To learn Fractions quickly and correctly, start by understanding the size of the parts, not by memorising four disconnected rules. A fraction names a quantity relative to a whole: the denominator tells us how many equal parts make one whole, and the numerator tells us how many of those parts we have. This eduKateSG Quick Notes guide teaches easy, accurate fraction methods with worked examples and a short practice routine for Primary and Secondary Mathematics.
- Identify the whole: check what the fraction is a part of.
- Compare sizes: use a number line, bar model or equivalent fractions.
- Add or subtract: first rename fractions with the same denominator.
- Multiply: multiply numerators and denominators, then simplify.
- Divide: multiply by the reciprocal of a non-zero divisor.
- Check: estimate whether the answer should be smaller, equal or larger.
First, picture the fraction. If a pizza is divided into four equal slices, 3/4 means three of the four pieces. The parts must be equal, and the whole must stay the same while you compare fractions. Draw 1/2 and 2/4 using bars of equal length. They cover the same amount, so 1/2 = 2/4. That is the meaning of equivalent fractions. Multiplying both numerator and denominator by the same non-zero number changes the name, not the value.
Second, learn why addition and subtraction require equal-sized parts. You cannot immediately add 1/2 and 1/3 by saying the answer is 2/5, because halves and thirds are different-sized pieces. Rename both using sixths: 1/2 = 3/6 and 1/3 = 2/6. Now the pieces match, so 1/2 + 1/3 = 5/6. The denominator stays 6 during that addition because we are counting sixths. When adding or subtracting fractions, the numerator counts how many of these equal pieces remain.
Third, connect multiplication and division to meaning. 2/3 × 3/5 asks for two-thirds of three-fifths, so it equals 6/15 = 2/5. Division asks how many groups of one size fit inside another: 3/4 ÷ 1/2 asks how many halves are in three-quarters. The answer is 3/2, or one and a half halves. The reciprocal procedure works for division by a non-zero fraction, but checking its meaning stops the rule from feeling like magic.
Worked example: subtract unlike fractions
Solve 3/4 − 1/6. The lowest common denominator is 12. Rename 3/4 as 9/12 and 1/6 as 2/12. Then 9/12 − 2/12 = 7/12. Check with a rough estimate: 3/4 is 0.75, and 1/6 is a little under 0.2, so an answer slightly over 0.55 makes sense. The exact 7/12 is about 0.583.
Use this short decision guide
- Same denominator? Add or subtract the numerators; keep that denominator, then simplify if possible.
- Different denominators? Find a common denominator before adding or subtracting.
- Multiplication? Multiply across, or cancel common factors first when valid.
- Division? Check the divisor is non-zero, invert that divisor and multiply.
- Mixed number? For multiplication and division, changing to an improper fraction usually makes the calculation clearer.
The five-minute drawing method
Use a simple bar for any fraction that feels abstract. Draw one identical whole for both quantities, split it into equal parts and shade what the numerator describes. For 2/3 versus 3/5, split each bar into fifteenths: 2/3 = 10/15, while 3/5 = 9/15. You can now see that 2/3 is larger than 3/5 by 1/15. The diagram does not replace formal working; it explains why the formal working is sensible.
Common fraction mistakes and the repair
- 1/2 + 1/3 = 2/5: incorrect; find sixths first, giving 5/6.
- Adding denominators during fraction addition: incorrect; equal-sized parts keep their shared denominator.
- Cancelling terms across an addition sign: invalid; cancellation belongs to common factors, not arbitrary terms.
- Forgetting which number to invert in division: invert the divisor, not the first fraction.
- Comparing different wholes: 1/2 of a large cake may exceed 3/4 of a small cake; state the whole.
Try it yourself
- Find 2/5 + 1/4.
- Find 3/4 of 20.
- Find 2/3 × 3/5 and give the simplest form.
Answers and why
1. 2/5 = 8/20 and 1/4 = 5/20, so the sum is 13/20. 2. Divide 20 into 4 equal groups (5 each), then take 3 groups: 15. 3. 6/15 simplifies to 2/5. Each answer can be checked by estimation or by a bar drawing.
How to remember fractions tomorrow
Do one addition, one multiplication and one comparison question from memory the next day. Explain why the denominator behaves differently in addition and multiplication. Keep any error in a tiny correction log with three parts: the incorrect step, the correct rule and a fresh question. That turns the answer into knowledge you can retrieve later.
Where to go next
Continue with the related lessons: Mathematics · Negative Numbers · Algebra.
For the longer learning pathway, read Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. This Quick Notes lesson is a separate, short learning resource; the reference page remains unchanged.