Sec 2 G1 Maths Tutor Bukit Timah

Did you know? A ratio can look straightforward until one quantity is written as a fraction and the other as a decimal. The problem is not that ratio has changed its meaning. It is that the two amounts must first be expressed in compatible units or representations. A strong Secondary 2 G1 Mathematics learner sees the same comparison behind 1/2:3/4, 0.5:0.75 and 2:3.

Quick answer: how do you simplify fractional and decimal ratios?

For a ratio containing fractions, multiply both terms by a common denominator. For a ratio containing decimals, multiply both terms by the same suitable power of ten. When both quantities are whole numbers, divide them by their greatest common factor. The order of the terms remains important: A:B is not interchangeable with B:A. If the two quantities are measured in different units, convert into one shared unit before simplifying.

This article supports families looking for a Secondary 2 G1 Mathematics tutor in Bukit Timah. The published 2027 SEC G1 Mathematics K110 syllabus explicitly includes ratios involving fractions and decimals as well as dividing a quantity in a given ratio. The MOE 2020 Secondary Two G1 content emphasises direct and inverse proportion, so this guide treats fractions and decimal ratios as prerequisite consolidation and a bridge rather than predicting they are the next chapter in every Secondary 2 school.

A one-minute ratio diagnostic

Ask a pupil to simplify 1/2:3/4, 0.4:0.6 and 250 mL:0.5 L. The correct answers are 2:3, 2:3 and 1:2. The first checks fraction equivalence, the second checks common decimal scaling, and the third checks unit conversion. If the learner can do the first two but not the third, do not prescribe a whole new ratio unit; begin by matching millilitres and litres.

Fractional ratio: two equivalent ways to see it

For 2/3:3/4, use twelfths. Two thirds equals eight twelfths and three quarters equals nine twelfths, so the ratio is 8:9. Alternatively divide 2/3 by 3/4, giving (2/3)×(4/3)=8/9; this is the same first-to-second quotient. The two methods agree because the ratio describes one number relative to the other, not a total.

The main trap is comparing only numerators. Two thirds and three quarters have different unit fractions: one is built from thirds and the other from quarters. Writing 2:3 would silently treat their denominators as equal. A fraction strip or common-denominator grid can make the correct parts visible before the child works independently with symbols.

Decimal ratio: multiply both amounts equally

For 0.45:0.6, multiply both by 100 to obtain 45:60, then reduce to 3:4. Multiplying only the first amount would change the ratio. A pupil who chooses 4.5:6 has also preserved the relationship, but the conventional requested form is usually simplest whole-number terms. The right response should keep the order and be checkable by the original quotient 0.45/0.6=0.75.

Part-to-part is not automatically part-to-whole

A juice:water ratio of 2:3 means two units of juice for every three of water. It does not mean that juice is two thirds of the whole. There are five ratio parts in the combined mixture, so juice is two fifths and water three fifths of the total. This distinction matters whenever a question asks for a fraction of the mixture or for amounts from a given total.

To share 40 items in ratio 3:5, count eight equal ratio parts. Each part is five items, so the two shares are 15 and 25. The independent check is twofold: the numbers must total forty and simplify back to 3:5. Passing only one condition is not enough. This habit transfers to classroom modelling, recipes and financial arithmetic.

75 original fractional and decimal ratio clinics

Use these examples in small groups. The first sets simplify fractions and decimals, then turn ratios into quantities and finally ask whether a contextual interpretation is justified. They are original mathematical teaching questions, not reused examination material or real business offers. Cover the worked answer first, identify which operation is needed and make the check aloud.

1. Fraction-ratio clinic 1: 1/2 to 1/3

Question: Simplify the ratio 1/2 : 1/3 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 6; the fractions are 3/6 and 2/6. Ratio therefore becomes 3:2, which reduces by common factor 1 to 3:2. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 1:1 would compare numerators as though a 2th and a 3th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 3:2, also simplifying to 3:2. Numerically the quotient of first to second is 1.5, equal to 3/2.

Independent transfer: If the second fraction doubled while the first stayed fixed, its share relative to the first would grow and the ratio would change. Ask the pupil to predict the direction before calculating a changed ratio.

2. Fraction-ratio clinic 2: 2/3 to 3/4

Question: Simplify the ratio 2/3 : 3/4 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 12; the fractions are 8/12 and 9/12. Ratio therefore becomes 8:9, which reduces by common factor 1 to 8:9. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 2:3 would compare numerators as though a 3th and a 4th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 8:9, also simplifying to 8:9. Numerically the quotient of first to second is 0.8888888888888888, equal to 8/9.

3. Fraction-ratio clinic 3: 3/5 to 1/2

Question: Simplify the ratio 3/5 : 1/2 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 10; the fractions are 6/10 and 5/10. Ratio therefore becomes 6:5, which reduces by common factor 1 to 6:5. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 3:1 would compare numerators as though a 5th and a 2th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 6:5, also simplifying to 6:5. Numerically the quotient of first to second is 1.2, equal to 6/5.

4. Fraction-ratio clinic 4: 2/5 to 3/10

Question: Simplify the ratio 2/5 : 3/10 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 10; the fractions are 4/10 and 3/10. Ratio therefore becomes 4:3, which reduces by common factor 1 to 4:3. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 2:3 would compare numerators as though a 5th and a 10th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 20:15, also simplifying to 4:3. Numerically the quotient of first to second is 1.3333333333333335, equal to 4/3.

5. Fraction-ratio clinic 5: 3/4 to 5/8

Question: Simplify the ratio 3/4 : 5/8 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 8; the fractions are 6/8 and 5/8. Ratio therefore becomes 6:5, which reduces by common factor 1 to 6:5. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 3:5 would compare numerators as though a 4th and a 8th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 24:20, also simplifying to 6:5. Numerically the quotient of first to second is 1.2, equal to 6/5.

6. Fraction-ratio clinic 6: 4/9 to 2/3

Question: Simplify the ratio 4/9 : 2/3 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 9; the fractions are 4/9 and 6/9. Ratio therefore becomes 4:6, which reduces by common factor 2 to 2:3. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 4:2 would compare numerators as though a 9th and a 3th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 12:18, also simplifying to 2:3. Numerically the quotient of first to second is 0.6666666666666666, equal to 2/3.

7. Fraction-ratio clinic 7: 5/6 to 1/4

Question: Simplify the ratio 5/6 : 1/4 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 12; the fractions are 10/12 and 3/12. Ratio therefore becomes 10:3, which reduces by common factor 1 to 10:3. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 5:1 would compare numerators as though a 6th and a 4th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 20:6, also simplifying to 10:3. Numerically the quotient of first to second is 3.3333333333333335, equal to 10/3.

8. Fraction-ratio clinic 8: 7/10 to 2/5

Question: Simplify the ratio 7/10 : 2/5 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 10; the fractions are 7/10 and 4/10. Ratio therefore becomes 7:4, which reduces by common factor 1 to 7:4. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 7:2 would compare numerators as though a 10th and a 5th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 35:20, also simplifying to 7:4. Numerically the quotient of first to second is 1.7499999999999998, equal to 7/4.

9. Fraction-ratio clinic 9: 1/6 to 2/9

Question: Simplify the ratio 1/6 : 2/9 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 18; the fractions are 3/18 and 4/18. Ratio therefore becomes 3:4, which reduces by common factor 1 to 3:4. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 1:2 would compare numerators as though a 6th and a 9th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 9:12, also simplifying to 3:4. Numerically the quotient of first to second is 0.75, equal to 3/4.

10. Fraction-ratio clinic 10: 5/12 to 7/18

Question: Simplify the ratio 5/12 : 7/18 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 36; the fractions are 15/36 and 14/36. Ratio therefore becomes 15:14, which reduces by common factor 1 to 15:14. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 5:7 would compare numerators as though a 12th and a 18th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 90:84, also simplifying to 15:14. Numerically the quotient of first to second is 1.0714285714285714, equal to 15/14.

11. Fraction-ratio clinic 11: 3/8 to 5/16

Question: Simplify the ratio 3/8 : 5/16 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 16; the fractions are 6/16 and 5/16. Ratio therefore becomes 6:5, which reduces by common factor 1 to 6:5. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 3:5 would compare numerators as though a 8th and a 16th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 48:40, also simplifying to 6:5. Numerically the quotient of first to second is 1.2, equal to 6/5.

12. Fraction-ratio clinic 12: 4/15 to 2/5

Question: Simplify the ratio 4/15 : 2/5 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 15; the fractions are 4/15 and 6/15. Ratio therefore becomes 4:6, which reduces by common factor 2 to 2:3. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 4:2 would compare numerators as though a 15th and a 5th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 20:30, also simplifying to 2:3. Numerically the quotient of first to second is 0.6666666666666666, equal to 2/3.

13. Fraction-ratio clinic 13: 7/12 to 5/6

Question: Simplify the ratio 7/12 : 5/6 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 12; the fractions are 7/12 and 10/12. Ratio therefore becomes 7:10, which reduces by common factor 1 to 7:10. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 7:5 would compare numerators as though a 12th and a 6th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 42:60, also simplifying to 7:10. Numerically the quotient of first to second is 0.7000000000000001, equal to 7/10.

14. Fraction-ratio clinic 14: 2/7 to 3/14

Question: Simplify the ratio 2/7 : 3/14 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 14; the fractions are 4/14 and 3/14. Ratio therefore becomes 4:3, which reduces by common factor 1 to 4:3. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 2:3 would compare numerators as though a 7th and a 14th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 28:21, also simplifying to 4:3. Numerically the quotient of first to second is 1.3333333333333333, equal to 4/3.

15. Fraction-ratio clinic 15: 5/9 to 7/12

Question: Simplify the ratio 5/9 : 7/12 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 36; the fractions are 20/36 and 21/36. Ratio therefore becomes 20:21, which reduces by common factor 1 to 20:21. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 5:7 would compare numerators as though a 9th and a 12th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 60:63, also simplifying to 20:21. Numerically the quotient of first to second is 0.9523809523809523, equal to 20/21.

16. Fraction-ratio clinic 16: 7/8 to 3/4

Question: Simplify the ratio 7/8 : 3/4 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 8; the fractions are 7/8 and 6/8. Ratio therefore becomes 7:6, which reduces by common factor 1 to 7:6. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 7:3 would compare numerators as though a 8th and a 4th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 28:24, also simplifying to 7:6. Numerically the quotient of first to second is 1.1666666666666667, equal to 7/6.

17. Fraction-ratio clinic 17: 9/10 to 3/5

Question: Simplify the ratio 9/10 : 3/5 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 10; the fractions are 9/10 and 6/10. Ratio therefore becomes 9:6, which reduces by common factor 3 to 3:2. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 9:3 would compare numerators as though a 10th and a 5th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 45:30, also simplifying to 3:2. Numerically the quotient of first to second is 1.5, equal to 3/2.

18. Fraction-ratio clinic 18: 2/3 to 4/9

Question: Simplify the ratio 2/3 : 4/9 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 9; the fractions are 6/9 and 4/9. Ratio therefore becomes 6:4, which reduces by common factor 2 to 3:2. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 2:4 would compare numerators as though a 3th and a 9th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 18:12, also simplifying to 3:2. Numerically the quotient of first to second is 1.5, equal to 3/2.

19. Fraction-ratio clinic 19: 5/8 to 1/3

Question: Simplify the ratio 5/8 : 1/3 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 24; the fractions are 15/24 and 8/24. Ratio therefore becomes 15:8, which reduces by common factor 1 to 15:8. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 5:1 would compare numerators as though a 8th and a 3th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 15:8, also simplifying to 15:8. Numerically the quotient of first to second is 1.875, equal to 15/8.

20. Fraction-ratio clinic 20: 3/7 to 2/21

Question: Simplify the ratio 3/7 : 2/21 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 21; the fractions are 9/21 and 2/21. Ratio therefore becomes 9:2, which reduces by common factor 1 to 9:2. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 3:2 would compare numerators as though a 7th and a 21th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 63:14, also simplifying to 9:2. Numerically the quotient of first to second is 4.5, equal to 9/2.

21. Fraction-ratio clinic 21: 7/15 to 4/5

Question: Simplify the ratio 7/15 : 4/5 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 15; the fractions are 7/15 and 12/15. Ratio therefore becomes 7:12, which reduces by common factor 1 to 7:12. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 7:4 would compare numerators as though a 15th and a 5th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 35:60, also simplifying to 7:12. Numerically the quotient of first to second is 0.5833333333333333, equal to 7/12.

22. Fraction-ratio clinic 22: 5/16 to 3/8

Question: Simplify the ratio 5/16 : 3/8 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 16; the fractions are 5/16 and 6/16. Ratio therefore becomes 5:6, which reduces by common factor 1 to 5:6. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 5:3 would compare numerators as though a 16th and a 8th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 40:48, also simplifying to 5:6. Numerically the quotient of first to second is 0.8333333333333334, equal to 5/6.

23. Fraction-ratio clinic 23: 1/4 to 3/5

Question: Simplify the ratio 1/4 : 3/5 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 20; the fractions are 5/20 and 12/20. Ratio therefore becomes 5:12, which reduces by common factor 1 to 5:12. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 1:3 would compare numerators as though a 4th and a 5th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 5:12, also simplifying to 5:12. Numerically the quotient of first to second is 0.4166666666666667, equal to 5/12.

24. Fraction-ratio clinic 24: 5/14 to 3/7

Question: Simplify the ratio 5/14 : 3/7 to the smallest positive whole-number terms. Explain what equal-sized unit you used.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both quantities need the same denominator before the pair can be read as whole-number parts. The least common denominator is 14; the fractions are 5/14 and 6/14. Ratio therefore becomes 5:6, which reduces by common factor 1 to 5:6. The ratio compares the first quantity with the second, not their sum.

The earliest incorrect step: Simply removing the two denominator numbers is invalid when they differ. Writing 5:3 would compare numerators as though a 14th and a 7th were the same-sized piece.

Check: Cross-multiplying the original fractions gives 35:42, also simplifying to 5:6. Numerically the quotient of first to second is 0.8333333333333334, equal to 5/6.

25. Decimal-ratio clinic 1: 0.3 to 0.6

Question: Simplify 0.3 : 0.6 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 10, making the comparison 3:6. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 3 to get 1:2.

The earliest incorrect step: Multiplying only 0.3 but not 0.6 by 10 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.3/0.6=0.5 and 1/2=0.5. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 1:2; only the units on the individual quantities change.

26. Decimal-ratio clinic 2: 0.25 to 0.75

Question: Simplify 0.25 : 0.75 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 25:75. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 25 to get 1:3.

The earliest incorrect step: Multiplying only 0.25 but not 0.75 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.25/0.75=0.3333333333333333 and 1/3=0.3333333333333333. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 1:3; only the units on the individual quantities change.

27. Decimal-ratio clinic 3: 0.4 to 1.2

Question: Simplify 0.4 : 1.2 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 10, making the comparison 4:12. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 4 to get 1:3.

The earliest incorrect step: Multiplying only 0.4 but not 1.2 by 10 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.4/1.2=0.33333333333333337 and 1/3=0.3333333333333333. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

28. Decimal-ratio clinic 4: 1.5 to 0.9

Question: Simplify 1.5 : 0.9 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 10, making the comparison 15:9. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 3 to get 5:3.

The earliest incorrect step: Multiplying only 1.5 but not 0.9 by 10 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 1.5/0.9=1.6666666666666665 and 5/3=1.6666666666666667. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 5:3; only the units on the individual quantities change.

29. Decimal-ratio clinic 5: 0.8 to 0.32

Question: Simplify 0.8 : 0.32 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 80:32. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 16 to get 5:2.

The earliest incorrect step: Multiplying only 0.8 but not 0.32 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.8/0.32=2.5 and 5/2=2.5. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 5:2; only the units on the individual quantities change.

30. Decimal-ratio clinic 6: 0.45 to 0.6

Question: Simplify 0.45 : 0.6 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 45:60. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 15 to get 3:4.

The earliest incorrect step: Multiplying only 0.45 but not 0.6 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.45/0.6=0.75 and 3/4=0.75. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 3:4; only the units on the individual quantities change.

31. Decimal-ratio clinic 7: 0.12 to 0.2

Question: Simplify 0.12 : 0.2 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 12:20. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 4 to get 3:5.

The earliest incorrect step: Multiplying only 0.12 but not 0.2 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.12/0.2=0.6 and 3/5=0.6. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 3:5; only the units on the individual quantities change.

32. Decimal-ratio clinic 8: 2.4 to 1.6

Question: Simplify 2.4 : 1.6 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 10, making the comparison 24:16. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 8 to get 3:2.

The earliest incorrect step: Multiplying only 2.4 but not 1.6 by 10 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 2.4/1.6=1.4999999999999998 and 3/2=1.5. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 3:2; only the units on the individual quantities change.

33. Decimal-ratio clinic 9: 0.075 to 0.125

Question: Simplify 0.075 : 0.125 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 1000, making the comparison 75:125. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 25 to get 3:5.

The earliest incorrect step: Multiplying only 0.075 but not 0.125 by 1000 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.075/0.125=0.6 and 3/5=0.6. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

34. Decimal-ratio clinic 10: 1.25 to 0.5

Question: Simplify 1.25 : 0.5 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 125:50. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 25 to get 5:2.

The earliest incorrect step: Multiplying only 1.25 but not 0.5 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 1.25/0.5=2.5 and 5/2=2.5. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

35. Decimal-ratio clinic 11: 3.6 to 0.9

Question: Simplify 3.6 : 0.9 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 10, making the comparison 36:9. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 9 to get 4:1.

The earliest incorrect step: Multiplying only 3.6 but not 0.9 by 10 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 3.6/0.9=4 and 4/1=4. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 4:1; only the units on the individual quantities change.

36. Decimal-ratio clinic 12: 0.04 to 0.14

Question: Simplify 0.04 : 0.14 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 4:14. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 2 to get 2:7.

The earliest incorrect step: Multiplying only 0.04 but not 0.14 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.04/0.14=0.2857142857142857 and 2/7=0.2857142857142857. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 2:7; only the units on the individual quantities change.

37. Decimal-ratio clinic 13: 2.75 to 1.1

Question: Simplify 2.75 : 1.1 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 275:110. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 55 to get 5:2.

The earliest incorrect step: Multiplying only 2.75 but not 1.1 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 2.75/1.1=2.5 and 5/2=2.5. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

38. Decimal-ratio clinic 14: 0.35 to 0.28

Question: Simplify 0.35 : 0.28 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 35:28. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 7 to get 5:4.

The earliest incorrect step: Multiplying only 0.35 but not 0.28 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.35/0.28=1.2499999999999998 and 5/4=1.25. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 5:4; only the units on the individual quantities change.

39. Decimal-ratio clinic 15: 0.625 to 0.375

Question: Simplify 0.625 : 0.375 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 1000, making the comparison 625:375. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 125 to get 5:3.

The earliest incorrect step: Multiplying only 0.625 but not 0.375 by 1000 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.625/0.375=1.6666666666666667 and 5/3=1.6666666666666667. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

40. Decimal-ratio clinic 16: 0.18 to 0.27

Question: Simplify 0.18 : 0.27 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 18:27. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 9 to get 2:3.

The earliest incorrect step: Multiplying only 0.18 but not 0.27 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.18/0.27=0.6666666666666666 and 2/3=0.6666666666666666. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

Independent transfer: Try expressing the same amounts in a different common unit, such as both in millilitres instead of litres. The ratio should stay 2:3; only the units on the individual quantities change.

41. Decimal-ratio clinic 17: 0.072 to 0.048

Question: Simplify 0.072 : 0.048 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 1000, making the comparison 72:48. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 24 to get 3:2.

The earliest incorrect step: Multiplying only 0.072 but not 0.048 by 1000 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.072/0.048=1.4999999999999998 and 3/2=1.5. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

42. Decimal-ratio clinic 18: 0.5 to 1.75

Question: Simplify 0.5 : 1.75 and explain why the decimal points can be cleared without changing the ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Multiply both components by 100, making the comparison 50:175. Both were scaled by the same positive multiplier, so their relationship is unchanged. Divide the whole-number terms by their common factor 25 to get 2:7.

The earliest incorrect step: Multiplying only 0.5 but not 1.75 by 100 changes the relationship. Rounding one entry to a nearby whole number before simplifying can also produce an incorrect ratio.

Check: Compare the quotients 0.5/1.75=0.2857142857142857 and 2/7=0.2857142857142857. They are equal. If your simplified pair had a different quotient, the scale operation was not applied equally.

43. Fractional ratio sharing 1: find both parts

Question: Two imaginary groups have a ratio of 1/2 : 1/4. Divide a total of 36 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 2:1. The parts total 3; each ratio part is 36/(3)=12 units. Group A receives 2×12=24 units; group B receives 1×12=12 units.

The earliest incorrect step: A common mistake is to multiply the whole by 1/2 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 24+12=36. Their ratio 24:12 simplifies to 2:1, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 72 units but the ratio stayed fixed, the shares would become 48 and 24 units. Explain why both amounts scale but the ratio itself does not.

44. Fractional ratio sharing 2: find both parts

Question: Two imaginary groups have a ratio of 2/3 : 1/2. Divide a total of 42 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 4:3. The parts total 7; each ratio part is 42/(7)=6 units. Group A receives 4×6=24 units; group B receives 3×6=18 units.

The earliest incorrect step: A common mistake is to multiply the whole by 2/3 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 24+18=42. Their ratio 24:18 simplifies to 4:3, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 84 units but the ratio stayed fixed, the shares would become 48 and 36 units. Explain why both amounts scale but the ratio itself does not.

45. Fractional ratio sharing 3: find both parts

Question: Two imaginary groups have a ratio of 3/4 : 1/2. Divide a total of 40 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 3:2. The parts total 5; each ratio part is 40/(5)=8 units. Group A receives 3×8=24 units; group B receives 2×8=16 units.

The earliest incorrect step: A common mistake is to multiply the whole by 3/4 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 24+16=40. Their ratio 24:16 simplifies to 3:2, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 80 units but the ratio stayed fixed, the shares would become 48 and 32 units. Explain why both amounts scale but the ratio itself does not.

46. Fractional ratio sharing 4: find both parts

Question: Two imaginary groups have a ratio of 1/3 : 2/5. Divide a total of 110 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 5:6. The parts total 11; each ratio part is 110/(11)=10 units. Group A receives 5×10=50 units; group B receives 6×10=60 units.

The earliest incorrect step: A common mistake is to multiply the whole by 1/3 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 50+60=110. Their ratio 50:60 simplifies to 5:6, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 220 units but the ratio stayed fixed, the shares would become 100 and 120 units. Explain why both amounts scale but the ratio itself does not.

47. Fractional ratio sharing 5: find both parts

Question: Two imaginary groups have a ratio of 5/6 : 1/2. Divide a total of 40 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 5:3. The parts total 8; each ratio part is 40/(8)=5 units. Group A receives 5×5=25 units; group B receives 3×5=15 units.

The earliest incorrect step: A common mistake is to multiply the whole by 5/6 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 25+15=40. Their ratio 25:15 simplifies to 5:3, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 80 units but the ratio stayed fixed, the shares would become 50 and 30 units. Explain why both amounts scale but the ratio itself does not.

48. Fractional ratio sharing 6: find both parts

Question: Two imaginary groups have a ratio of 3/5 : 1/4. Divide a total of 136 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 12:5. The parts total 17; each ratio part is 136/(17)=8 units. Group A receives 12×8=96 units; group B receives 5×8=40 units.

The earliest incorrect step: A common mistake is to multiply the whole by 3/5 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 96+40=136. Their ratio 96:40 simplifies to 12:5, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 272 units but the ratio stayed fixed, the shares would become 192 and 80 units. Explain why both amounts scale but the ratio itself does not.

49. Fractional ratio sharing 7: find both parts

Question: Two imaginary groups have a ratio of 7/10 : 1/2. Divide a total of 72 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 7:5. The parts total 12; each ratio part is 72/(12)=6 units. Group A receives 7×6=42 units; group B receives 5×6=30 units.

The earliest incorrect step: A common mistake is to multiply the whole by 7/10 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 42+30=72. Their ratio 42:30 simplifies to 7:5, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 144 units but the ratio stayed fixed, the shares would become 84 and 60 units. Explain why both amounts scale but the ratio itself does not.

50. Fractional ratio sharing 8: find both parts

Question: Two imaginary groups have a ratio of 1/8 : 3/4. Divide a total of 28 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 1:6. The parts total 7; each ratio part is 28/(7)=4 units. Group A receives 1×4=4 units; group B receives 6×4=24 units.

The earliest incorrect step: A common mistake is to multiply the whole by 1/8 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 4+24=28. Their ratio 4:24 simplifies to 1:6, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 56 units but the ratio stayed fixed, the shares would become 8 and 48 units. Explain why both amounts scale but the ratio itself does not.

51. Fractional ratio sharing 9: find both parts

Question: Two imaginary groups have a ratio of 5/9 : 1/3. Divide a total of 48 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 5:3. The parts total 8; each ratio part is 48/(8)=6 units. Group A receives 5×6=30 units; group B receives 3×6=18 units.

The earliest incorrect step: A common mistake is to multiply the whole by 5/9 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 30+18=48. Their ratio 30:18 simplifies to 5:3, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 96 units but the ratio stayed fixed, the shares would become 60 and 36 units. Explain why both amounts scale but the ratio itself does not.

52. Fractional ratio sharing 10: find both parts

Question: Two imaginary groups have a ratio of 3/8 : 5/16. Divide a total of 33 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 6:5. The parts total 11; each ratio part is 33/(11)=3 units. Group A receives 6×3=18 units; group B receives 5×3=15 units.

The earliest incorrect step: A common mistake is to multiply the whole by 3/8 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 18+15=33. Their ratio 18:15 simplifies to 6:5, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 66 units but the ratio stayed fixed, the shares would become 36 and 30 units. Explain why both amounts scale but the ratio itself does not.

53. Fractional ratio sharing 11: find both parts

Question: Two imaginary groups have a ratio of 2/7 : 3/14. Divide a total of 35 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 4:3. The parts total 7; each ratio part is 35/(7)=5 units. Group A receives 4×5=20 units; group B receives 3×5=15 units.

The earliest incorrect step: A common mistake is to multiply the whole by 2/7 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 20+15=35. Their ratio 20:15 simplifies to 4:3, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 70 units but the ratio stayed fixed, the shares would become 40 and 30 units. Explain why both amounts scale but the ratio itself does not.

54. Fractional ratio sharing 12: find both parts

Question: Two imaginary groups have a ratio of 3/4 : 5/8. Divide a total of 77 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 6:5. The parts total 11; each ratio part is 77/(11)=7 units. Group A receives 6×7=42 units; group B receives 5×7=35 units.

Check: The two shares total 42+35=77. Their ratio 42:35 simplifies to 6:5, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 154 units but the ratio stayed fixed, the shares would become 84 and 70 units. Explain why both amounts scale but the ratio itself does not.

55. Fractional ratio sharing 13: find both parts

Question: Two imaginary groups have a ratio of 7/12 : 5/6. Divide a total of 68 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 7:10. The parts total 17; each ratio part is 68/(17)=4 units. Group A receives 7×4=28 units; group B receives 10×4=40 units.

The earliest incorrect step: A common mistake is to multiply the whole by 7/12 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 28+40=68. Their ratio 28:40 simplifies to 7:10, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 136 units but the ratio stayed fixed, the shares would become 56 and 80 units. Explain why both amounts scale but the ratio itself does not.

56. Fractional ratio sharing 14: find both parts

Question: Two imaginary groups have a ratio of 4/15 : 2/5. Divide a total of 50 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 2:3. The parts total 5; each ratio part is 50/(5)=10 units. Group A receives 2×10=20 units; group B receives 3×10=30 units.

The earliest incorrect step: A common mistake is to multiply the whole by 4/15 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 20+30=50. Their ratio 20:30 simplifies to 2:3, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 100 units but the ratio stayed fixed, the shares would become 40 and 60 units. Explain why both amounts scale but the ratio itself does not.

57. Fractional ratio sharing 15: find both parts

Question: Two imaginary groups have a ratio of 5/16 : 3/8. Divide a total of 88 units between them using that ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First simplify the fractional ratio to 5:6. The parts total 11; each ratio part is 88/(11)=8 units. Group A receives 5×8=40 units; group B receives 6×8=48 units.

The earliest incorrect step: A common mistake is to multiply the whole by 5/16 as though that fraction were Group A’s fraction of the total. The original ratio compares two parts; it does not give either part of the whole directly.

Check: The two shares total 40+48=88. Their ratio 40:48 simplifies to 5:6, which matches the original fractional comparison. Both conditions must hold.

Independent transfer: If the total doubled to 176 units but the ratio stayed fixed, the shares would become 80 and 96 units. Explain why both amounts scale but the ratio itself does not.

58. Context check 1: Same ratio, different actual amounts

Question: A is 0.5 L and B is 0.75 L. Compare A:B and decide whether amounts A=1 L, B=1.5 L have the same ratio.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. First 0.5:0.75 becomes 2:3; the second 1:1.5 also becomes 2:3. Both describe the same relative amounts, although the actual totals differ.

The earliest incorrect step: Do not confuse equivalent ratios with equal physical quantities.

Check: Multiply each pair by two and see that only absolute size changes.

Independent transfer: For a changed-number version, identify which quantity comes first, write units beside each amount, and ask whether the ratio is part-to-part or a fraction of a whole. The answer should remain explainable without relying on a formula memorised out of context.

59. Context check 2: Ratio between parts is not fraction of total

Question: Juice:water ratio is 2:3. What fraction of the combined mixture is juice?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. There are five combined parts and two are juice. Juice fraction of total is 2/5, not 2/3.

The earliest incorrect step: A part-to-part ratio has a different denominator from a part-to-whole fraction.

Check: Check water is 3/5, and the fractions add to one.

60. Context check 3: Convert millilitres to litres before ratio

Question: A bottle contains 250 mL of liquid A and 0.5 L of liquid B. Find A:B.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. 0.5 L=500 mL, so ratio 250:500=1:2.

The earliest incorrect step: Using the raw numerals 250:0.5 compares unlike units.

Check: Check both volumes in litres: 0.25:0.5=1:2.

61. Context check 4: Simplify a ratio with both quantities negative?

Question: Compare -4 and -6 as signed values; should a real-world size ratio be treated like the positive ratio 2:3?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. A standard positive-quantity ratio description applies to nonnegative magnitudes. For signed quantities, the comparison and interpretation must be stated explicitly rather than silently treating debts and amounts as positive lengths.

The earliest incorrect step: Dropping negative signs automatically can change the meaning of a financial balance.

Check: Discuss whether the problem asks for magnitudes or signed numerical quotients before simplifying.

62. Context check 5: Comparing two rates is not always ratio sharing

Question: Two fictional machines produce 12 and 18 items per hour. What is the output-rate ratio?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. The rates share a unit, so 12:18 simplifies to 2:3. This compares production rates; it does not by itself state how many items are currently held.

The earliest incorrect step: Treating a rate ratio as a division of an existing total without a total being given.

Check: Multiply both rates by one hour to see the 12:18 outputs under equal durations.

63. Context check 6: Equivalent ratio with a decimal multiplier

Question: Find an equivalent ratio to 3:5 where first term is 4.5.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Scale 3 by 1.5 to get 4.5, so scale 5 equally: 7.5. Ratio is 4.5:7.5.

The earliest incorrect step: Scaling only one ratio component changes the comparison.

Check: Divide 4.5/7.5=0.6, equal to 3/5.

64. Context check 7: Order of terms changes meaning

Question: If red:blue=2:5, is blue:red=2:5?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. No. Blue:red is 5:2; the order of labelled quantities reverses.

The earliest incorrect step: Reading ratios as sets of unordered numbers loses which quantity is first.

Check: Put the labels beside each component before simplifying.

65. Context check 8: Zero ratio parts need real-world interpretation

Question: If A:B=0:3, what does it say about nonnegative quantities?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. A is zero and B is positive in the usual ratio setting. The ratio simplifies to 0:1; it is not division by zero.

The earliest incorrect step: Calling it impossible simply because a zero appears can misinterpret a valid zero part.

Check: Distinguish 0:3 from 3:0; the latter cannot be interpreted as an ordinary finite quotient A/B.

66. Context check 9: Find total from one fractional ratio share

Question: The ratio A:B is 3:5 and A contains 21 items. Find total.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. One ratio part is 21/3=7; B has 35 and total is 56.

The earliest incorrect step: Dividing 21 by 8 treats the known A amount as the whole rather than a part.

Check: Check A=3/8 of 56=21.

67. Context check 10: Ratio from percentages

Question: A hypothetical class contains 40% Group A and 60% Group B. Find A:B.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. 40:60 simplifies to 2:3. The two percentages share the same whole.

The earliest incorrect step: Computing 40/100:60 gives unlike denominators.

Check: Use a fictional total of 10: A=4, B=6.

68. Context check 11: Convert mixed number before forming ratio

Question: Find 1½ : 2¼ in whole-number terms.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Write 1.5:2.25 or 6/4:9/4, giving 6:9=2:3.

The earliest incorrect step: Treating 1½ as 1.2 or 2¼ as 2.4 changes the ratio.

Check: Multiply both quantities by four to make six and nine.

69. Context check 12: Ratio with zero whole?

Question: A question says ‘divide zero items in ratio 2:3’. What shares result?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both groups receive zero under the proportion formula. However, the ratio of the resulting actual shares is 0:0, which does not exhibit the given nonzero ratio as an ordinary quotient; specify the degenerate total.

The earliest incorrect step: Claiming a 2:3 ratio has been directly observed in two zero quantities overstates the meaning.

Check: Check the total is zero and the arithmetic gives zero in both groups.

70. Context check 13: Ratios after part added

Question: A:B is initially 2:3 with A=8 and B=12. Two more units are added only to A. What is the new ratio?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. New amounts are 10 and 12, so ratio simplifies to 5:6. The original 2:3 ratio no longer holds.

The earliest incorrect step: Adding the same two to both sides despite the story naming only A changes the scenario.

Check: Check the new amounts and simplify their direct ratio.

71. Context check 14: A ratio can compare lengths at different units

Question: A plan line is 4 cm; the corresponding actual length is 2 m. What is drawing:actual length ratio using matching units?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Two metres is 200 cm, so drawing:actual is 4:200=1:50.

The earliest incorrect step: Writing 4:2 without unit conversion produces the wrong dimensionless scale.

Check: Check 1 cm drawing maps to 50 cm real.

72. Context check 15: Ratio is not an arithmetic difference

Question: The ratio of two amounts is 2:5 and their total is 42. Find the difference.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Seven parts make 42, so one part is 6. The amounts are 12 and 30; their difference is 18.

The earliest incorrect step: Treating the ratio entries two and five as the actual amounts would give difference three.

Check: Check difference is three ratio parts × six each.

73. Context check 16: More than 100 percent as ratio

Question: Quantity A is 125% of B. Give A:B in simplest integer terms.

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. A/B=1.25=5/4, so A:B=5:4.

The earliest incorrect step: Assuming percent cannot exceed 100 fails when comparing one quantity to another.

Check: At B=80, A=100, whose ratio is 5:4.

74. Context check 17: Compare two ratio plans with different totals

Question: Plan A divides 40 into 1:3, Plan B divides 60 into 1:3. Are the shares the same?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. Both have the same ratio, but Plan A shares are 10 and 30 while Plan B shares are 15 and 45.

The earliest incorrect step: Equating relative shares with absolute amounts ignores different totals.

Check: Both first shares remain one quarter of their own totals.

75. Context check 18: Scale ratio by negative multiplier?

Question: In a physical quantities ratio 3:4, should multiplying both by -2 be used to report negative amounts?

Did you know? A ratio compares like quantities before they are rewritten in whole-number form. In ordinary positive-quantity ratios, equivalent positive pairs are used. Signed quantities require a clear model; do not use a negative multiplier to turn nonnegative counts negative.

The earliest incorrect step: Blind algebra can produce a pair with no valid real-world count meaning.

Check: Ask whether the context permits signed values, such as financial net balances.

Five recurring mistakes worth diagnosing

Only one ratio term gets scaled

When 0.3:0.6 becomes 3:0.6, the first quantity was multiplied by ten while the second was not. The resulting ratio is not equivalent. A tutor can write two arrows labelled ×10, one for each term, and ask whether the first-to-second quotient stayed constant. The corrected 3:6 simplifies to 1:2.

Unlike fractional units are ignored

A learner simplifies 1/2:3/4 to 1:3 by removing denominators. Re-express as quarters to obtain 2/4:3/4=2:3. Alternatively take the quotient (1/2)/(3/4)=2/3. The two representations should agree, confirming that the denominators could not simply be discarded.

Units were never matched

If 250 mL:0.5 L is written 250:0.5, the numeral comparison is meaningless as a dimensionless length-or-volume ratio. Convert 0.5 L to 500 mL or 250 mL to 0.25 L. Either route gives 1:2. The error is unit preparation rather than a lack of whole-number ratio skill.

A part-to-part ratio is mistaken for a percentage of total

A:B=2:3 means A is two parts for every three parts of B, but A as a fraction of the whole is 2/5. A tutor can label five equal boxes and shade two for A. Repeating the question with the known total included checks whether the student uses the correct denominator.

A reversed ratio silently changes the answer

A:B=2:5 is not B:A=2:5. The latter is 5:2 when the order is reversed. Write names beside the two values before reducing, especially in word problems about ingredients or class groups. A correct numerical pair in the wrong order can answer a different comparison.

How a focused three-pupil Mathematics lesson can approach ratios

The immutable eduKateSG Clementi three-pupil Maths tutorial reference describes 90-minute sessions near Sixth Avenue MRT. One pupil might need fractional-unit repair, another decimal scaling, and a third part-to-whole word problems. A useful lesson keeps the core concept shared while each learner works at an appropriate entry point. The source is a programme-format reference, not confirmation of current Secondary 2 G1 slots.

10 minutes: retrieve fraction equivalence and converting common metric units.
20 minutes: show equal-unit models for a fractional ratio and a decimal ratio.
25 minutes: individually simplify mixed ratios and diagnose the first wrong line.
15 minutes: turn one ratio into actual shares with a given total.
10 minutes: independent varied problems without chapter or method cues.
10 minutes: reconstruct the original ratio and schedule a short retest.

Five short parent–student practice sessions

Day 1: simplify three fraction ratios by common denominators.
Day 2: simplify three decimal ratios by scaling both terms equally.
Day 3: convert millilitres and litres before comparing quantities.
Day 4: divide one whole into the specified ratio and check both sum and comparison.
Day 5: practise a mixed question with the term order reversed and explain the difference.

A parent can ask ‘What are we comparing and in which order?’ or ‘Did you multiply both terms by the same amount?’ before checking the final answer. That short reasoning check helps without turning ratio practice into a long copying exercise. If the child is uncertain with unlike fractions, return to equivalent fractions first.

G1 Mathematics and the syllabus-year distinction

SEAB lists K110 as G1 Mathematics for the 2027 Secondary Education Certificate, with 4046 as the earlier reference code. The published N2 Ratio and Proportion strand covers ratios involving fractions and decimals. That 2027 document is a graduating-cohort specification, not a statement that a 2026 Secondary 2 child sits the paper next year. The school’s current chapter sequence and the eventual official examination year remain decisive.

Frequently asked questions

Why do we multiply fractions in ratios by a common denominator?

It expresses both quantities in the same-sized unit parts, making their comparison a whole-number ratio before further reduction.

Can ratios contain decimals?

Yes. They express valid comparisons. Questions often ask for simplest whole-number form, achieved by scaling both terms equally.

What is the difference between 2:3 and 2/3 of a whole?

The ratio is a comparison between two parts. When those are all the parts, the first quantity is 2/5 of their combined total.

How should I handle a ratio mixing mL and L?

Convert both quantities to a common unit before simplifying. Do not compare the bare numbers from incompatible units.

Is every ratio question direct proportion?

No. Some ask for part-to-part comparison, some for sharing a fixed total, and others for a relationship in which one quantity varies directly. Read the exact relationship rather than applying one rule automatically.

Does tuition guarantee better grades?

No. Ask whether the student can independently simplify a new fractional ratio, interpret it correctly and check a changed-total sharing question after a delay.

Official curriculum and accuracy notes

Reader note: All quantities, groups and ratios used here are original instructional examples. They do not describe actual school performance, business offers or examination questions. The order of ratio terms and common physical units are part of the mathematical meaning. Confirm tuition opening, syllabus match, fees, venue and individual suitability directly.

Bring one fraction-ratio error from the pupil’s current worksheet to eduKate Singapore and ask how the tutor would test the meaning of the ratio before increasing question difficulty.

Secondary 2 G1 Mathematics Tutor Bukit Timah | Ratios with Fractions and Decimals