← 3-1 · Dividend equals divisor times quotient · Division as the Inverse of Multiplication

Dividend equals divisor times quotient · 10 practice problems

3.OA.B.63.OA.A.4

Generated variants — 10

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

Variant 1 answer: A = 3, B = 6

The sum of $A$ and $B$ is $9$ . Find the value of $A$ and the value of $B$ .

$9 \div A = B \div 2$

Show solution

Understand

Two numbers A and B add up to 9, and they make the equation 9 divided by A equal to B divided by 2. We must find each of A and B.

Givens

A plus B equals 9.
9 divided by A equals B divided by 2.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 9, we can list the few sensible pairs and check which one keeps both sides of 9 divided by A equal to B divided by 2. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 9 nicely only for certain values, focus on pairs where A is a number 9 can be divided by, with B being 9 minus A. That gives the pairs (A=1,B=8), (A=3,B=6).

(A,B) \in \{(1,8),(3,6)\}

Grade 3 sense: dividing 9 evenly only works for the factors of 9, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=3, B=6): 9 divided by 3 is 3, and 6 divided by 2 is 3, so both sides equal 3. The other pairs fail to balance. Only A = 3, B = 6 works.

9 \div 3 = 3 \quad \text{and} \quad 6 \div 2 = 3

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 3.

Answer: A = 3, B = 6

Review

Check both conditions: A + B = 3 + 6 = 9 (correct), and 9 divided by 3 equals 3 while 6 divided by 2 equals 3, so the two sides match. Both requirements hold.

Rewrite 9 divided by A equals B divided by 2 as a cross-product: 9 times 2 equals A times B, so A times B = 18. Find two numbers that add to 9 and multiply to 18, which are 3 and 6; since A must divide 9, A = 3 and B = 6.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 9 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 9 and make 9 divide nicely, and only 3 and 6 keep both sides equal!

Variant 2 answer: A = 6, B = 6

The sum of $A$ and $B$ is $12$ . Find the value of $A$ and the value of $B$ .

$18 \div A = B \div 2$

Show solution

Understand

Two numbers A and B add up to 12, and they make the equation 18 divided by A equal to B divided by 2. We must find each of A and B.

Givens

A plus B equals 12.
18 divided by A equals B divided by 2.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 12, we can list the few sensible pairs and check which one keeps both sides of 18 divided by A equal to B divided by 2. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 18 nicely only for certain values, focus on pairs where A is a number 18 can be divided by, with B being 12 minus A. That gives the pairs (A=1,B=11), (A=2,B=10), (A=3,B=9), (A=6,B=6), (A=9,B=3).

(A,B) \in \{(1,11),(2,10),(3,9),(6,6),(9,3)\}

Grade 3 sense: dividing 18 evenly only works for the factors of 18, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=6, B=6): 18 divided by 6 is 3, and 6 divided by 2 is 3, so both sides equal 3. The other pairs fail to balance. Only A = 6, B = 6 works.

18 \div 6 = 3 \quad \text{and} \quad 6 \div 2 = 3

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 3.

Answer: A = 6, B = 6

Review

Check both conditions: A + B = 6 + 6 = 12 (correct), and 18 divided by 6 equals 3 while 6 divided by 2 equals 3, so the two sides match. Both requirements hold.

Rewrite 18 divided by A equals B divided by 2 as a cross-product: 18 times 2 equals A times B, so A times B = 36. Find two numbers that add to 12 and multiply to 36, which are 6 and 6; since A must divide 18, A = 6 and B = 6.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 18 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 12 and make 18 divide nicely, and only 6 and 6 keep both sides equal!

Variant 3 answer: A = 7, B = 4

The sum of $A$ and $B$ is $11$ . Find the value of $A$ and the value of $B$ .

$14 \div A = B \div 2$

Show solution

Understand

Two numbers A and B add up to 11, and they make the equation 14 divided by A equal to B divided by 2. We must find each of A and B.

Givens

A plus B equals 11.
14 divided by A equals B divided by 2.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 11, we can list the few sensible pairs and check which one keeps both sides of 14 divided by A equal to B divided by 2. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 14 nicely only for certain values, focus on pairs where A is a number 14 can be divided by, with B being 11 minus A. That gives the pairs (A=1,B=10), (A=2,B=9), (A=7,B=4).

(A,B) \in \{(1,10),(2,9),(7,4)\}

Grade 3 sense: dividing 14 evenly only works for the factors of 14, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=7, B=4): 14 divided by 7 is 2, and 4 divided by 2 is 2, so both sides equal 2. The other pairs fail to balance. Only A = 7, B = 4 works.

14 \div 7 = 2 \quad \text{and} \quad 4 \div 2 = 2

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 2.

Answer: A = 7, B = 4

Review

Check both conditions: A + B = 7 + 4 = 11 (correct), and 14 divided by 7 equals 2 while 4 divided by 2 equals 2, so the two sides match. Both requirements hold.

Rewrite 14 divided by A equals B divided by 2 as a cross-product: 14 times 2 equals A times B, so A times B = 28. Find two numbers that add to 11 and multiply to 28, which are 7 and 4; since A must divide 14, A = 7 and B = 4.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 14 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 11 and make 14 divide nicely, and only 7 and 4 keep both sides equal!

Variant 4 answer: A = 2, B = 6

The sum of $A$ and $B$ is $8$ . Find the value of $A$ and the value of $B$ .

$4 \div A = B \div 3$

Show solution

Understand

Two numbers A and B add up to 8, and they make the equation 4 divided by A equal to B divided by 3. We must find each of A and B.

Givens

A plus B equals 8.
4 divided by A equals B divided by 3.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 8, we can list the few sensible pairs and check which one keeps both sides of 4 divided by A equal to B divided by 3. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 4 nicely only for certain values, focus on pairs where A is a number 4 can be divided by, with B being 8 minus A. That gives the pairs (A=1,B=7), (A=2,B=6), (A=4,B=4).

(A,B) \in \{(1,7),(2,6),(4,4)\}

Grade 3 sense: dividing 4 evenly only works for the factors of 4, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=2, B=6): 4 divided by 2 is 2, and 6 divided by 3 is 2, so both sides equal 2. The other pairs fail to balance. Only A = 2, B = 6 works.

4 \div 2 = 2 \quad \text{and} \quad 6 \div 3 = 2

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 2.

Answer: A = 2, B = 6

Review

Check both conditions: A + B = 2 + 6 = 8 (correct), and 4 divided by 2 equals 2 while 6 divided by 3 equals 2, so the two sides match. Both requirements hold.

Rewrite 4 divided by A equals B divided by 3 as a cross-product: 4 times 3 equals A times B, so A times B = 12. Find two numbers that add to 8 and multiply to 12, which are 2 and 6; since A must divide 4, A = 2 and B = 6.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 4 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 8 and make 4 divide nicely, and only 2 and 6 keep both sides equal!

Variant 5 answer: A = 3, B = 4

The sum of $A$ and $B$ is $7$ . Find the value of $A$ and the value of $B$ .

$6 \div A = B \div 2$

Show solution

Understand

Two numbers A and B add up to 7, and they make the equation 6 divided by A equal to B divided by 2. We must find each of A and B.

Givens

A plus B equals 7.
6 divided by A equals B divided by 2.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 7, we can list the few sensible pairs and check which one keeps both sides of 6 divided by A equal to B divided by 2. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 6 nicely only for certain values, focus on pairs where A is a number 6 can be divided by, with B being 7 minus A. That gives the pairs (A=1,B=6), (A=2,B=5), (A=3,B=4), (A=6,B=1).

(A,B) \in \{(1,6),(2,5),(3,4),(6,1)\}

Grade 3 sense: dividing 6 evenly only works for the factors of 6, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=3, B=4): 6 divided by 3 is 2, and 4 divided by 2 is 2, so both sides equal 2. The other pairs fail to balance. Only A = 3, B = 4 works.

6 \div 3 = 2 \quad \text{and} \quad 4 \div 2 = 2

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 2.

Answer: A = 3, B = 4

Review

Check both conditions: A + B = 3 + 4 = 7 (correct), and 6 divided by 3 equals 2 while 4 divided by 2 equals 2, so the two sides match. Both requirements hold.

Rewrite 6 divided by A equals B divided by 2 as a cross-product: 6 times 2 equals A times B, so A times B = 12. Find two numbers that add to 7 and multiply to 12, which are 3 and 4; since A must divide 6, A = 3 and B = 4.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 6 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 7 and make 6 divide nicely, and only 3 and 4 keep both sides equal!

Variant 6 answer: A = 4, B = 6

The sum of $A$ and $B$ is $10$ . Find the value of $A$ and the value of $B$ .

$8 \div A = B \div 3$

Show solution

Understand

Two numbers A and B add up to 10, and they make the equation 8 divided by A equal to B divided by 3. We must find each of A and B.

Givens

A plus B equals 10.
8 divided by A equals B divided by 3.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 10, we can list the few sensible pairs and check which one keeps both sides of 8 divided by A equal to B divided by 3. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 8 nicely only for certain values, focus on pairs where A is a number 8 can be divided by, with B being 10 minus A. That gives the pairs (A=1,B=9), (A=2,B=8), (A=4,B=6), (A=8,B=2).

(A,B) \in \{(1,9),(2,8),(4,6),(8,2)\}

Grade 3 sense: dividing 8 evenly only works for the factors of 8, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=4, B=6): 8 divided by 4 is 2, and 6 divided by 3 is 2, so both sides equal 2. The other pairs fail to balance. Only A = 4, B = 6 works.

8 \div 4 = 2 \quad \text{and} \quad 6 \div 3 = 2

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 2.

Answer: A = 4, B = 6

Review

Check both conditions: A + B = 4 + 6 = 10 (correct), and 8 divided by 4 equals 2 while 6 divided by 3 equals 2, so the two sides match. Both requirements hold.

Rewrite 8 divided by A equals B divided by 3 as a cross-product: 8 times 3 equals A times B, so A times B = 24. Find two numbers that add to 10 and multiply to 24, which are 4 and 6; since A must divide 8, A = 4 and B = 6.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 8 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 10 and make 8 divide nicely, and only 4 and 6 keep both sides equal!

Variant 7 answer: A = 3, B = 8

The sum of $A$ and $B$ is $11$ . Find the value of $A$ and the value of $B$ .

$12 \div A = B \div 2$

Show solution

Understand

Two numbers A and B add up to 11, and they make the equation 12 divided by A equal to B divided by 2. We must find each of A and B.

Givens

A plus B equals 11.
12 divided by A equals B divided by 2.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 11, we can list the few sensible pairs and check which one keeps both sides of 12 divided by A equal to B divided by 2. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 12 nicely only for certain values, focus on pairs where A is a number 12 can be divided by, with B being 11 minus A. That gives the pairs (A=1,B=10), (A=2,B=9), (A=3,B=8), (A=4,B=7), (A=6,B=5).

(A,B) \in \{(1,10),(2,9),(3,8),(4,7),(6,5)\}

Grade 3 sense: dividing 12 evenly only works for the factors of 12, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=3, B=8): 12 divided by 3 is 4, and 8 divided by 2 is 4, so both sides equal 4. The other pairs fail to balance. Only A = 3, B = 8 works.

12 \div 3 = 4 \quad \text{and} \quad 8 \div 2 = 4

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 4.

Answer: A = 3, B = 8

Review

Check both conditions: A + B = 3 + 8 = 11 (correct), and 12 divided by 3 equals 4 while 8 divided by 2 equals 4, so the two sides match. Both requirements hold.

Rewrite 12 divided by A equals B divided by 2 as a cross-product: 12 times 2 equals A times B, so A times B = 24. Find two numbers that add to 11 and multiply to 24, which are 3 and 8; since A must divide 12, A = 3 and B = 8.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 12 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 11 and make 12 divide nicely, and only 3 and 8 keep both sides equal!

Variant 8 answer: A = 5, B = 4

The sum of $A$ and $B$ is $9$ . Find the value of $A$ and the value of $B$ .

$10 \div A = B \div 2$

Show solution

Understand

Two numbers A and B add up to 9, and they make the equation 10 divided by A equal to B divided by 2. We must find each of A and B.

Givens

A plus B equals 9.
10 divided by A equals B divided by 2.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 9, we can list the few sensible pairs and check which one keeps both sides of 10 divided by A equal to B divided by 2. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 10 nicely only for certain values, focus on pairs where A is a number 10 can be divided by, with B being 9 minus A. That gives the pairs (A=1,B=8), (A=2,B=7), (A=5,B=4).

(A,B) \in \{(1,8),(2,7),(5,4)\}

Grade 3 sense: dividing 10 evenly only works for the factors of 10, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=5, B=4): 10 divided by 5 is 2, and 4 divided by 2 is 2, so both sides equal 2. The other pairs fail to balance. Only A = 5, B = 4 works.

10 \div 5 = 2 \quad \text{and} \quad 4 \div 2 = 2

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 2.

Answer: A = 5, B = 4

Review

Check both conditions: A + B = 5 + 4 = 9 (correct), and 10 divided by 5 equals 2 while 4 divided by 2 equals 2, so the two sides match. Both requirements hold.

Rewrite 10 divided by A equals B divided by 2 as a cross-product: 10 times 2 equals A times B, so A times B = 20. Find two numbers that add to 9 and multiply to 20, which are 5 and 4; since A must divide 10, A = 5 and B = 4.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 10 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 9 and make 10 divide nicely, and only 5 and 4 keep both sides equal!

Variant 9 answer: A = 8, B = 6

The sum of $A$ and $B$ is $14$ . Find the value of $A$ and the value of $B$ .

$16 \div A = B \div 3$

Show solution

Understand

Two numbers A and B add up to 14, and they make the equation 16 divided by A equal to B divided by 3. We must find each of A and B.

Givens

A plus B equals 14.
16 divided by A equals B divided by 3.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 14, we can list the few sensible pairs and check which one keeps both sides of 16 divided by A equal to B divided by 3. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 16 nicely only for certain values, focus on pairs where A is a number 16 can be divided by, with B being 14 minus A. That gives the pairs (A=1,B=13), (A=2,B=12), (A=4,B=10), (A=8,B=6).

(A,B) \in \{(1,13),(2,12),(4,10),(8,6)\}

Grade 3 sense: dividing 16 evenly only works for the factors of 16, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=8, B=6): 16 divided by 8 is 2, and 6 divided by 3 is 2, so both sides equal 2. The other pairs fail to balance. Only A = 8, B = 6 works.

16 \div 8 = 2 \quad \text{and} \quad 6 \div 3 = 2

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 2.

Answer: A = 8, B = 6

Review

Check both conditions: A + B = 8 + 6 = 14 (correct), and 16 divided by 8 equals 2 while 6 divided by 3 equals 2, so the two sides match. Both requirements hold.

Rewrite 16 divided by A equals B divided by 3 as a cross-product: 16 times 3 equals A times B, so A times B = 48. Find two numbers that add to 14 and multiply to 48, which are 8 and 6; since A must divide 16, A = 8 and B = 6.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 16 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 14 and make 16 divide nicely, and only 8 and 6 keep both sides equal!

Variant 10 answer: A = 5, B = 6

The sum of $A$ and $B$ is $11$ . Find the value of $A$ and the value of $B$ .

$15 \div A = B \div 2$

Show solution

Understand

Two numbers A and B add up to 11, and they make the equation 15 divided by A equal to B divided by 2. We must find each of A and B.

Givens

A plus B equals 11.
15 divided by A equals B divided by 2.

Unknowns

The value of A.
The value of B.

Constraints

A and B are whole numbers that make both division expressions sensible.
A cannot be 0 because we divide by A.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Because A and B are small whole numbers that sum to 11, we can list the few sensible pairs and check which one keeps both sides of 15 divided by A equal to B divided by 2. Guess and check fits this bounded set perfectly.

Execute

#2 Make a Systematic List 3.OA.A.4

Since A divides 15 nicely only for certain values, focus on pairs where A is a number 15 can be divided by, with B being 11 minus A. That gives the pairs (A=1,B=10), (A=3,B=8), (A=5,B=6).

(A,B) \in \{(1,10),(3,8),(5,6)\}

Grade 3 sense: dividing 15 evenly only works for the factors of 15, so only a few A values are worth trying.

#6 Guess and Check 3.OA.B.6

Compute both sides for each pair. For (A=5, B=6): 15 divided by 5 is 3, and 6 divided by 2 is 3, so both sides equal 3. The other pairs fail to balance. Only A = 5, B = 6 works.

15 \div 5 = 3 \quad \text{and} \quad 6 \div 2 = 3

Grade 3 division: the equation balances only when both quotients come out to the same whole number, 3.

Answer: A = 5, B = 6

Review

Check both conditions: A + B = 5 + 6 = 11 (correct), and 15 divided by 5 equals 3 while 6 divided by 2 equals 3, so the two sides match. Both requirements hold.

Rewrite 15 divided by A equals B divided by 2 as a cross-product: 15 times 2 equals A times B, so A times B = 30. Find two numbers that add to 11 and multiply to 30, which are 5 and 6; since A must divide 15, A = 5 and B = 6.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the candidate values of A that divide 15 evenly.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that both sides of the equation give the same quotient.

💡 Try the few pairs that add to 11 and make 15 divide nicely, and only 5 and 6 keep both sides equal!