← Reverse the operations to find the start · Work Backwards to Recover a Start Value

Reverse the operations to find the start · 10 practice problems

3.OA.D.83.OA.A.4

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Generated variants — 10

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

Variant 1 easy

Mia thought of a number. She multiplied it by $3$ , added $4$ , and then divided the result by $7$ , which gave $4$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 3$ , then $+4$ , then $\div 7$ to reach $4$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 3, then 4 was added, then the result was divided by 7, ending at 4. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 3, then add 4, then divide by 7
The final result after all three steps is 4
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-7 first, then undo add-4, then undo multiply-by-3

2 · Planchoose the strategy

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

The end result (4) is given and the start is unknown, which is the classic signal to work backwards. The diagram even labels each inverse operation, so we apply the opposite of each step in reverse order. A quick forward check confirms the answer.

3 · Execute3 carry out the plan

1Undo the divide-by-7

#11 Work Backwards 3.OA.A.4

The last forward step divided by 7 to reach 4, so its inverse is multiply by 7. Multiplying 4 by 7 recovers the value before that step.

4 \times 7 = 28

Multiplying undoes dividing, so we climb back up from 4 to 28.

2Undo the add-4

#11 Work Backwards 3.OA.D.8

Before dividing, 4 had been added. The inverse of adding 4 is subtracting 4, so take 4 away from 28.

28 - 4 = 24

Subtraction reverses addition, peeling off the 4 that was put on.

3Undo the multiply-by-3

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 3. The inverse is divide by 3, so divide 24 by 3 to recover the original number.

24 \div 3 = 8

Dividing undoes multiplying, returning us to the very first number.

Answer: 8

4 · Reviewdoes it hold up?

Run it forward to check: 8 times 3 is 24, plus 4 is 28, divided by 7 is 4 — exactly the given result. So 8 is correct.

Another way: Guess and check (tool 6): try a start of 8, push it through multiply 3, add 4, divide 7, and you land on 4, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-3 and divide-by-7 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 2 easy

Mia thought of a number. She multiplied it by $2$ , added $8$ , and then divided the result by $5$ , which gave $4$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 2$ , then $+8$ , then $\div 5$ to reach $4$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 2, then 8 was added, then the result was divided by 5, ending at 4. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 2, then add 8, then divide by 5
The final result after all three steps is 4
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-5 first, then undo add-8, then undo multiply-by-2

2 · Planchoose the strategy

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

3 · Execute3 carry out the plan

1Undo the divide-by-5

#11 Work Backwards 3.OA.A.4

The last forward step divided by 5 to reach 4, so its inverse is multiply by 5. Multiplying 4 by 5 recovers the value before that step.

4 \times 5 = 20

Multiplying undoes dividing, so we climb back up from 4 to 20.

2Undo the add-8

#11 Work Backwards 3.OA.D.8

Before dividing, 8 had been added. The inverse of adding 8 is subtracting 8, so take 8 away from 20.

20 - 8 = 12

Subtraction reverses addition, peeling off the 8 that was put on.

3Undo the multiply-by-2

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 2. The inverse is divide by 2, so divide 12 by 2 to recover the original number.

12 \div 2 = 6

Dividing undoes multiplying, returning us to the very first number.

Answer: 6

4 · Reviewdoes it hold up?

Run it forward to check: 6 times 2 is 12, plus 8 is 20, divided by 5 is 4 — exactly the given result. So 6 is correct.

Another way: Guess and check (tool 6): try a start of 6, push it through multiply 2, add 8, divide 5, and you land on 4, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-2 and divide-by-5 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 3 easy

Mia thought of a number. She multiplied it by $2$ , added $6$ , and then divided the result by $3$ , which gave $8$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 2$ , then $+6$ , then $\div 3$ to reach $8$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 2, then 6 was added, then the result was divided by 3, ending at 8. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 2, then add 6, then divide by 3
The final result after all three steps is 8
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-3 first, then undo add-6, then undo multiply-by-2

2 · Planchoose the strategy

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

The end result (8) is given and the start is unknown, which is the classic signal to work backwards. The diagram even labels each inverse operation, so we apply the opposite of each step in reverse order. A quick forward check confirms the answer.

3 · Execute3 carry out the plan

1Undo the divide-by-3

#11 Work Backwards 3.OA.A.4

The last forward step divided by 3 to reach 8, so its inverse is multiply by 3. Multiplying 8 by 3 recovers the value before that step.

8 \times 3 = 24

Multiplying undoes dividing, so we climb back up from 8 to 24.

2Undo the add-6

#11 Work Backwards 3.OA.D.8

Before dividing, 6 had been added. The inverse of adding 6 is subtracting 6, so take 6 away from 24.

24 - 6 = 18

Subtraction reverses addition, peeling off the 6 that was put on.

3Undo the multiply-by-2

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 2. The inverse is divide by 2, so divide 18 by 2 to recover the original number.

18 \div 2 = 9

Dividing undoes multiplying, returning us to the very first number.

Answer: 9

4 · Reviewdoes it hold up?

Run it forward to check: 9 times 2 is 18, plus 6 is 24, divided by 3 is 8 — exactly the given result. So 9 is correct.

Another way: Guess and check (tool 6): try a start of 9, push it through multiply 2, add 6, divide 3, and you land on 8, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-2 and divide-by-3 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 4 medium

Mia thought of a number. She multiplied it by $5$ , added $10$ , and then divided the result by $7$ , which gave $10$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 5$ , then $+10$ , then $\div 7$ to reach $10$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 5, then 10 was added, then the result was divided by 7, ending at 10. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 5, then add 10, then divide by 7
The final result after all three steps is 10
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-7 first, then undo add-10, then undo multiply-by-5

2 · Planchoose the strategy

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

The end result (10) is given and the start is unknown, which is the classic signal to work backwards. The diagram even labels each inverse operation, so we apply the opposite of each step in reverse order. A quick forward check confirms the answer.

3 · Execute3 carry out the plan

1Undo the divide-by-7

#11 Work Backwards 3.OA.A.4

The last forward step divided by 7 to reach 10, so its inverse is multiply by 7. Multiplying 10 by 7 recovers the value before that step.

10 \times 7 = 70

Multiplying undoes dividing, so we climb back up from 10 to 70.

2Undo the add-10

#11 Work Backwards 3.OA.D.8

Before dividing, 10 had been added. The inverse of adding 10 is subtracting 10, so take 10 away from 70.

70 - 10 = 60

Subtraction reverses addition, peeling off the 10 that was put on.

3Undo the multiply-by-5

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 5. The inverse is divide by 5, so divide 60 by 5 to recover the original number.

60 \div 5 = 12

Dividing undoes multiplying, returning us to the very first number.

Answer: 12

4 · Reviewdoes it hold up?

Run it forward to check: 12 times 5 is 60, plus 10 is 70, divided by 7 is 10 — exactly the given result. So 12 is correct.

Another way: Guess and check (tool 6): try a start of 12, push it through multiply 5, add 10, divide 7, and you land on 10, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-5 and divide-by-7 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 5 medium

Mia thought of a number. She multiplied it by $3$ , added $11$ , and then divided the result by $4$ , which gave $8$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 3$ , then $+11$ , then $\div 4$ to reach $8$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 3, then 11 was added, then the result was divided by 4, ending at 8. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 3, then add 11, then divide by 4
The final result after all three steps is 8
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-4 first, then undo add-11, then undo multiply-by-3

2 · Planchoose the strategy

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

3 · Execute3 carry out the plan

1Undo the divide-by-4

#11 Work Backwards 3.OA.A.4

The last forward step divided by 4 to reach 8, so its inverse is multiply by 4. Multiplying 8 by 4 recovers the value before that step.

8 \times 4 = 32

Multiplying undoes dividing, so we climb back up from 8 to 32.

2Undo the add-11

#11 Work Backwards 3.OA.D.8

Before dividing, 11 had been added. The inverse of adding 11 is subtracting 11, so take 11 away from 32.

32 - 11 = 21

Subtraction reverses addition, peeling off the 11 that was put on.

3Undo the multiply-by-3

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 3. The inverse is divide by 3, so divide 21 by 3 to recover the original number.

21 \div 3 = 7

Dividing undoes multiplying, returning us to the very first number.

Answer: 7

4 · Reviewdoes it hold up?

Run it forward to check: 7 times 3 is 21, plus 11 is 32, divided by 4 is 8 — exactly the given result. So 7 is correct.

Another way: Guess and check (tool 6): try a start of 7, push it through multiply 3, add 11, divide 4, and you land on 8, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-3 and divide-by-4 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 6 medium

Mia thought of a number. She multiplied it by $6$ , added $12$ , and then divided the result by $2$ , which gave $15$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 6$ , then $+12$ , then $\div 2$ to reach $15$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 6, then 12 was added, then the result was divided by 2, ending at 15. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 6, then add 12, then divide by 2
The final result after all three steps is 15
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-2 first, then undo add-12, then undo multiply-by-6

2 · Planchoose the strategy

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

The end result (15) is given and the start is unknown, which is the classic signal to work backwards. The diagram even labels each inverse operation, so we apply the opposite of each step in reverse order. A quick forward check confirms the answer.

3 · Execute3 carry out the plan

1Undo the divide-by-2

#11 Work Backwards 3.OA.A.4

The last forward step divided by 2 to reach 15, so its inverse is multiply by 2. Multiplying 15 by 2 recovers the value before that step.

15 \times 2 = 30

Multiplying undoes dividing, so we climb back up from 15 to 30.

2Undo the add-12

#11 Work Backwards 3.OA.D.8

Before dividing, 12 had been added. The inverse of adding 12 is subtracting 12, so take 12 away from 30.

30 - 12 = 18

Subtraction reverses addition, peeling off the 12 that was put on.

3Undo the multiply-by-6

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 6. The inverse is divide by 6, so divide 18 by 6 to recover the original number.

18 \div 6 = 3

Dividing undoes multiplying, returning us to the very first number.

Answer: 3

4 · Reviewdoes it hold up?

Run it forward to check: 3 times 6 is 18, plus 12 is 30, divided by 2 is 15 — exactly the given result. So 3 is correct.

Another way: Guess and check (tool 6): try a start of 3, push it through multiply 6, add 12, divide 2, and you land on 15, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-6 and divide-by-2 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 7 medium

Mia thought of a number. She multiplied it by $9$ , added $18$ , and then divided the result by $6$ , which gave $6$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 9$ , then $+18$ , then $\div 6$ to reach $6$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 9, then 18 was added, then the result was divided by 6, ending at 6. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 9, then add 18, then divide by 6
The final result after all three steps is 6
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-6 first, then undo add-18, then undo multiply-by-9

2 · Planchoose the strategy

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

The end result (6) is given and the start is unknown, which is the classic signal to work backwards. The diagram even labels each inverse operation, so we apply the opposite of each step in reverse order. A quick forward check confirms the answer.

3 · Execute3 carry out the plan

1Undo the divide-by-6

#11 Work Backwards 3.OA.A.4

The last forward step divided by 6 to reach 6, so its inverse is multiply by 6. Multiplying 6 by 6 recovers the value before that step.

6 \times 6 = 36

Multiplying undoes dividing, so we climb back up from 6 to 36.

2Undo the add-18

#11 Work Backwards 3.OA.D.8

Before dividing, 18 had been added. The inverse of adding 18 is subtracting 18, so take 18 away from 36.

36 - 18 = 18

Subtraction reverses addition, peeling off the 18 that was put on.

3Undo the multiply-by-9

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 9. The inverse is divide by 9, so divide 18 by 9 to recover the original number.

18 \div 9 = 2

Dividing undoes multiplying, returning us to the very first number.

Answer: 2

4 · Reviewdoes it hold up?

Run it forward to check: 2 times 9 is 18, plus 18 is 36, divided by 6 is 6 — exactly the given result. So 2 is correct.

Another way: Guess and check (tool 6): try a start of 2, push it through multiply 9, add 18, divide 6, and you land on 6, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-9 and divide-by-6 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 8 hard

Mia thought of a number. She multiplied it by $5$ , added $20$ , and then divided the result by $8$ , which gave $5$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 5$ , then $+20$ , then $\div 8$ to reach $5$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 5, then 20 was added, then the result was divided by 8, ending at 5. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 5, then add 20, then divide by 8
The final result after all three steps is 5
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-8 first, then undo add-20, then undo multiply-by-5

2 · Planchoose the strategy

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

The end result (5) is given and the start is unknown, which is the classic signal to work backwards. The diagram even labels each inverse operation, so we apply the opposite of each step in reverse order. A quick forward check confirms the answer.

3 · Execute3 carry out the plan

1Undo the divide-by-8

#11 Work Backwards 3.OA.A.4

The last forward step divided by 8 to reach 5, so its inverse is multiply by 8. Multiplying 5 by 8 recovers the value before that step.

5 \times 8 = 40

Multiplying undoes dividing, so we climb back up from 5 to 40.

2Undo the add-20

#11 Work Backwards 3.OA.D.8

Before dividing, 20 had been added. The inverse of adding 20 is subtracting 20, so take 20 away from 40.

40 - 20 = 20

Subtraction reverses addition, peeling off the 20 that was put on.

3Undo the multiply-by-5

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 5. The inverse is divide by 5, so divide 20 by 5 to recover the original number.

20 \div 5 = 4

Dividing undoes multiplying, returning us to the very first number.

Answer: 4

4 · Reviewdoes it hold up?

Run it forward to check: 4 times 5 is 20, plus 20 is 40, divided by 8 is 5 — exactly the given result. So 4 is correct.

Another way: Guess and check (tool 6): try a start of 4, push it through multiply 5, add 20, divide 8, and you land on 5, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-5 and divide-by-8 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 9 hard

Mia thought of a number. She multiplied it by $4$ , added $40$ , and then divided the result by $8$ , which gave $10$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 4$ , then $+40$ , then $\div 8$ to reach $10$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 4, then 40 was added, then the result was divided by 8, ending at 10. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 4, then add 40, then divide by 8
The final result after all three steps is 10
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-8 first, then undo add-40, then undo multiply-by-4

2 · Planchoose the strategy

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

3 · Execute3 carry out the plan

1Undo the divide-by-8

#11 Work Backwards 3.OA.A.4

The last forward step divided by 8 to reach 10, so its inverse is multiply by 8. Multiplying 10 by 8 recovers the value before that step.

10 \times 8 = 80

Multiplying undoes dividing, so we climb back up from 10 to 80.

2Undo the add-40

#11 Work Backwards 3.OA.D.8

Before dividing, 40 had been added. The inverse of adding 40 is subtracting 40, so take 40 away from 80.

80 - 40 = 40

Subtraction reverses addition, peeling off the 40 that was put on.

3Undo the multiply-by-4

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 4. The inverse is divide by 4, so divide 40 by 4 to recover the original number.

40 \div 4 = 10

Dividing undoes multiplying, returning us to the very first number.

Answer: 10

4 · Reviewdoes it hold up?

Run it forward to check: 10 times 4 is 40, plus 40 is 80, divided by 8 is 10 — exactly the given result. So 10 is correct.

Another way: Guess and check (tool 6): try a start of 10, push it through multiply 4, add 40, divide 8, and you land on 10, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-4 and divide-by-8 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Variant 10 hard

Mia thought of a number. She multiplied it by $4$ , added $50$ , and then divided the result by $5$ , which gave $14$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 4$ , then $+50$ , then $\div 5$ to reach $14$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

A starting number was multiplied by 4, then 50 was added, then the result was divided by 5, ending at 14. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 4, then add 50, then divide by 5
The final result after all three steps is 14
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-5 first, then undo add-50, then undo multiply-by-4

2 · Planchoose the strategy

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

The end result (14) is given and the start is unknown, which is the classic signal to work backwards. The diagram even labels each inverse operation, so we apply the opposite of each step in reverse order. A quick forward check confirms the answer.

3 · Execute3 carry out the plan

1Undo the divide-by-5

#11 Work Backwards 3.OA.A.4

The last forward step divided by 5 to reach 14, so its inverse is multiply by 5. Multiplying 14 by 5 recovers the value before that step.

14 \times 5 = 70

Multiplying undoes dividing, so we climb back up from 14 to 70.

2Undo the add-50

#11 Work Backwards 3.OA.D.8

Before dividing, 50 had been added. The inverse of adding 50 is subtracting 50, so take 50 away from 70.

70 - 50 = 20

Subtraction reverses addition, peeling off the 50 that was put on.

3Undo the multiply-by-4

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 4. The inverse is divide by 4, so divide 20 by 4 to recover the original number.

20 \div 4 = 5

Dividing undoes multiplying, returning us to the very first number.

Answer: 5

4 · Reviewdoes it hold up?

Run it forward to check: 5 times 4 is 20, plus 50 is 70, divided by 5 is 14 — exactly the given result. So 5 is correct.

Another way: Guess and check (tool 6): try a start of 5, push it through multiply 4, add 50, divide 5, and you land on 14, confirming the answer without reversing.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-4 and divide-by-5 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡Takeaway. To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!

Where next?

← Back to the problem: Reverse the operations to find the start