← Recover the unknown, then recompute · Work Backwards to Recover a Start Value

Recover the unknown, then recompute · 12 practice problems

3.OA.A.43.OA.B.6

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Generated variants — 12

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

Variant 1 easy

A certain number was supposed to be multiplied by $7$ , but by mistake it was multiplied by $3$ , giving $24$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 7, but it was multiplied by 3 instead, giving 24. We must first find that number, then multiply it by 7 the correct way.

Givens

The wrong calculation multiplied the number by 3 and got 24.
The correct calculation should multiply the number by 7.

Unknowns

The hidden number.
The result of the correct calculation (number x 7).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 24 to recover the hidden number, then go forward to do the correct multiplication by 7.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 3 gives 24, find the number by asking what times 3 makes 24. That is 8.

\square \times 3 = 24 \Rightarrow \square = 8

Finding the missing factor of 24 with 3 is the unknown-factor idea: 8 times 3 is 24.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 8 by 7 as the problem intended.

8 \times 7 = 56

Once the number is known, the correct answer is just the fact 8 times 7.

Answer: 56

4 · Reviewdoes it hold up?

The hidden number 8 checks out because 8 x 3 = 24, matching the wrong result, and the correct answer 56 is larger than 24, which makes sense since multiplying by 7 instead of 3 gives more.

Another way: You could divide directly: 24 divided by 3 is 8, then 8 x 7 = 56, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 8 x 7 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 3 = 24 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 2 easy

A certain number was supposed to be multiplied by $4$ , but by mistake it was multiplied by $9$ , giving $27$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 4, but it was multiplied by 9 instead, giving 27. We must first find that number, then multiply it by 4 the correct way.

Givens

The wrong calculation multiplied the number by 9 and got 27.
The correct calculation should multiply the number by 4.

Unknowns

The hidden number.
The result of the correct calculation (number x 4).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 27 to recover the hidden number, then go forward to do the correct multiplication by 4.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 9 gives 27, find the number by asking what times 9 makes 27. That is 3.

\square \times 9 = 27 \Rightarrow \square = 3

Finding the missing factor of 27 with 9 is the unknown-factor idea: 3 times 9 is 27.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 3 by 4 as the problem intended.

3 \times 4 = 12

Once the number is known, the correct answer is just the fact 3 times 4.

Answer: 12

4 · Reviewdoes it hold up?

The hidden number 3 checks out because 3 x 9 = 27, matching the wrong result, and the correct answer 12 is smaller than 27, which makes sense since multiplying by 4 instead of 9 gives less.

Another way: You could divide directly: 27 divided by 9 is 3, then 3 x 4 = 12, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 3 x 4 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 9 = 27 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 3 easy

A certain number was supposed to be multiplied by $3$ , but by mistake it was multiplied by $7$ , giving $28$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 3, but it was multiplied by 7 instead, giving 28. We must first find that number, then multiply it by 3 the correct way.

Givens

The wrong calculation multiplied the number by 7 and got 28.
The correct calculation should multiply the number by 3.

Unknowns

The hidden number.
The result of the correct calculation (number x 3).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 28 to recover the hidden number, then go forward to do the correct multiplication by 3.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 7 gives 28, find the number by asking what times 7 makes 28. That is 4.

\square \times 7 = 28 \Rightarrow \square = 4

Finding the missing factor of 28 with 7 is the unknown-factor idea: 4 times 7 is 28.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 4 by 3 as the problem intended.

4 \times 3 = 12

Once the number is known, the correct answer is just the fact 4 times 3.

Answer: 12

4 · Reviewdoes it hold up?

The hidden number 4 checks out because 4 x 7 = 28, matching the wrong result, and the correct answer 12 is smaller than 28, which makes sense since multiplying by 3 instead of 7 gives less.

Another way: You could divide directly: 28 divided by 7 is 4, then 4 x 3 = 12, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 4 x 3 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 7 = 28 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 4 easy

A certain number was supposed to be multiplied by $9$ , but by mistake it was multiplied by $5$ , giving $30$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 9, but it was multiplied by 5 instead, giving 30. We must first find that number, then multiply it by 9 the correct way.

Givens

The wrong calculation multiplied the number by 5 and got 30.
The correct calculation should multiply the number by 9.

Unknowns

The hidden number.
The result of the correct calculation (number x 9).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 30 to recover the hidden number, then go forward to do the correct multiplication by 9.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 5 gives 30, find the number by asking what times 5 makes 30. That is 6.

\square \times 5 = 30 \Rightarrow \square = 6

Finding the missing factor of 30 with 5 is the unknown-factor idea: 6 times 5 is 30.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 6 by 9 as the problem intended.

6 \times 9 = 54

Once the number is known, the correct answer is just the fact 6 times 9.

Answer: 54

4 · Reviewdoes it hold up?

The hidden number 6 checks out because 6 x 5 = 30, matching the wrong result, and the correct answer 54 is larger than 30, which makes sense since multiplying by 9 instead of 5 gives more.

Another way: You could divide directly: 30 divided by 5 is 6, then 6 x 9 = 54, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 6 x 9 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 5 = 30 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 5 medium

A certain number was supposed to be multiplied by $8$ , but by mistake it was multiplied by $5$ , giving $35$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 8, but it was multiplied by 5 instead, giving 35. We must first find that number, then multiply it by 8 the correct way.

Givens

The wrong calculation multiplied the number by 5 and got 35.
The correct calculation should multiply the number by 8.

Unknowns

The hidden number.
The result of the correct calculation (number x 8).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 35 to recover the hidden number, then go forward to do the correct multiplication by 8.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 5 gives 35, find the number by asking what times 5 makes 35. That is 7.

\square \times 5 = 35 \Rightarrow \square = 7

Finding the missing factor of 35 with 5 is the unknown-factor idea: 7 times 5 is 35.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 7 by 8 as the problem intended.

7 \times 8 = 56

Once the number is known, the correct answer is just the fact 7 times 8.

Answer: 56

4 · Reviewdoes it hold up?

The hidden number 7 checks out because 7 x 5 = 35, matching the wrong result, and the correct answer 56 is larger than 35, which makes sense since multiplying by 8 instead of 5 gives more.

Another way: You could divide directly: 35 divided by 5 is 7, then 7 x 8 = 56, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 7 x 8 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 5 = 35 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 6 medium

A certain number was supposed to be multiplied by $6$ , but by mistake it was multiplied by $4$ , giving $36$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 6, but it was multiplied by 4 instead, giving 36. We must first find that number, then multiply it by 6 the correct way.

Givens

The wrong calculation multiplied the number by 4 and got 36.
The correct calculation should multiply the number by 6.

Unknowns

The hidden number.
The result of the correct calculation (number x 6).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 36 to recover the hidden number, then go forward to do the correct multiplication by 6.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 4 gives 36, find the number by asking what times 4 makes 36. That is 9.

\square \times 4 = 36 \Rightarrow \square = 9

Finding the missing factor of 36 with 4 is the unknown-factor idea: 9 times 4 is 36.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 9 by 6 as the problem intended.

9 \times 6 = 54

Once the number is known, the correct answer is just the fact 9 times 6.

Answer: 54

4 · Reviewdoes it hold up?

The hidden number 9 checks out because 9 x 4 = 36, matching the wrong result, and the correct answer 54 is larger than 36, which makes sense since multiplying by 6 instead of 4 gives more.

Another way: You could divide directly: 36 divided by 4 is 9, then 9 x 6 = 54, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 9 x 6 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 4 = 36 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 7 medium

A certain number was supposed to be multiplied by $2$ , but by mistake it was multiplied by $8$ , giving $40$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 2, but it was multiplied by 8 instead, giving 40. We must first find that number, then multiply it by 2 the correct way.

Givens

The wrong calculation multiplied the number by 8 and got 40.
The correct calculation should multiply the number by 2.

Unknowns

The hidden number.
The result of the correct calculation (number x 2).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 40 to recover the hidden number, then go forward to do the correct multiplication by 2.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 8 gives 40, find the number by asking what times 8 makes 40. That is 5.

\square \times 8 = 40 \Rightarrow \square = 5

Finding the missing factor of 40 with 8 is the unknown-factor idea: 5 times 8 is 40.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 5 by 2 as the problem intended.

5 \times 2 = 10

Once the number is known, the correct answer is just the fact 5 times 2.

Answer: 10

4 · Reviewdoes it hold up?

The hidden number 5 checks out because 5 x 8 = 40, matching the wrong result, and the correct answer 10 is smaller than 40, which makes sense since multiplying by 2 instead of 8 gives less.

Another way: You could divide directly: 40 divided by 8 is 5, then 5 x 2 = 10, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 5 x 2 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 8 = 40 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 8 medium

A certain number was supposed to be multiplied by $8$ , but by mistake it was multiplied by $7$ , giving $42$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 8, but it was multiplied by 7 instead, giving 42. We must first find that number, then multiply it by 8 the correct way.

Givens

The wrong calculation multiplied the number by 7 and got 42.
The correct calculation should multiply the number by 8.

Unknowns

The hidden number.
The result of the correct calculation (number x 8).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 42 to recover the hidden number, then go forward to do the correct multiplication by 8.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 7 gives 42, find the number by asking what times 7 makes 42. That is 6.

\square \times 7 = 42 \Rightarrow \square = 6

Finding the missing factor of 42 with 7 is the unknown-factor idea: 6 times 7 is 42.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 6 by 8 as the problem intended.

6 \times 8 = 48

Once the number is known, the correct answer is just the fact 6 times 8.

Answer: 48

4 · Reviewdoes it hold up?

The hidden number 6 checks out because 6 x 7 = 42, matching the wrong result, and the correct answer 48 is larger than 42, which makes sense since multiplying by 8 instead of 7 gives more.

Another way: You could divide directly: 42 divided by 7 is 6, then 6 x 8 = 48, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 6 x 8 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 7 = 42 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 9 hard

A certain number was supposed to be multiplied by $3$ , but by mistake it was multiplied by $6$ , giving $42$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 3, but it was multiplied by 6 instead, giving 42. We must first find that number, then multiply it by 3 the correct way.

Givens

The wrong calculation multiplied the number by 6 and got 42.
The correct calculation should multiply the number by 3.

Unknowns

The hidden number.
The result of the correct calculation (number x 3).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 42 to recover the hidden number, then go forward to do the correct multiplication by 3.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 6 gives 42, find the number by asking what times 6 makes 42. That is 7.

\square \times 6 = 42 \Rightarrow \square = 7

Finding the missing factor of 42 with 6 is the unknown-factor idea: 7 times 6 is 42.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 7 by 3 as the problem intended.

7 \times 3 = 21

Once the number is known, the correct answer is just the fact 7 times 3.

Answer: 21

4 · Reviewdoes it hold up?

The hidden number 7 checks out because 7 x 6 = 42, matching the wrong result, and the correct answer 21 is smaller than 42, which makes sense since multiplying by 3 instead of 6 gives less.

Another way: You could divide directly: 42 divided by 6 is 7, then 7 x 3 = 21, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 7 x 3 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 6 = 42 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 10 hard

A certain number was supposed to be multiplied by $5$ , but by mistake it was multiplied by $8$ , giving $48$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 5, but it was multiplied by 8 instead, giving 48. We must first find that number, then multiply it by 5 the correct way.

Givens

The wrong calculation multiplied the number by 8 and got 48.
The correct calculation should multiply the number by 5.

Unknowns

The hidden number.
The result of the correct calculation (number x 5).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 48 to recover the hidden number, then go forward to do the correct multiplication by 5.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 8 gives 48, find the number by asking what times 8 makes 48. That is 6.

\square \times 8 = 48 \Rightarrow \square = 6

Finding the missing factor of 48 with 8 is the unknown-factor idea: 6 times 8 is 48.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 6 by 5 as the problem intended.

6 \times 5 = 30

Once the number is known, the correct answer is just the fact 6 times 5.

Answer: 30

4 · Reviewdoes it hold up?

The hidden number 6 checks out because 6 x 8 = 48, matching the wrong result, and the correct answer 30 is smaller than 48, which makes sense since multiplying by 5 instead of 8 gives less.

Another way: You could divide directly: 48 divided by 8 is 6, then 6 x 5 = 30, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 6 x 5 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 8 = 48 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 11 hard

A certain number was supposed to be multiplied by $9$ , but by mistake it was multiplied by $6$ , giving $48$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 9, but it was multiplied by 6 instead, giving 48. We must first find that number, then multiply it by 9 the correct way.

Givens

The wrong calculation multiplied the number by 6 and got 48.
The correct calculation should multiply the number by 9.

Unknowns

The hidden number.
The result of the correct calculation (number x 9).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 48 to recover the hidden number, then go forward to do the correct multiplication by 9.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 6 gives 48, find the number by asking what times 6 makes 48. That is 8.

\square \times 6 = 48 \Rightarrow \square = 8

Finding the missing factor of 48 with 6 is the unknown-factor idea: 8 times 6 is 48.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 8 by 9 as the problem intended.

8 \times 9 = 72

Once the number is known, the correct answer is just the fact 8 times 9.

Answer: 72

4 · Reviewdoes it hold up?

The hidden number 8 checks out because 8 x 6 = 48, matching the wrong result, and the correct answer 72 is larger than 48, which makes sense since multiplying by 9 instead of 6 gives more.

Another way: You could divide directly: 48 divided by 6 is 8, then 8 x 9 = 72, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 8 x 9 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 6 = 48 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Variant 12 hard

A certain number was supposed to be multiplied by $5$ , but by mistake it was multiplied by $9$ , giving $72$ . What is the result of the correct calculation?

Your answer

Show solution

1 · Understandwhat's really being asked

A number should have been multiplied by 5, but it was multiplied by 9 instead, giving 72. We must first find that number, then multiply it by 5 the correct way.

Givens

The wrong calculation multiplied the number by 9 and got 72.
The correct calculation should multiply the number by 5.

Unknowns

The hidden number.
The result of the correct calculation (number x 5).

Constraints

The same hidden number is used in both the wrong and the correct calculation.

2 · Planchoose the strategy

#11 Work Backwards

The wrong result is given, so work backwards from 72 to recover the hidden number, then go forward to do the correct multiplication by 5.

3 · Execute2 carry out the plan

1Recover the hidden number

#11 Work Backwards 3.OA.B.6

Since the number times 9 gives 72, find the number by asking what times 9 makes 72. That is 8.

\square \times 9 = 72 \Rightarrow \square = 8

Finding the missing factor of 72 with 9 is the unknown-factor idea: 8 times 9 is 72.

2Do the correct calculation

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 8 by 5 as the problem intended.

8 \times 5 = 40

Once the number is known, the correct answer is just the fact 8 times 5.

Answer: 40

4 · Reviewdoes it hold up?

The hidden number 8 checks out because 8 x 9 = 72, matching the wrong result, and the correct answer 40 is smaller than 72, which makes sense since multiplying by 5 instead of 9 gives less.

Another way: You could divide directly: 72 divided by 9 is 8, then 8 x 5 = 40, using division to undo the multiplication.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 8 x 5 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 9 = 72 as a missing factor.

💡Takeaway. Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!

Where next?

← Back to the problem: Recover the unknown, then recompute