Number series, related numbers and complete the sum
Three number puzzles from verbal reasoning papers: the next number in a series, the middle number in a group, and the number that makes a sum balance. Each one hides a rule, and the sums are ones you already know.
Rush doesn’t practise number series, related numbers and complete the sum yet. These games practise related skills: Sequences and Order & Algebra.
Chapter 1
What are these three number puzzles?
Verbal reasoning papers set three kinds of number puzzle. The sums in them are easy. The hard part is spotting the rule behind the numbers.
Each question has five options. Once you have the rule, the answer takes seconds.
Number series. A row of numbers that follows a rule. You find the next one, or a missing one.
Related numbers. Groups of three. In each group the middle number is made from the two outside it, the same way every time. You find the missing middle.
Complete the sum. A sum with an equals sign and a gap. You find the number that makes both sides equal.
The maths pages on number sequences and algebra teach the same skills.
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Number series
Which number comes next?
4, 6, 10, 16, 24, ?
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Related numbers
Which middle number is missing?
(2 [4] 2) (7 [21] 3) (8 [?] 5)
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Complete the sum
Which number completes the sum?
20 + 4 × 5 = 8 × ?
Chapter 2
Solve each one in 3 steps.
The three questions from Chapter 1, one step at a time.
Number series
The question
Which number comes next?
4, 6, 10, 16, 24, ?
- 32
- 34
- 36
- 40
- 48
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Step 1
Write the gaps.
Write the gap between each pair of numbers: 4 to 6 is +2, then +4, +6 and +8. The gaps aren’t the same, so look at how they change.
The gaps grow.
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Step 2
Find the gap between the gaps.
Do the same with the gaps. 2 to 4 is +2, 4 to 6 is +2, and 6 to 8 is +2. Each gap is 2 more than the one before.
The gaps go up by 2 each time.
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Step 3
Add the next gap.
After +8 comes +10. Add it to the last number: 24 + 10 = 34.
24 + 10 = 34.
So it’s 34. The gaps go 2, 4, 6, 8, so the next gap is 10.
Related numbers
The question
The middle number in each group is made from the two outside it, in the same way every time. Which number is missing?
(2 [4] 2) (7 [21] 3) (8 [?] 5)
- 3
- 13
- 16
- 40
- 45
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Step 1
Try each sign on the first group.
Use the two outside numbers, 2 and 2, and see which sign makes the middle number, 4. Adding works, and so does multiplying.
Two signs fit. One is a trap.
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Step 2
Test both on the second group.
A rule only counts if it works every time. 7 + 3 = 10, not 21, so adding is out. 7 × 3 = 21, so the rule is ×.
Only × works on both groups.
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Step 3
Use it on the third group.
Multiply the two outside numbers: 8 × 5 = 40.
The missing middle is 40.
So it’s 40. The rule is ×: 2 × 2 = 4, 7 × 3 = 21 and 8 × 5 = 40.
Complete the sum
The question
Which number completes the sum?
20 + 4 × 5 = 8 × ?
- 3
- 5
- 15
- 32
- 40
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Step 1
Work out the full side.
The left side has no gap, so start there. × comes before +, even though it’s written second: 4 × 5 = 20, then 20 + 20 = 40.
The left side is 40.
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Step 2
Make the other side match.
The right side must make 40 too. 8 times what is 40? Undo the × with ÷: 40 ÷ 8 = 5.
Level means equal: 8 × ? must be 40.
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Step 3
Put it back in to check.
Write the whole sum with 5 in the gap and work out both sides. 40 = 40, so 5 is right.
Both sides match, so 5 is right.
So it’s 5. The left side is 40, and 8 × 5 = 40.
Three traps in these questions
| The mistake | Do this instead |
|---|---|
| Adding the last gap again. 32 adds 8 again, but the gaps grow by 2 each time. | Write the gaps. If they change, write how they change. |
| Using a rule that fits one group. 13 adds 8 and 5. 2 + 2 is 4, but 7 + 3 isn’t 21. | Test the rule on the second group before you use it. |
| Doing the + before the ×. 15 comes from 24 × 5 = 120, then 120 ÷ 8. | × and ÷ come before + and −. Then find the missing number. |
Remember: A rule has to fit every number you’re given.
Chapter 3
The harder ones.
Papers make each kind harder in a few ways. Each card is one of them: how it works, an example and the trap.
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Two series taking turns
Gaps that jump up and down? Read every other number.
In 1, 3, 2, 6, 3, 9, ? the numbers in odd places go 1, 2, 3 and the ones between go 3, 6, 9. The next place is an odd one, so it’s 4. Trap: 12, the next number in the wrong series.
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Times or divide
Gaps that grow fast point to × or ÷.
2, 6, 18, 54 has gaps 4, 12, 36. Each number is 3 times the last, so next is 54 × 3 = 162. Trap: 90, from adding 36 again. Halving works the same way: 96, 48, 24, 12, 6.
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Two steps
No single sign works? Try two steps, like adding and then doubling.
In (5 [14] 2) (3 [20] 7) (4 [?] 8), adding, then doubling, fits both groups, so it’s 24. Trap: 5 × 2 + 4 = 14 fits the first group, but 3 × 7 + 4 is 25, not 20.
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A middle smaller than both
When the middle is smaller than the outside numbers, try ÷ and − first.
In (36 [4] 9) (40 [5] 8) (42 [?] 6), the left number ÷ the right gives the middle, so it’s 7. Keep the order: left first. Trap: 36, from 42 − 6. Taking away gives 27 for the first group, not 4.
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Partners go from the left
× and ÷ are partners, and so are + and −. When partners meet, work from the left.
In 24 ÷ 4 × 2 = 3 × ?, the left side is 24 ÷ 4 = 6, then 6 × 2 = 12. 3 × 4 = 12, so it’s 4. Trap: 1, from doing 4 × 2 first. The same goes for 15 − 5 + 2, which is 12.
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A gap after − or ÷
Ask what you take away, or divide by, to get the other side.
In 45 ÷ 5 = 15 − ?, the left side is 9. 15 take away what leaves 9? Count up from 9 to 15: it’s 6. Trap: 24, from adding 9 to 15. With ÷ it works the same: in 36 ÷ ? = 2 × 6, the answer is 3.
Learn the squares by heart
A square number is a number times itself: 4 × 4 = 16. A series of squares has gaps 3, 5, 7, 9. Some related numbers rules square a number too.
Chapter 4
Try it yourself.
Easier ones first. Try each one before you open its answer.
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Question 1Level 1, Number series
Which number comes next?
45, 38, 31, 24, ?
- 10
- 16
- 17
- 18
- 31
Show answer
Answer: 17.
How to answer it
- Every gap is −7: 45 − 38 = 7, 38 − 31 = 7 and 31 − 24 = 7.
- Take 7 from the last number: 24 − 7 = 17.
Why the other answers are wrong
- 10That takes 7 away twice. The next number is one gap on from 24.
- 16That takes away 8. Every gap is 7.
- 18That takes away 6. Counting back from 24, don’t count the 24 itself: 23, 22, 21, 20, 19, 18, 17.
- 3131 is already in the series, before 24. The numbers go down.
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Question 2Level 1, Related numbers
The middle number in each group is made from the two outside it, in the same way every time. Which number is missing?
(3 [8] 5) (6 [10] 4) (9 [?] 2)
- 7
- 10
- 11
- 13
- 18
Show answer
Answer: 11.
How to answer it
- 3 + 5 = 8 and 6 + 4 = 10. The rule is: add the two outside numbers.
- So 9 + 2 = 11.
Why the other answers are wrong
- 7That takes 2 from 9. Taking away doesn’t fit the first group: 3 and 5 can’t make 8 that way.
- 10That is the middle number of the second group, copied. The third group needs its own.
- 13That adds 9 and 4, but the 4 belongs to the second group. Use 9 and 2.
- 18That multiplies 9 by 2. Multiplying doesn’t fit: 3 × 5 is 15, not 8.
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Question 3Level 1, Complete the sum
Which number completes the sum?
7 × 4 = 20 + ?
- 7
- 8
- 9
- 28
- 48
Show answer
Answer: 8.
How to answer it
- Work out the left side: 7 × 4 = 28.
- 20 plus what makes 28? 28 − 20 = 8. Check: 20 + 8 = 28.
Why the other answers are wrong
- 77 is copied from the sum. 20 + 7 is 27, one short of 28.
- 920 + 9 is 29, one too many. Count on from 20 to 28 and you make 8 jumps.
- 28That is the left side’s total, not the missing number.
- 48That adds 20 to 28. The missing number goes with 20 to make 28, so take 20 away.
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Question 4Level 2, Number series
Which number comes next?
2, 30, 4, 27, 6, 24, ?
- 8
- 10
- 21
- 22
- 26
Show answer
Answer: 8.
How to answer it
- The gaps jump up and down, so read every other number. 2, 4, 6 go up by 2. 30, 27, 24 go down by 3.
- The two series take turns, and 24 was the second series’ turn. So the first series comes next: 6 + 2 = 8.
Why the other answers are wrong
- 10That adds 4 to 6. The first series goes up by 2 each time.
- 21That is the next number in the second series. The series take turns, so it’s the first series’ turn.
- 22That takes 2 from 24. The second series goes down by 3, and it isn’t its turn.
- 26That adds 2 to 24, mixing up the two series. The +2 belongs to 2, 4, 6.
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Question 5Level 2, Related numbers
The middle number in each group is made from the two outside it, in the same way every time. Which number is missing?
(4 [9] 1) (3 [10] 4) (5 [?] 3)
- 8
- 11
- 13
- 15
- 16
Show answer
Answer: 13.
How to answer it
- No single sign works on the first group: 4 + 1 is 5 and 4 × 1 is 4, but the middle is 9. Try two steps.
- Double the left number, then add the right one: 4 × 2 + 1 = 9, and 3 × 2 + 4 = 10. It works on both groups.
- So 5 × 2 + 3 = 13.
Why the other answers are wrong
- 8That adds 5 and 3. Adding alone doesn’t fit: 4 + 1 is 5, not 9.
- 11That doubles the right number, 3, and adds 5. It’s the left number that doubles: 4 × 2 + 1 = 9.
- 15That multiplies 5 by 3. Multiplying doesn’t fit: 4 × 1 is 4, not 9.
- 16That adds, then doubles: 5 + 3 = 8, and 8 × 2 = 16. That rule gives 10 for the first group, not 9.
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Question 6Level 2, Complete the sum
Which number completes the sum?
40 − 10 ÷ 2 = 5 × ?
- 3
- 6
- 7
- 35
- 175
Show answer
Answer: 7.
How to answer it
- ÷ comes before −, so 10 ÷ 2 = 5, then 40 − 5 = 35.
- 5 times what is 35? 35 ÷ 5 = 7. Check: 5 × 7 = 35.
Why the other answers are wrong
- 3That does the − first: 40 − 10 = 30, 30 ÷ 2 = 15, and 15 ÷ 5 = 3. The ÷ comes first.
- 6That leaves out the ÷ 2: 40 − 10 = 30, and 30 ÷ 5 = 6.
- 35That is the left side’s total, not the missing number.
- 175That multiplies 35 by 5. The missing number times 5 makes 35, so divide: 35 ÷ 5.
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Question 7Level 3, Number series
Which number comes next?
2, 3, 5, 9, 17, ?
- 25
- 29
- 33
- 34
- 35
Show answer
Answer: 33.
How to answer it
- The gaps are 1, 2, 4, 8. Each gap is double the one before.
- So the next gap is 16, and 17 + 16 = 33. You can also see it as double the last number, take away 1: 17 × 2 − 1 = 33.
Why the other answers are wrong
- 25That adds 8 again. The gaps double, so after 8 comes 16.
- 29That adds 12, as if the gaps grow by 4. They go 1, 2, 4, 8: each one doubles.
- 34That doubles 17 but doesn’t take 1 away. 9 doubled is 18, and the series has 17.
- 35That doubles 17 and adds 1. 9 doubled, add 1, is 19, but the series has 17.
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Question 8Level 3, Related numbers
The middle number in each group is made from the two outside it, in the same way every time. Which number is missing?
(9 [15] 4) (10 [24] 2) (12 [?] 5)
- 7
- 17
- 19
- 21
- 51
Show answer
Answer: 21.
How to answer it
- No single sign makes 15 from 9 and 4. 9 + 4 + 2 = 15 fits the first group, but 10 + 2 + 2 is 14, not 24, so that rule is out.
- Take away, then times by 3: 9 − 4 = 5 and 5 × 3 = 15. 10 − 2 = 8 and 8 × 3 = 24. It works on both groups.
- So 12 − 5 = 7, and 7 × 3 = 21.
Why the other answers are wrong
- 7That stops after taking away. The rule then times by 3.
- 17That adds 12 and 5. Adding doesn’t fit: 9 + 4 is 13, not 15.
- 19That adds the outside numbers, then 2. It fits the first group, but 10 + 2 + 2 is 14, not 24.
- 51That adds, then times by 3: 12 + 5 = 17, and 17 × 3 = 51. The rule takes away first: 9 − 4 = 5, and 5 × 3 = 15.
For parents.
What parents ask us about number series, related numbers and complete the sum.
Questions parents ask
What are number series, related numbers and complete the sum?
Three number puzzles from verbal reasoning papers. A number series asks for the next number, or a missing one, in a row that follows a rule. Related numbers give groups like (2 [10] 5), where the middle number is made from the outer two, and ask for a missing middle. Complete the sum asks for the number that makes both sides of a sum equal.
Are these questions maths or verbal reasoning?
They are set in verbal reasoning papers, but the sums are school maths: times tables, adding and taking away. What they test is spotting a rule. The maths pages on number sequences and algebra practise the same skills.
What is the quickest way to find a related numbers rule?
Try +, −, × and ÷ on the outer numbers of the first group, in that order. Then test any rule that works on the second group before using it, because two rules often fit the first group. If no single sign works, try two steps, such as adding and then doubling.
Why do children get complete the sum wrong when they know their tables?
Usually it’s the order. In 20 + 4 × 5, the × comes first, so it’s 40, not 120. The other slip is stopping too soon and giving the total of one side, which is nearly always one of the options.
Practise the related skills in Rush.
- SequencesIt asks for the next number in a series against the clock, the same gaps that solve number series.
- Order & AlgebraIt asks for missing numbers in sums and the order of operations, the two things complete the sum needs.
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