11 plus Equations & Missing Numbers questions

What an Equations & Missing Numbers question looks like

Equations & Missing Numbers is the algebra strand of 11 plus maths, and it is gentler than the word algebra suggests. Every question hides a number behind a letter or a question mark and asks what belongs there. What marks this type out from the other work a child does with unknown numbers is that the letter stays in the question to the end, so the answer is sometimes an expression such as 2x minus 7 rather than a number.

The live questions come in a handful of recognisable shapes. Some give a line of maths and ask for the letter: a − 9 = 10, or 3n + 4 = 19, or 5a − 7 = 28. Some hide the number behind a question mark, such as 13 × ? = 156 or (360 ÷ ?) + 5 = 25. Some give a value and ask what an expression comes to, like 3a + 2 when a is 5, or a pay formula in pence that must be turned into pounds. Some ask what a bracket is the same as, or ask for the letter in an equation containing one.

Others tell a short story, about stickers or bags of apples or the girls and boys in a class, and the child builds the line of maths before solving it. A few are shape questions where the unknown is a side: the width of a rectangle from its perimeter, the side of a regular pentagon, the long side of a kite. A few ask for the expression itself, and one has the letter on both sides. The numbers stay small and the arithmetic is Year 5 arithmetic. The difficulty sits in deciding which operation to undo, and in which order.

Example one: find the letter

What is the value of n if 3n + 4 = 19?

3n means 3 lots of n. A number has been multiplied by 3, then 4 added, and the result is 19. Undo the last thing first: take 4 off both sides, which leaves 3n = 15. Now undo the multiplying. 15 split into 3 equal lots is 5.

The answer is 5. Checking takes a second: 3 lots of 5 is 15, and 15 add 4 is 19.

23 is the tempting wrong answer, because 19 + 4 = 23. A child moving quickly sees a 4 and adds it. The rule is that you do the opposite of whatever the question did, and it added, so you take away.

Example two: find the letter, then do something with it

If 5 lots of x equal 60, what are 3 lots of x?

Two things are being asked, and the second is easy to miss. First find x: 60 split into 5 equal lots is 12. Now answer what was actually asked, 3 lots of x, which is 3 × 12 = 36.

The answer is 36.

Two wrong answers are waiting. 12 is there for the child who finds x and stops, which feels like finishing because a number has appeared. 20 is there for the child who goes straight at the total and works out 60 ÷ 3, skipping x. Both come from answering a slightly different question.

Example three: choose the expression

Ben picks x apples. Ella picks four fewer than Ben. Which expression shows how many they picked altogether? The options are 2x − 4, x − 4, 2x, x + 4 and 2x + 4.

Nothing here works out to a number, because nobody has said what x is, and that is the point. Write down each person on their own. Ben is x. Ella is four fewer, so Ella is x − 4. Altogether means add the two: x + (x − 4). Two lots of x is 2x, and the 4 still comes off.

The answer is 2x − 4.

2x tempts because it is right about there being two children and wrong about Ella picking fewer. x − 4 tempts because it is Ella’s amount on its own, a real quantity but not the one asked for.

How to answer it

1. Find the letter or the question mark, and decide what the question wants: a number, or an expression. If none of the options is a number, no arithmetic will help.
2. If it is a story, turn it into a line of maths first. Write what each person or thing is worth, using the letter, then join them up.
3. Peel the operations off in reverse order, doing the opposite of each one. Adding undoes taking away, dividing undoes multiplying. Whatever you do to one side, do to the other.
4. Read the question again before choosing. It may want 2 lots of x, or x halved, or the cost of five bags, not the letter you just found. Check the units too, because one sort ends in pence and asks for pounds.
5. Put your answer back into the original line and see that it works. This catches almost every slip on this type, and takes ten seconds.

Common mistakes

• Doing the same operation instead of the opposite one. If the question added 4, you take 4 away. Every set contains an option built from going the wrong way, usually the roundest-looking number there.
• Stopping halfway. In 5a − 7 = 28, adding 7 gives 35, and 35 is sitting there in the options. Finding x when the question asked for 3 lots of x is the same error in different clothes.
• Only multiplying the first thing in a bracket. 4(n + 5) is 4n + 20, not 4n + 5, because the 4 multiplies everything inside. Buried in a longer expression, the slip is harder to spot.
• Reading 3a as 3 + a. A number written next to a letter means multiply. When a is 5, 3a + 2 is 17, not 10.
• Losing the sign in an expression. Four fewer means take four away, so 2x − 4, never 2x + 4. In a list of five options the two look almost identical.

How to practise 11 plus Equations & Missing Numbers questions

The most useful habit here is the cheapest: after every answer, put it back into the original line and check it balances. A child who does that stops losing marks to the half-finished answer and the flipped sign, which between them account for most of the wrong options here. It also helps to write the line of maths first and solve it second, because doing both at once is where sign errors creep in. Short and regular beats long and occasional, the pattern we set out in how much 11 plus practice a child needs each day. 11+ Daily has an Equations & Missing Numbers project alongside the other maths topics, on laptop, tablet or phone, all listed 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