HSPT: Missing numbers: solve from the outside in
Learn to find a missing number inside parentheses by first finding what the whole group must equal.
How do you find a missing number that sits inside parentheses?
Short answer
Undo the operation outside the parentheses to find the value of the whole group. Then solve the small question inside the group.
Know first: Order of operations, Multiplication and division facts
Step 1 of 5
Treat the parentheses as one number
In a question like 6 × (2 + □) = 54, the 6 multiplies the whole group in parentheses, not just the 2. So start by asking what the group must equal.
If 6 times the group is 54, the group is 54 ÷ 6 = 9. Now the question is simple: 2 plus what makes 9? The box holds 7.
Step 2 of 5
Find the group, then the box
Worked example: Multiplying a group
What value fills the box? 2 × (9 + □) = 40
- A11
- B31
- C20
- D29
- 1
Find the group
2 × (group) = 40, so the group is 40 ÷ 2 = 20.
- 2
Solve inside
9 + □ = 20, so □ = 11.
- 3
Check
2 × (9 + 11) = 2 × 20 = 40.
- 4
See why the other choices are there
20 stops at the group's value. 31 subtracts 9 from 40 and ignores the multiplication. 29 adds 9 to the group instead of subtracting it.
Answer
11
Find the wrong step
A student solves 3 × (□ + 8) = 54.
- 1
Subtracts 8 from 54 first and gets 46.
What went wrong
The 3 multiplies the whole group, so 8 isn't being added to the final 54. Undo the outside operation, the multiplication, first.
- 2
Divides 46 by 3 and gets a number that is not whole.
- 3
Picks the closest choice.
The fix
Divide first: 54 ÷ 3 = 18. Then □ + 8 = 18, so □ = 10. Check: 3 × 18 = 54.
Step 3 of 5
Undo the outside, then the inside
Check yourself · Question 1
What value fills the box? 6 × (□ + 5) = 78
Answer: B
The group is 78 ÷ 6 = 13. Then □ + 5 = 13, so □ = 8. Check: 6 × (8 + 5) = 6 × 13 = 78.
Trap answer: 13
Stopping at the group
- Why it looks right
- Dividing 78 by 6 is the hard part, and once it's done the question feels finished.
- Why it's wrong
- 13 is what the whole group equals. The box is only part of that group.
- The right answer
- 8, because 8 + 5 = 13 and 6 × 13 = 78.
Check yourself · Question 2
What value fills the box? (□ − 2) × 7 = 63
Answer: D
The group is 63 ÷ 7 = 9. Then □ − 2 = 9, so □ = 9 + 2 = 11. Check: (11 − 2) × 7 = 9 × 7 = 63.
Common mistake
"I undo the number next to the box first, since it's closest."
- Why it's tempting
- The number beside the box looks like the most direct thing to remove.
- Do this instead
- The operation outside the parentheses touches the whole group, so undo it first. Then deal with the number inside.
Step 4 of 5
What to remember
Remember
- A number outside parentheses acts on the whole group.
- Undo the outside operation first to find what the group equals.
- Then solve inside the group, and check by putting your answer back in.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
What did the whole group equal in the practice question, and which choice is that group value?