← Reverse the operation order to find the unknown · Work Backwards to Recover a Start Value

Reverse the operation order to find the unknown · 12 practice problems

5.OA.A.13.OA.A.4

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 9

Find the number that makes □\square correct.

45÷(14−□)×8−31=4145 \div (14 - \square) \times 8 - 31 = 41

Show solution
1 · Understandwhat's really being asked

An expression takes 45, divides it by (14 minus a hidden number), multiplies by 8, then subtracts 31, and the result is 41. We need the hidden number in the box.

Givens
  • The expression is 45 / (14 - box) x 8 - 31.
  • Its value is 41.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 14 - box must divide 45 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 41 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 31. Adding it back tells us what the expression was worth just before that.
41+31=7241 + 31 = 72
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 8

#11 Work Backwards 3.OA.A.4
Before multiplying by 8, the value was 8 times smaller, so divide.
72÷8=972 \div 8 = 9
Division undoes the times-8.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
45 divided by the parentheses gave 9, so the parentheses must hold 45 divided by 9.
45÷9=545 \div 9 = 5
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 14 minus the box equals 5, so the box is what is missing from 14.
14−□=5⇒□=14−5=914 - \square = 5 \Rightarrow \square = 14 - 5 = 9
The hidden number is 9.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 9 back in and follow the original order of operations from left to right.
14−9=5, 45÷5=9, 9×8=72, 72−31=4114 - 9 = 5,\ 45 \div 5 = 9,\ 9 \times 8 = 72,\ 72 - 31 = 41
The forward run lands on the given value, so the box is right.
Answer: 9
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 41 exactly, and 5 divides 45 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 14 - box divides 45 evenly narrows the candidates fast, and 9 is the one that also hits 41.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 2 easy answer: 5

Find the number that makes □\square correct.

48÷(11−□)×6−22=2648 \div (11 - \square) \times 6 - 22 = 26

Show solution
1 · Understandwhat's really being asked

An expression takes 48, divides it by (11 minus a hidden number), multiplies by 6, then subtracts 22, and the result is 26. We need the hidden number in the box.

Givens
  • The expression is 48 / (11 - box) x 6 - 22.
  • Its value is 26.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 11 - box must divide 48 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 26 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 22. Adding it back tells us what the expression was worth just before that.
26+22=4826 + 22 = 48
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 6

#11 Work Backwards 3.OA.A.4
Before multiplying by 6, the value was 6 times smaller, so divide.
48÷6=848 \div 6 = 8
Division undoes the times-6.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
48 divided by the parentheses gave 8, so the parentheses must hold 48 divided by 8.
48÷8=648 \div 8 = 6
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 11 minus the box equals 6, so the box is what is missing from 11.
11−□=6⇒□=11−6=511 - \square = 6 \Rightarrow \square = 11 - 6 = 5
The hidden number is 5.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 5 back in and follow the original order of operations from left to right.
11−5=6, 48÷6=8, 8×6=48, 48−22=2611 - 5 = 6,\ 48 \div 6 = 8,\ 8 \times 6 = 48,\ 48 - 22 = 26
The forward run lands on the given value, so the box is right.
Answer: 5
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 26 exactly, and 6 divides 48 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 11 - box divides 48 evenly narrows the candidates fast, and 5 is the one that also hits 26.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 3 easy answer: 9

Find the number that makes □\square correct.

56÷(16−□)×4−21=1156 \div (16 - \square) \times 4 - 21 = 11

Show solution
1 · Understandwhat's really being asked

An expression takes 56, divides it by (16 minus a hidden number), multiplies by 4, then subtracts 21, and the result is 11. We need the hidden number in the box.

Givens
  • The expression is 56 / (16 - box) x 4 - 21.
  • Its value is 11.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 16 - box must divide 56 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 11 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 21. Adding it back tells us what the expression was worth just before that.
11+21=3211 + 21 = 32
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 4

#11 Work Backwards 3.OA.A.4
Before multiplying by 4, the value was 4 times smaller, so divide.
32÷4=832 \div 4 = 8
Division undoes the times-4.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
56 divided by the parentheses gave 8, so the parentheses must hold 56 divided by 8.
56÷8=756 \div 8 = 7
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 16 minus the box equals 7, so the box is what is missing from 16.
16−□=7⇒□=16−7=916 - \square = 7 \Rightarrow \square = 16 - 7 = 9
The hidden number is 9.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 9 back in and follow the original order of operations from left to right.
16−9=7, 56÷7=8, 8×4=32, 32−21=1116 - 9 = 7,\ 56 \div 7 = 8,\ 8 \times 4 = 32,\ 32 - 21 = 11
The forward run lands on the given value, so the box is right.
Answer: 9
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 11 exactly, and 7 divides 56 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 16 - box divides 56 evenly narrows the candidates fast, and 9 is the one that also hits 11.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 4 easy answer: 7

Find the number that makes □\square correct.

60÷(12−□)×4−15=3360 \div (12 - \square) \times 4 - 15 = 33

Show solution
1 · Understandwhat's really being asked

An expression takes 60, divides it by (12 minus a hidden number), multiplies by 4, then subtracts 15, and the result is 33. We need the hidden number in the box.

Givens
  • The expression is 60 / (12 - box) x 4 - 15.
  • Its value is 33.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 12 - box must divide 60 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 33 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 15. Adding it back tells us what the expression was worth just before that.
33+15=4833 + 15 = 48
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 4

#11 Work Backwards 3.OA.A.4
Before multiplying by 4, the value was 4 times smaller, so divide.
48÷4=1248 \div 4 = 12
Division undoes the times-4.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
60 divided by the parentheses gave 12, so the parentheses must hold 60 divided by 12.
60÷12=560 \div 12 = 5
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 12 minus the box equals 5, so the box is what is missing from 12.
12−□=5⇒□=12−5=712 - \square = 5 \Rightarrow \square = 12 - 5 = 7
The hidden number is 7.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 7 back in and follow the original order of operations from left to right.
12−7=5, 60÷5=12, 12×4=48, 48−15=3312 - 7 = 5,\ 60 \div 5 = 12,\ 12 \times 4 = 48,\ 48 - 15 = 33
The forward run lands on the given value, so the box is right.
Answer: 7
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 33 exactly, and 5 divides 60 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 12 - box divides 60 evenly narrows the candidates fast, and 7 is the one that also hits 33.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 5 medium answer: 4

Find the number that makes □\square correct.

64÷(12−□)×7−40=1664 \div (12 - \square) \times 7 - 40 = 16

Show solution
1 · Understandwhat's really being asked

An expression takes 64, divides it by (12 minus a hidden number), multiplies by 7, then subtracts 40, and the result is 16. We need the hidden number in the box.

Givens
  • The expression is 64 / (12 - box) x 7 - 40.
  • Its value is 16.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 12 - box must divide 64 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 16 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 40. Adding it back tells us what the expression was worth just before that.
16+40=5616 + 40 = 56
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 7

#11 Work Backwards 3.OA.A.4
Before multiplying by 7, the value was 7 times smaller, so divide.
56÷7=856 \div 7 = 8
Division undoes the times-7.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
64 divided by the parentheses gave 8, so the parentheses must hold 64 divided by 8.
64÷8=864 \div 8 = 8
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 12 minus the box equals 8, so the box is what is missing from 12.
12−□=8⇒□=12−8=412 - \square = 8 \Rightarrow \square = 12 - 8 = 4
The hidden number is 4.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 4 back in and follow the original order of operations from left to right.
12−4=8, 64÷8=8, 8×7=56, 56−40=1612 - 4 = 8,\ 64 \div 8 = 8,\ 8 \times 7 = 56,\ 56 - 40 = 16
The forward run lands on the given value, so the box is right.
Answer: 4
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 16 exactly, and 8 divides 64 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 12 - box divides 64 evenly narrows the candidates fast, and 4 is the one that also hits 16.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 6 medium answer: 4

Find the number that makes □\square correct.

72÷(10−□)×3−27=972 \div (10 - \square) \times 3 - 27 = 9

Show solution
1 · Understandwhat's really being asked

An expression takes 72, divides it by (10 minus a hidden number), multiplies by 3, then subtracts 27, and the result is 9. We need the hidden number in the box.

Givens
  • The expression is 72 / (10 - box) x 3 - 27.
  • Its value is 9.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 10 - box must divide 72 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 9 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 27. Adding it back tells us what the expression was worth just before that.
9+27=369 + 27 = 36
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 3

#11 Work Backwards 3.OA.A.4
Before multiplying by 3, the value was 3 times smaller, so divide.
36÷3=1236 \div 3 = 12
Division undoes the times-3.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
72 divided by the parentheses gave 12, so the parentheses must hold 72 divided by 12.
72÷12=672 \div 12 = 6
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 10 minus the box equals 6, so the box is what is missing from 10.
10−□=6⇒□=10−6=410 - \square = 6 \Rightarrow \square = 10 - 6 = 4
The hidden number is 4.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 4 back in and follow the original order of operations from left to right.
10−4=6, 72÷6=12, 12×3=36, 36−27=910 - 4 = 6,\ 72 \div 6 = 12,\ 12 \times 3 = 36,\ 36 - 27 = 9
The forward run lands on the given value, so the box is right.
Answer: 4
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 9 exactly, and 6 divides 72 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 10 - box divides 72 evenly narrows the candidates fast, and 4 is the one that also hits 9.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 7 medium answer: 5

Find the number that makes □\square correct.

75÷(20−□)×4−13=775 \div (20 - \square) \times 4 - 13 = 7

Show solution
1 · Understandwhat's really being asked

An expression takes 75, divides it by (20 minus a hidden number), multiplies by 4, then subtracts 13, and the result is 7. We need the hidden number in the box.

Givens
  • The expression is 75 / (20 - box) x 4 - 13.
  • Its value is 7.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 20 - box must divide 75 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 7 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 13. Adding it back tells us what the expression was worth just before that.
7+13=207 + 13 = 20
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 4

#11 Work Backwards 3.OA.A.4
Before multiplying by 4, the value was 4 times smaller, so divide.
20÷4=520 \div 4 = 5
Division undoes the times-4.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
75 divided by the parentheses gave 5, so the parentheses must hold 75 divided by 5.
75÷5=1575 \div 5 = 15
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 20 minus the box equals 15, so the box is what is missing from 20.
20−□=15⇒□=20−15=520 - \square = 15 \Rightarrow \square = 20 - 15 = 5
The hidden number is 5.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 5 back in and follow the original order of operations from left to right.
20−5=15, 75÷15=5, 5×4=20, 20−13=720 - 5 = 15,\ 75 \div 15 = 5,\ 5 \times 4 = 20,\ 20 - 13 = 7
The forward run lands on the given value, so the box is right.
Answer: 5
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 7 exactly, and 15 divides 75 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 20 - box divides 75 evenly narrows the candidates fast, and 5 is the one that also hits 7.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 8 medium answer: 8

Find the number that makes □\square correct.

84÷(15−□)×2−19=584 \div (15 - \square) \times 2 - 19 = 5

Show solution
1 · Understandwhat's really being asked

An expression takes 84, divides it by (15 minus a hidden number), multiplies by 2, then subtracts 19, and the result is 5. We need the hidden number in the box.

Givens
  • The expression is 84 / (15 - box) x 2 - 19.
  • Its value is 5.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 15 - box must divide 84 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 5 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 19. Adding it back tells us what the expression was worth just before that.
5+19=245 + 19 = 24
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 2

#11 Work Backwards 3.OA.A.4
Before multiplying by 2, the value was 2 times smaller, so divide.
24÷2=1224 \div 2 = 12
Division undoes the times-2.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
84 divided by the parentheses gave 12, so the parentheses must hold 84 divided by 12.
84÷12=784 \div 12 = 7
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 15 minus the box equals 7, so the box is what is missing from 15.
15−□=7⇒□=15−7=815 - \square = 7 \Rightarrow \square = 15 - 7 = 8
The hidden number is 8.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 8 back in and follow the original order of operations from left to right.
15−8=7, 84÷7=12, 12×2=24, 24−19=515 - 8 = 7,\ 84 \div 7 = 12,\ 12 \times 2 = 24,\ 24 - 19 = 5
The forward run lands on the given value, so the box is right.
Answer: 8
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 5 exactly, and 7 divides 84 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 15 - box divides 84 evenly narrows the candidates fast, and 8 is the one that also hits 5.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 9 hard answer: 4

Find the number that makes □\square correct.

90÷(13−□)×3−17=1390 \div (13 - \square) \times 3 - 17 = 13

Show solution
1 · Understandwhat's really being asked

An expression takes 90, divides it by (13 minus a hidden number), multiplies by 3, then subtracts 17, and the result is 13. We need the hidden number in the box.

Givens
  • The expression is 90 / (13 - box) x 3 - 17.
  • Its value is 13.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 13 - box must divide 90 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 13 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 17. Adding it back tells us what the expression was worth just before that.
13+17=3013 + 17 = 30
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 3

#11 Work Backwards 3.OA.A.4
Before multiplying by 3, the value was 3 times smaller, so divide.
30÷3=1030 \div 3 = 10
Division undoes the times-3.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
90 divided by the parentheses gave 10, so the parentheses must hold 90 divided by 10.
90÷10=990 \div 10 = 9
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 13 minus the box equals 9, so the box is what is missing from 13.
13−□=9⇒□=13−9=413 - \square = 9 \Rightarrow \square = 13 - 9 = 4
The hidden number is 4.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 4 back in and follow the original order of operations from left to right.
13−4=9, 90÷9=10, 10×3=30, 30−17=1313 - 4 = 9,\ 90 \div 9 = 10,\ 10 \times 3 = 30,\ 30 - 17 = 13
The forward run lands on the given value, so the box is right.
Answer: 4
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 13 exactly, and 9 divides 90 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 13 - box divides 90 evenly narrows the candidates fast, and 4 is the one that also hits 13.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 10 hard answer: 6

Find the number that makes □\square correct.

96÷(14−□)×5−33=2796 \div (14 - \square) \times 5 - 33 = 27

Show solution
1 · Understandwhat's really being asked

An expression takes 96, divides it by (14 minus a hidden number), multiplies by 5, then subtracts 33, and the result is 27. We need the hidden number in the box.

Givens
  • The expression is 96 / (14 - box) x 5 - 33.
  • Its value is 27.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 14 - box must divide 96 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 27 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 33. Adding it back tells us what the expression was worth just before that.
27+33=6027 + 33 = 60
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 5

#11 Work Backwards 3.OA.A.4
Before multiplying by 5, the value was 5 times smaller, so divide.
60÷5=1260 \div 5 = 12
Division undoes the times-5.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
96 divided by the parentheses gave 12, so the parentheses must hold 96 divided by 12.
96÷12=896 \div 12 = 8
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 14 minus the box equals 8, so the box is what is missing from 14.
14−□=8⇒□=14−8=614 - \square = 8 \Rightarrow \square = 14 - 8 = 6
The hidden number is 6.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 6 back in and follow the original order of operations from left to right.
14−6=8, 96÷8=12, 12×5=60, 60−33=2714 - 6 = 8,\ 96 \div 8 = 12,\ 12 \times 5 = 60,\ 60 - 33 = 27
The forward run lands on the given value, so the box is right.
Answer: 6
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 27 exactly, and 8 divides 96 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 14 - box divides 96 evenly narrows the candidates fast, and 6 is the one that also hits 27.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 11 hard answer: 6

Find the number that makes □\square correct.

108÷(18−□)×3−25=2108 \div (18 - \square) \times 3 - 25 = 2

Show solution
1 · Understandwhat's really being asked

An expression takes 108, divides it by (18 minus a hidden number), multiplies by 3, then subtracts 25, and the result is 2. We need the hidden number in the box.

Givens
  • The expression is 108 / (18 - box) x 3 - 25.
  • Its value is 2.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 18 - box must divide 108 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 2 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 25. Adding it back tells us what the expression was worth just before that.
2+25=272 + 25 = 27
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 3

#11 Work Backwards 3.OA.A.4
Before multiplying by 3, the value was 3 times smaller, so divide.
27÷3=927 \div 3 = 9
Division undoes the times-3.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
108 divided by the parentheses gave 9, so the parentheses must hold 108 divided by 9.
108÷9=12108 \div 9 = 12
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 18 minus the box equals 12, so the box is what is missing from 18.
18−□=12⇒□=18−12=618 - \square = 12 \Rightarrow \square = 18 - 12 = 6
The hidden number is 6.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 6 back in and follow the original order of operations from left to right.
18−6=12, 108÷12=9, 9×3=27, 27−25=218 - 6 = 12,\ 108 \div 12 = 9,\ 9 \times 3 = 27,\ 27 - 25 = 2
The forward run lands on the given value, so the box is right.
Answer: 6
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 2 exactly, and 12 divides 108 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 18 - box divides 108 evenly narrows the candidates fast, and 6 is the one that also hits 2.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.
Variant 12 hard answer: 9

Find the number that makes □\square correct.

120÷(17−□)×4−26=34120 \div (17 - \square) \times 4 - 26 = 34

Show solution
1 · Understandwhat's really being asked

An expression takes 120, divides it by (17 minus a hidden number), multiplies by 4, then subtracts 26, and the result is 34. We need the hidden number in the box.

Givens
  • The expression is 120 / (17 - box) x 4 - 26.
  • Its value is 34.
  • The parentheses are worked out first, then division and multiplication left to right, then the subtraction.
Unknowns
  • The number hidden in the box.
Constraints
  • 17 - box must divide 120 evenly.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Going forward from the box means guessing. Instead start at 34 and undo the operations in reverse order, each time using the opposite operation, until only the box is left.

3 · Execute5 carry out the plan

1Undo the last subtraction

#11 Work Backwards 5.OA.A.1
The final move was subtracting 26. Adding it back tells us what the expression was worth just before that.
34+26=6034 + 26 = 60
Undoing runs the recipe backwards, so subtraction becomes addition.

2Undo the multiplication by 4

#11 Work Backwards 3.OA.A.4
Before multiplying by 4, the value was 4 times smaller, so divide.
60÷4=1560 \div 4 = 15
Division undoes the times-4.

3Undo the division to open the parentheses

#11 Work Backwards 3.OA.A.4
120 divided by the parentheses gave 15, so the parentheses must hold 120 divided by 15.
120÷15=8120 \div 15 = 8
In a division, the divisor is the dividend split by the quotient.

4Find the box

#11 Work Backwards 3.OA.A.4
The parentheses say 17 minus the box equals 8, so the box is what is missing from 17.
17−□=8⇒□=17−8=917 - \square = 8 \Rightarrow \square = 17 - 8 = 9
The hidden number is 9.

5Run it forwards to be sure

#6 Guess and Check 5.OA.A.1
Put 9 back in and follow the original order of operations from left to right.
17−9=8, 120÷8=15, 15×4=60, 60−26=3417 - 9 = 8,\ 120 \div 8 = 15,\ 15 \times 4 = 60,\ 60 - 26 = 34
The forward run lands on the given value, so the box is right.
Answer: 9
4 · Reviewdoes it hold up?

The forward check in the last step reproduces 34 exactly, and 8 divides 120 evenly, so nothing was rounded along the way.

Another way: Guess and check: trying boxes until 17 - box divides 120 evenly narrows the candidates fast, and 9 is the one that also hits 34.

Standardsmin grade 5
  • 5.OA.A.1 Use parentheses and brackets in numerical expressions and evaluate them in order — Reading the expression in the correct order so it can be undone in reverse.
  • 3.OA.A.4 Determine the unknown whole number in a multiplication or division equation — Recovering the unknown divisor and the unknown inside the parentheses.
💡Takeaway. To find a hidden number, start from the answer and undo each step in reverse, swapping every operation for its opposite.