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Equivalent fractions and simplifying
Using the highest common factor to reduce a fraction to its lowest terms.
A fraction is a part of a whole. Lots of different fractions can stand for the same amount: half a pizza is the same amount of pizza as two quarters. In this lesson you will learn:
- what the top and bottom numbers of a fraction mean
- what equivalent fractions are, and how to spot them on a fraction wall
- how to make equivalent fractions by multiplying the top and bottom by the same number
- how to find a missing number in two equivalent fractions
- how to simplify a fraction to its simplest form, using the highest common factor
1. What a fraction means
This pizza is cut into 4 equal slices. 3 of them are shaded. So 34 of the pizza is shaded.
numerator: the top number. It counts the parts you have.
denominator: the bottom number. It tells you how many equal parts make the whole.
Pizza maker: cut the pizza, then tap slices to shade them
2. Equivalent fractions on a fraction wall
Each row of the wall is one whole, cut into equal bricks. Follow the red line down: every row where the line lands exactly on the edge of a brick shows an equal fraction.
12 = 24 = 36 = 48 = 510 = 612. The thirds don’t land exactly on the line, so no number of thirds is equal to 12.
Try it: tap any brick. The wall shades that many bricks in its row and shows every fraction that is equal to it. Try 23, 34 and 25.
3. Making equivalent fractions
Example: 23 of a chocolate bar
23: 2 of 3 pieces
Cut each piece into 4: 8 of 12 pieces
Cutting every piece into 4 makes 4 times as many pieces altogether (3 × 4 = 12) and 4 times as many shaded pieces (2 × 4 = 8). The amount of chocolate does not change.
Cut the bar again: the shaded amount never changes
Cut each piece into:
4. Finding a missing number
Going up: multiply
- The bottoms are both known: 4 × 5 = 20.
- Do the same to the top: 3 × 5 = 15.
Going down: divide
- The tops are both known: 18 ÷ 6 = 3.
- Do the same to the bottom: 24 ÷ 6 = 4.
Check by going back the other way: 15 ÷ 5 = 3 and 20 ÷ 5 = 4, so 1520 is the same as 34.
Missing number workshop
- First find what the known pair was multiplied or divided by, then do the same to find the missing number.
- Two in a row with no slips moves you up a level. Level 5 has three fractions in a row.
5. Simplifying to the simplest form
Way 1: step by step
Simplify 2436
- Both are even, so divide by 2.2436 = 1218
- Both are still even: divide by 2 again.1218 = 69
- 6 and 9 are both in the 3 times table.69 = 23
- Only 1 goes into 2 and 3. Stop.23
Way 2: one step with the HCF
The highest common factor of 24 and 36 is 12 (Lesson 1.6). Divide both by 12:
The same answer in one step. If you can’t spot the HCF, Way 1 always gets there.
Year 6 tip: simplify first to find a missing number
69 = ?12 There is no whole number that multiplies 9 to make 12.
- Simplify first: 69 = 23.
- Now 3 × 4 = 12, so 2 × 4 = 8.
Simplifying steps: divide until you can’t
6. Equivalent or not? And fractions in stories
Are 68 and 912 equivalent?
- 68 = 34 (divide by 2)
- 912 = 34 (divide by 3)
A fraction from a story
There are 30 children in a class. 12 walk to school. What fraction of the class walk to school?
- The part goes on top, the whole goes on the bottom: 1230.
- The HCF of 12 and 30 is 6: 1230 = 25.
- The rest do not walk: 30 − 12 = 18, so 1830 = 35 do not walk.
Odd one out: three of these fractions are equal. Which one is not?
7. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Use rough paper and write each step. Move up when you are getting most of them right.
The mistakes that cost marks
- Adding the same number to the top and bottom. Writing 23 = 34 because 1 was added to both. only multiplying or dividing both by the same number keeps a fraction the same size.
- Doing something to only one number. Writing 34 = 320. whatever you do to the bottom, do to the top as well: 34 = 1520.
- Stopping too soon. Giving 1218 as the simplest form of 2436. check again: 12 and 18 are both even. Keep dividing until only 1 goes into both, giving 23.
- Using the shaded and unshaded parts. Writing 31 for a pizza with 3 of 4 slices shaded. the bottom number is all the equal parts in the whole: 34.
- Answering the wrong part of a story. Giving the fraction who walk when the question asks who do not. read the question again, and take the part it asks about away from the whole if you need to.
- Leaving the answer unsimplified. Writing 1230 when the question says “in its simplest form”. always finish by checking: does any number except 1 go into both?
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 3.1.
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