6 and 10:15. The paired quantities have the same multiplicative relationship, but the totals are different: 5, 10 and 25 parts at the respective scales. Equivalent ratios preserve relative proportions, not necessarily actual counts.

Suppose a recipe uses rice and water in a ratio of 2:3 by a stated volume measure. With 4 cups of rice, the corresponding water amount is 6 cups. Scaling both parts by two preserves the ratio. Increasing only the rice amount would change the ratio.

Do not transfer a practical recipe ratio to all foods or every cooking method without checking context. Actual cooking depends on ingredient, equipment and technique. The mathematical example teaches proportional scaling, not a universal cooking rule.

9. Percent of a group versus percent of a subgroup

Consider a school with 100 students. Forty are in a club; 15 of those forty have volunteered at an event. Then:

• 15 out of the 40 club members volunteered: 37.5% of club members.
• 15 out of all 100 school students volunteered through this club: 15% of the school.

Both statements are compatible. One uses the club as the denominator; the other uses the entire school. A student who reads “15 students volunteered” and automatically selects 15% may be correct only if the intended base is 100.

The same confusion appears in public health, marketing, surveys and news. “Twenty percent of surveyed households” is not automatically twenty percent of all households in a country. Ask who or what was included in the denominator.

10. A comparison table: which representation answers which question?

This table makes visible why an answer can look mathematically familiar but still fail the question. The numbers are linked; the requested relationship determines which representation is correct.

11. What to do when the child knows the method but still gets the base wrong

Ask the student to explain each number in ordinary language. What does the 30 represent? The whole club. What does the 18 represent? The girls only. What does 2/3 compare? Boys with girls. What does 2/5 compare? Boys with the whole club.

Then change the question without changing the underlying relation. Give the number of girls instead of the total; or give the final discounted price instead of the original. If the child keeps reusing the first calculation, they may have memorised a procedure without recognising which whole the problem supplies.

For a general approach to unfamiliar mathematical stories, use How to Solve Mathematics Word Problems Without Guessing the Formula. If a concept remains unstable, see Find the Missing Prerequisite Before Relearning a Chapter.

12. Is this different at G1, G2 and G3?

G1, G2 and G3 are subject levels broadly mapped from the former N(T), N(A) and Express standards under Full SBB. A student may take different subjects at different levels. The mathematical principles of ratio, fraction and percentage retain their meaning across settings, while the syllabus content, abstract demand and assessment format differ.

For the 2027 SEC, Mathematics appears in official subject directories as G1 K110, G2 K210 and G3 K310. A learner can benefit from a simpler illustration even when studying a more demanding subject level, but the practice must eventually return to the required syllabus.

The illustrations here are not a claim that each topic or question form appears in every G-level paper. Match the examples to the child’s school year and subject teacher’s instruction.

13. A fifteen-minute parent practice session

1. Draw a bar of five equal parts and label two parts “boys” and three “girls”. Ask for the ratio and fraction of the whole.
2. Give a total of 30 and ask for actual numbers in each group.
3. Change the given information to 18 girls, and ask the student what the 18 represents.
4. Give $80 with a 20% discount; find the final price.
5. Reverse the problem: final price $64 after 20% off; find the original price.
6. Ask for one sentence explaining why the base changed between the two questions.

The timings are flexible; stop if the child no longer understands the relationship. Success is measured by the ability to explain a fresh situation independently, not by completing the largest number of questions.

14. A quick correction template

When a fraction or percentage answer is wrong, ask the student to complete four fields: the given base, the part compared, the required relation, and the calculation that now follows. For the 20% discount question, those fields are “original $80”, “discount $16”, “find final price”, and “80 − 16 = 64”.

For the reverse discount problem, the base is an unknown original price, while the known $64 represents 80% of it. This prevents the child from treating $64 as the original 100% and multiplying the wrong base by 20%.

15. Frequently asked questions

How do I know which number is the whole in a percentage question?

Identify what represents 100% at the stage being discussed. It might be the original price, entire group or remaining amount. The words in the question determine the base.

Why is a 2:3 ratio not the same as two-thirds of the total?

In a part-to-part ratio of 2:3, the total has five parts. The first quantity is 2/5 of the whole, while 2/3 compares it with the second quantity.

Why doesn’t a 20% increase followed by a 20% decrease return to the original?

The increase and decrease use different bases. From $100, the increase adds $20; from $120, the decrease removes $24, leaving $96.

How do you solve a reverse-percentage problem?

Identify what percentage of the original the known amount represents. If $64 is 80% of the original, divide $64 by 0.8 to find $80.

Are students at G1 allowed to learn more advanced ratio questions?

Extra exploration can be useful when foundations are secure, but the required curriculum and assessment should remain appropriate to the actual subject level and year. Do not substitute arbitrary harder papers for teaching.

Will more practice papers solve denominator confusion?

Only if practice includes understanding and correcting the specific base error. First use bars, diagrams and contrasting examples, then check transfer on new questions.

Continue reading

• Mathematics word problems: read, model, solve and check
• How to use formula sheets without guessing
• How to interpret chart percentages and denominators
• Full SBB G-level master guide
• Official 2027 SEC syllabus directory

Editorial review: 7 November 2026. Calculations and word problems are original examples, independently checked for consistency. Full SBB and 2027 Mathematics syllabus information follows MOE and SEAB official materials. This page is not a claim that every G-level paper tests identical ratio or percentage content.