11 plus equivalent fractions and simplifying questions

What a Fractions: Basics & Equivalents question looks like

This is the fraction topic that sits underneath all the others in 11 plus maths, and what marks it out from its neighbours is that the fraction itself is the object of the question. Fractions of Amounts works out what a fraction of a quantity comes to, Fractions of Shapes reads one off a divided diagram, and conversion questions rewrite it as a decimal or a percentage. Here the fraction stays a fraction: whether two are worth the same, whether one can be cut down further, and how a handful of them order by size.

Every question gives five options and one is right, and most of the bank sits in four shapes: an equivalent pair with a number missing, such as 3/5 = 9/?, for the missing top or bottom number; a fraction to put in its simplest form, or a list holding one option that is already simplest; which fraction, or which pair, is genuinely equivalent; and which of two fractions is larger, or four or five of them written out in order.

Around those sit word problems that put the same skills into a real situation, a few that attach a small table, a bar chart, a number line or a Venn diagram to read the numbers from, and a few put as a claim to judge: Marcus says that 2/5 and 4/10 are not equivalent, is he correct? Each option there carries a reason as well as a yes or a no, and the reason has to hold up too.

Example one: the missing denominator in 3/5 = 9/?

Start with the pair you can see both of, the two top numbers. 3 has become 9, so it has been multiplied by 3. An equivalent fraction is made by doing the same to the top and the bottom, so the bottom is multiplied by 3 too: 5 x 3 = 15. The answer is 15.

The wrong option that catches most children is 25, from multiplying 5 by the number already in front of them rather than by the multiplier just worked out. 20 catches the child who noticed a jump but never checked its size, and 10 the child who doubled out of habit. All three go away with one discipline: find the multiplier, then use that and nothing else.

Example two: simplify 18/24 to its lowest terms

Find the biggest number that goes into both 18 and 24. 2 and 3 both go, but 6 is the biggest. Divide both parts by it: 18 divided by 6 is 3, and 24 divided by 6 is 4. The answer is 3/4.

Two of the wrong options are not wrong about the value at all. 9/12 comes from dividing both numbers by 2 and stopping, and 6/8 from dividing both by 3 and stopping. Each is genuinely equal to 18/24, and each is wrong only because the question asked for lowest terms: that child has the method and stopped a step early. The option to worry about is 2/3, which is not equal to 18/24 at all, because it comes from dividing the top and the bottom by different numbers.

Example three: 1/2, 3/8, 5/8 and 1/4 in order from smallest to largest

Fractions with different bottom numbers cannot be compared as they stand, so give them all the same bottom number. Eighths work here, because both 2 and 4 go into 8. 1/2 is 4/8 and 1/4 is 2/8, while 3/8 and 5/8 are in eighths already. Now the top numbers go in order: 2/8, 3/8, 4/8, 5/8, which written back is 1/4, 3/8, 1/2, 5/8.

The tempting wrong answer starts 1/4, 3/8 correctly and then puts 5/8 ahead of 1/2. It comes from converting the quarter, stopping there and judging the half by eye, and by eye a half looks like the big one because it has the smallest bottom number. Another option leaves all four in the order the question listed them, which is the same error underneath: comparing before converting.

How to answer it

1. Read what the question actually wants. An equivalent, the simplest form, or an order. Option lists here often hold two or three fractions worth exactly the same amount, so “in its simplest form” is doing real work.
2. For a missing number, find the multiplier from the pair you can see. Ask what the top was multiplied or divided by to become the new top, then do exactly the same to the bottom.
3. For simplifying, divide by the biggest number that goes into both, then look again. If the biggest is hard to spot, halve twice, or divide by 3 and keep going. The answer is finished only when nothing but 1 goes into both parts.
4. For comparing or ordering, give everything the same bottom number. Look for a number all the bottoms go into: 4 and 8 give 8, 5 and 10 give 10, 6 and 9 and 18 give 18. Then put the top numbers in order.
5. Read the direction once more before choosing. Smallest to largest and largest to smallest both come up, and the exact reverse of the right answer is sometimes there as an option.

Common mistakes

• Adding instead of multiplying. Turning 1/2 into eighths, the bottom doubles twice, 2 to 4 to 8, rather than gaining 6. A child who adds 6 to the top as well gets 7/8, nowhere near a half.
• Stopping one step short when simplifying. 18/24 taken to 9/12 or 6/8, or 8/12 taken to 4/6. Each is a correct equivalent and none of them is in lowest terms.
• Changing the top and the bottom by different amounts. This is the error behind most of the fake equivalent pairs: 4/7 and 8/21 doubles the top and triples the bottom, 3/4 and 9/16 triples the top and quadruples the bottom.
• Judging size by the top number alone, or the bottom alone. 7/10 is not bigger than 3/4 because 7 beats 3, and out of 1/2, 1/3, 1/4 and 1/5 the smallest is 1/5.

How to practise 11 plus Fractions: Basics & Equivalents questions

The most useful thing here is a habit rather than more questions: whenever your child gives a fraction as an answer, ask whether it can be cut down further. Most of the lost marks in this topic are a step short rather than plain wrong. Times tables do more work than anything else, because finding the biggest number that goes into both parts is recall, not reasoning. Beyond that, short and regular beats long and occasional, which is the pattern we set out in how much 11 plus practice a child needs each day, and the rest of eleven plus fraction work is built on top of this topic. 11+ Daily has a Fractions: Basics & Equivalents project, on laptop, tablet or phone, and the full list sits on our 11 plus maths practice page. Every question is original to 11+ Daily, nothing is a past paper and nothing is licensed from another publisher, and the bank is re-checked in rounds.

Questions parents ask