← Symbolize the unknown, then build an expression · Work Backwards to Recover a Start Value

Symbolize the unknown, then build an expression · 12 practice problems

5.OA.A.15.OA.A.2

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Generated variants — 12

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

Variant 1 easy

Take a certain number, subtract 4, then multiply by 7. Add to that the quotient of 20 divided by 5, and the result is 81. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 4 taken off it, the answer is multiplied by 7, and then 20 divided by 5 is added on. The end result is 81. We need the hidden number.

Givens

Subtract 4 from the number, then multiply by 7.
Add the quotient of 20 divided by 5.
The final result is 81.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 81.

(\square - 4) \times 7 + 20 \div 5 = 81

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

20 divided by 5 needs no unknown, so work it out now.

20 \div 5 = 4

The expression is now the product plus 4.

3Undo the 'add 4'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 4, so subtract it back.

81 - 4 = 77

Before that addition, the value was 77.

4Undo the 'multiply by 7'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

77 \div 7 = 11

The parentheses held 11.

5Undo the 'subtract 4'

#11 Work Backwards 5.OA.A.1

Adding 4 back returns the number we started from.

11 + 4 = 15

The certain number is 15.

Answer: 15

4 · Reviewdoes it hold up?

Run the recipe forwards on 15: 15 - 4 = 11, 11 x 7 = 77, 77 + 4 = 81. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 2 easy

Take a certain number, subtract 3, then multiply by 11. Add to that the quotient of 16 divided by 8, and the result is 112. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 3 taken off it, the answer is multiplied by 11, and then 16 divided by 8 is added on. The end result is 112. We need the hidden number.

Givens

Subtract 3 from the number, then multiply by 11.
Add the quotient of 16 divided by 8.
The final result is 112.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 112.

(\square - 3) \times 11 + 16 \div 8 = 112

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

16 divided by 8 needs no unknown, so work it out now.

16 \div 8 = 2

The expression is now the product plus 2.

3Undo the 'add 2'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 2, so subtract it back.

112 - 2 = 110

Before that addition, the value was 110.

4Undo the 'multiply by 11'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

110 \div 11 = 10

The parentheses held 10.

5Undo the 'subtract 3'

#11 Work Backwards 5.OA.A.1

Adding 3 back returns the number we started from.

10 + 3 = 13

The certain number is 13.

Answer: 13

4 · Reviewdoes it hold up?

Run the recipe forwards on 13: 13 - 3 = 10, 10 x 11 = 110, 110 + 2 = 112. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 3 easy

Take a certain number, subtract 5, then multiply by 4. Add to that the quotient of 18 divided by 6, and the result is 115. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 5 taken off it, the answer is multiplied by 4, and then 18 divided by 6 is added on. The end result is 115. We need the hidden number.

Givens

Subtract 5 from the number, then multiply by 4.
Add the quotient of 18 divided by 6.
The final result is 115.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 115.

(\square - 5) \times 4 + 18 \div 6 = 115

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

18 divided by 6 needs no unknown, so work it out now.

18 \div 6 = 3

The expression is now the product plus 3.

3Undo the 'add 3'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 3, so subtract it back.

115 - 3 = 112

Before that addition, the value was 112.

4Undo the 'multiply by 4'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

112 \div 4 = 28

The parentheses held 28.

5Undo the 'subtract 5'

#11 Work Backwards 5.OA.A.1

Adding 5 back returns the number we started from.

28 + 5 = 33

The certain number is 33.

Answer: 33

4 · Reviewdoes it hold up?

Run the recipe forwards on 33: 33 - 5 = 28, 28 x 4 = 112, 112 + 3 = 115. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 4 easy

Take a certain number, subtract 2, then multiply by 8. Add to that the quotient of 45 divided by 9, and the result is 125. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 2 taken off it, the answer is multiplied by 8, and then 45 divided by 9 is added on. The end result is 125. We need the hidden number.

Givens

Subtract 2 from the number, then multiply by 8.
Add the quotient of 45 divided by 9.
The final result is 125.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 125.

(\square - 2) \times 8 + 45 \div 9 = 125

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

45 divided by 9 needs no unknown, so work it out now.

45 \div 9 = 5

The expression is now the product plus 5.

3Undo the 'add 5'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 5, so subtract it back.

125 - 5 = 120

Before that addition, the value was 120.

4Undo the 'multiply by 8'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

120 \div 8 = 15

The parentheses held 15.

5Undo the 'subtract 2'

#11 Work Backwards 5.OA.A.1

Adding 2 back returns the number we started from.

15 + 2 = 17

The certain number is 17.

Answer: 17

4 · Reviewdoes it hold up?

Run the recipe forwards on 17: 17 - 2 = 15, 15 x 8 = 120, 120 + 5 = 125. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 5 medium

Take a certain number, subtract 4, then multiply by 6. Add to that the quotient of 28 divided by 4, and the result is 127. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 4 taken off it, the answer is multiplied by 6, and then 28 divided by 4 is added on. The end result is 127. We need the hidden number.

Givens

Subtract 4 from the number, then multiply by 6.
Add the quotient of 28 divided by 4.
The final result is 127.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 127.

(\square - 4) \times 6 + 28 \div 4 = 127

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

28 divided by 4 needs no unknown, so work it out now.

28 \div 4 = 7

The expression is now the product plus 7.

3Undo the 'add 7'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 7, so subtract it back.

127 - 7 = 120

Before that addition, the value was 120.

4Undo the 'multiply by 6'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

120 \div 6 = 20

The parentheses held 20.

5Undo the 'subtract 4'

#11 Work Backwards 5.OA.A.1

Adding 4 back returns the number we started from.

20 + 4 = 24

The certain number is 24.

Answer: 24

4 · Reviewdoes it hold up?

Run the recipe forwards on 24: 24 - 4 = 20, 20 x 6 = 120, 120 + 7 = 127. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 6 medium

Take a certain number, subtract 7, then multiply by 3. Add to that the quotient of 30 divided by 6, and the result is 140. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 7 taken off it, the answer is multiplied by 3, and then 30 divided by 6 is added on. The end result is 140. We need the hidden number.

Givens

Subtract 7 from the number, then multiply by 3.
Add the quotient of 30 divided by 6.
The final result is 140.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 140.

(\square - 7) \times 3 + 30 \div 6 = 140

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

30 divided by 6 needs no unknown, so work it out now.

30 \div 6 = 5

The expression is now the product plus 5.

3Undo the 'add 5'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 5, so subtract it back.

140 - 5 = 135

Before that addition, the value was 135.

4Undo the 'multiply by 3'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

135 \div 3 = 45

The parentheses held 45.

5Undo the 'subtract 7'

#11 Work Backwards 5.OA.A.1

Adding 7 back returns the number we started from.

45 + 7 = 52

The certain number is 52.

Answer: 52

4 · Reviewdoes it hold up?

Run the recipe forwards on 52: 52 - 7 = 45, 45 x 3 = 135, 135 + 5 = 140. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 7 medium

Take a certain number, subtract 9, then multiply by 5. Add to that the quotient of 42 divided by 7, and the result is 156. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 9 taken off it, the answer is multiplied by 5, and then 42 divided by 7 is added on. The end result is 156. We need the hidden number.

Givens

Subtract 9 from the number, then multiply by 5.
Add the quotient of 42 divided by 7.
The final result is 156.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 156.

(\square - 9) \times 5 + 42 \div 7 = 156

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

42 divided by 7 needs no unknown, so work it out now.

42 \div 7 = 6

The expression is now the product plus 6.

3Undo the 'add 6'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 6, so subtract it back.

156 - 6 = 150

Before that addition, the value was 150.

4Undo the 'multiply by 5'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

150 \div 5 = 30

The parentheses held 30.

5Undo the 'subtract 9'

#11 Work Backwards 5.OA.A.1

Adding 9 back returns the number we started from.

30 + 9 = 39

The certain number is 39.

Answer: 39

4 · Reviewdoes it hold up?

Run the recipe forwards on 39: 39 - 9 = 30, 30 x 5 = 150, 150 + 6 = 156. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 8 medium

Take a certain number, subtract 2, then multiply by 6. Add to that the quotient of 12 divided by 3, and the result is 160. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 2 taken off it, the answer is multiplied by 6, and then 12 divided by 3 is added on. The end result is 160. We need the hidden number.

Givens

Subtract 2 from the number, then multiply by 6.
Add the quotient of 12 divided by 3.
The final result is 160.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 160.

(\square - 2) \times 6 + 12 \div 3 = 160

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

12 divided by 3 needs no unknown, so work it out now.

12 \div 3 = 4

The expression is now the product plus 4.

3Undo the 'add 4'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 4, so subtract it back.

160 - 4 = 156

Before that addition, the value was 156.

4Undo the 'multiply by 6'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

156 \div 6 = 26

The parentheses held 26.

5Undo the 'subtract 2'

#11 Work Backwards 5.OA.A.1

Adding 2 back returns the number we started from.

26 + 2 = 28

The certain number is 28.

Answer: 28

4 · Reviewdoes it hold up?

Run the recipe forwards on 28: 28 - 2 = 26, 26 x 6 = 156, 156 + 4 = 160. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 9 hard

Take a certain number, subtract 3, then multiply by 9. Add to that the quotient of 35 divided by 7, and the result is 167. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 3 taken off it, the answer is multiplied by 9, and then 35 divided by 7 is added on. The end result is 167. We need the hidden number.

Givens

Subtract 3 from the number, then multiply by 9.
Add the quotient of 35 divided by 7.
The final result is 167.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 167.

(\square - 3) \times 9 + 35 \div 7 = 167

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

35 divided by 7 needs no unknown, so work it out now.

35 \div 7 = 5

The expression is now the product plus 5.

3Undo the 'add 5'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 5, so subtract it back.

167 - 5 = 162

Before that addition, the value was 162.

4Undo the 'multiply by 9'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

162 \div 9 = 18

The parentheses held 18.

5Undo the 'subtract 3'

#11 Work Backwards 5.OA.A.1

Adding 3 back returns the number we started from.

18 + 3 = 21

The certain number is 21.

Answer: 21

4 · Reviewdoes it hold up?

Run the recipe forwards on 21: 21 - 3 = 18, 18 x 9 = 162, 162 + 5 = 167. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 10 hard

Take a certain number, subtract 6, then multiply by 5. Add to that the quotient of 24 divided by 8, and the result is 203. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 6 taken off it, the answer is multiplied by 5, and then 24 divided by 8 is added on. The end result is 203. We need the hidden number.

Givens

Subtract 6 from the number, then multiply by 5.
Add the quotient of 24 divided by 8.
The final result is 203.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 203.

(\square - 6) \times 5 + 24 \div 8 = 203

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

24 divided by 8 needs no unknown, so work it out now.

24 \div 8 = 3

The expression is now the product plus 3.

3Undo the 'add 3'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 3, so subtract it back.

203 - 3 = 200

Before that addition, the value was 200.

4Undo the 'multiply by 5'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

200 \div 5 = 40

The parentheses held 40.

5Undo the 'subtract 6'

#11 Work Backwards 5.OA.A.1

Adding 6 back returns the number we started from.

40 + 6 = 46

The certain number is 46.

Answer: 46

4 · Reviewdoes it hold up?

Run the recipe forwards on 46: 46 - 6 = 40, 40 x 5 = 200, 200 + 3 = 203. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 11 hard

Take a certain number, subtract 8, then multiply by 4. Add to that the quotient of 27 divided by 3, and the result is 217. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 8 taken off it, the answer is multiplied by 4, and then 27 divided by 3 is added on. The end result is 217. We need the hidden number.

Givens

Subtract 8 from the number, then multiply by 4.
Add the quotient of 27 divided by 3.
The final result is 217.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 217.

(\square - 8) \times 4 + 27 \div 3 = 217

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

27 divided by 3 needs no unknown, so work it out now.

27 \div 3 = 9

The expression is now the product plus 9.

3Undo the 'add 9'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 9, so subtract it back.

217 - 9 = 208

Before that addition, the value was 208.

4Undo the 'multiply by 4'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

208 \div 4 = 52

The parentheses held 52.

5Undo the 'subtract 8'

#11 Work Backwards 5.OA.A.1

Adding 8 back returns the number we started from.

52 + 8 = 60

The certain number is 60.

Answer: 60

4 · Reviewdoes it hold up?

Run the recipe forwards on 60: 60 - 8 = 52, 52 x 4 = 208, 208 + 9 = 217. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Variant 12 hard

Take a certain number, subtract 5, then multiply by 7. Add to that the quotient of 50 divided by 5, and the result is 220. Find the certain number.

Your answer

Show solution

1 · Understandwhat's really being asked

A hidden number has 5 taken off it, the answer is multiplied by 7, and then 50 divided by 5 is added on. The end result is 220. We need the hidden number.

Givens

Subtract 5 from the number, then multiply by 7.
Add the quotient of 50 divided by 5.
The final result is 220.

Unknowns

The certain number the recipe starts from.

Constraints

The steps happen in the order the sentence gives them.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Write the words as one expression so the order of operations is fixed and visible, then peel the steps off from the outside in, each with its opposite.

3 · Execute5 carry out the plan

1Write the calculation as one expression

#13 Convert to Algebra 5.OA.A.2

Call the hidden number a box. The sentence becomes a single expression that equals 220.

(\square - 5) \times 7 + 50 \div 5 = 220

Written out, the order stops being a matter of memory.

2Simplify the part you already know

#7 Identify Subproblems 5.OA.A.1

50 divided by 5 needs no unknown, so work it out now.

50 \div 5 = 10

The expression is now the product plus 10.

3Undo the 'add 10'

#11 Work Backwards 5.OA.A.1

The last thing done was adding 10, so subtract it back.

220 - 10 = 210

Before that addition, the value was 210.

4Undo the 'multiply by 7'

#11 Work Backwards 5.OA.A.1

Division undoes multiplication.

210 \div 7 = 30

The parentheses held 30.

5Undo the 'subtract 5'

#11 Work Backwards 5.OA.A.1

Adding 5 back returns the number we started from.

30 + 5 = 35

The certain number is 35.

Answer: 35

4 · Reviewdoes it hold up?

Run the recipe forwards on 35: 35 - 5 = 30, 30 x 7 = 210, 210 + 10 = 220. It lands on the given result.

Another way: Guess and check works too, but each guess costs three steps; undoing costs three steps once and cannot miss.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Reading the expression in the right order so it can be undone in reverse.
5.OA.A.2 Write simple expressions that record calculations with numbers — Turning the sentence into a single expression with a box for the unknown.

💡Takeaway. Write the words as one expression first. Then finding the unknown is just walking the steps backwards, swapping each for its opposite.

Where next?

← Back to the problem: Symbolize the unknown, then build an expression