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Recover side length from number of cut pieces · 9 practice problems

3.OA.A.33.MD.D.8

Generated variants — 9

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

Variant 1 answer: 30 cm

A rectangular sheet of paper is $9\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $3\ \text{cm}$ on a side, giving $6$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 9 cm wide. It is cut with no leftover into 3 cm by 3 cm squares, making 6 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 9 cm wide (its top edge is 9 cm).
Each cut square is 3 cm by 3 cm.
Cutting the whole rectangle gives exactly 6 squares with no paper left over.
The figure shows the 9 cm width on top and one 3 cm by 3 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 3 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 3 cm squares fit across the 9 cm width (one row), then use the total of 6 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 3 cm wide and the rectangle is 9 cm wide, so divide to find how many squares fit in one row across the top.

9 \div 3 = 3

How many equal 3 cm pieces fit in 9 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 6 squares in all and 3 squares per row, so divide to find how many rows of squares there are.

6 \div 3 = 2

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 3 cm tall and there are 2 rows, so multiply to get the rectangle's height (its other side).

2 \times 3 = 6

Stacking 2 squares each 3 cm tall gives a height of 6 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 9 cm by 6 cm. Add the four sides, or double the sum of width and height.

2 \times (9 + 6) = 2 \times 15 = 30

A rectangle's perimeter is twice the length plus twice the width.

Answer: 30 cm

Review

The rectangle is 9 cm by 6 cm. Its area is 9 times 6 equals 54 square cm, and 6 squares of area 3 times 3 equals 9 each also total 54 square cm, so the pieces fit exactly. The perimeter 9 + 6 + 9 + 6 = 30 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 3 that makes 3 squares per row times the rows equal 6; only 2 rows (6 cm) works, giving the same 30 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 9 cm by 6 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!

Variant 2 answer: 64 cm

A rectangular sheet of paper is $20\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $4\ \text{cm}$ on a side, giving $15$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 20 cm wide. It is cut with no leftover into 4 cm by 4 cm squares, making 15 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 20 cm wide (its top edge is 20 cm).
Each cut square is 4 cm by 4 cm.
Cutting the whole rectangle gives exactly 15 squares with no paper left over.
The figure shows the 20 cm width on top and one 4 cm by 4 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 4 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 4 cm squares fit across the 20 cm width (one row), then use the total of 15 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 4 cm wide and the rectangle is 20 cm wide, so divide to find how many squares fit in one row across the top.

20 \div 4 = 5

How many equal 4 cm pieces fit in 20 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 15 squares in all and 5 squares per row, so divide to find how many rows of squares there are.

15 \div 5 = 3

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 4 cm tall and there are 3 rows, so multiply to get the rectangle's height (its other side).

3 \times 4 = 12

Stacking 3 squares each 4 cm tall gives a height of 12 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 20 cm by 12 cm. Add the four sides, or double the sum of width and height.

2 \times (20 + 12) = 2 \times 32 = 64

A rectangle's perimeter is twice the length plus twice the width.

Answer: 64 cm

Review

The rectangle is 20 cm by 12 cm. Its area is 20 times 12 equals 240 square cm, and 15 squares of area 4 times 4 equals 16 each also total 240 square cm, so the pieces fit exactly. The perimeter 20 + 12 + 20 + 12 = 64 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 4 that makes 5 squares per row times the rows equal 15; only 3 rows (12 cm) works, giving the same 64 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 20 cm by 12 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!

Variant 3 answer: 72 cm

A rectangular sheet of paper is $24\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $6\ \text{cm}$ on a side, giving $8$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 24 cm wide. It is cut with no leftover into 6 cm by 6 cm squares, making 8 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 24 cm wide (its top edge is 24 cm).
Each cut square is 6 cm by 6 cm.
Cutting the whole rectangle gives exactly 8 squares with no paper left over.
The figure shows the 24 cm width on top and one 6 cm by 6 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 6 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 6 cm squares fit across the 24 cm width (one row), then use the total of 8 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 6 cm wide and the rectangle is 24 cm wide, so divide to find how many squares fit in one row across the top.

24 \div 6 = 4

How many equal 6 cm pieces fit in 24 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 8 squares in all and 4 squares per row, so divide to find how many rows of squares there are.

8 \div 4 = 2

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 6 cm tall and there are 2 rows, so multiply to get the rectangle's height (its other side).

2 \times 6 = 12

Stacking 2 squares each 6 cm tall gives a height of 12 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 24 cm by 12 cm. Add the four sides, or double the sum of width and height.

2 \times (24 + 12) = 2 \times 36 = 72

A rectangle's perimeter is twice the length plus twice the width.

Answer: 72 cm

Review

The rectangle is 24 cm by 12 cm. Its area is 24 times 12 equals 288 square cm, and 8 squares of area 6 times 6 equals 36 each also total 288 square cm, so the pieces fit exactly. The perimeter 24 + 12 + 24 + 12 = 72 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 6 that makes 4 squares per row times the rows equal 8; only 2 rows (12 cm) works, giving the same 72 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 24 cm by 12 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!

Variant 4 answer: 42 cm

A rectangular sheet of paper is $12\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $3\ \text{cm}$ on a side, giving $12$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 12 cm wide. It is cut with no leftover into 3 cm by 3 cm squares, making 12 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 12 cm wide (its top edge is 12 cm).
Each cut square is 3 cm by 3 cm.
Cutting the whole rectangle gives exactly 12 squares with no paper left over.
The figure shows the 12 cm width on top and one 3 cm by 3 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 3 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 3 cm squares fit across the 12 cm width (one row), then use the total of 12 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 3 cm wide and the rectangle is 12 cm wide, so divide to find how many squares fit in one row across the top.

12 \div 3 = 4

How many equal 3 cm pieces fit in 12 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 12 squares in all and 4 squares per row, so divide to find how many rows of squares there are.

12 \div 4 = 3

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 3 cm tall and there are 3 rows, so multiply to get the rectangle's height (its other side).

3 \times 3 = 9

Stacking 3 squares each 3 cm tall gives a height of 9 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 12 cm by 9 cm. Add the four sides, or double the sum of width and height.

2 \times (12 + 9) = 2 \times 21 = 42

A rectangle's perimeter is twice the length plus twice the width.

Answer: 42 cm

Review

The rectangle is 12 cm by 9 cm. Its area is 12 times 9 equals 108 square cm, and 12 squares of area 3 times 3 equals 9 each also total 108 square cm, so the pieces fit exactly. The perimeter 12 + 9 + 12 + 9 = 42 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 3 that makes 4 squares per row times the rows equal 12; only 3 rows (9 cm) works, giving the same 42 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 12 cm by 9 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!

Variant 5 answer: 42 cm

A rectangular sheet of paper is $15\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $3\ \text{cm}$ on a side, giving $10$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 15 cm wide. It is cut with no leftover into 3 cm by 3 cm squares, making 10 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 15 cm wide (its top edge is 15 cm).
Each cut square is 3 cm by 3 cm.
Cutting the whole rectangle gives exactly 10 squares with no paper left over.
The figure shows the 15 cm width on top and one 3 cm by 3 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 3 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 3 cm squares fit across the 15 cm width (one row), then use the total of 10 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 3 cm wide and the rectangle is 15 cm wide, so divide to find how many squares fit in one row across the top.

15 \div 3 = 5

How many equal 3 cm pieces fit in 15 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 10 squares in all and 5 squares per row, so divide to find how many rows of squares there are.

10 \div 5 = 2

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 3 cm tall and there are 2 rows, so multiply to get the rectangle's height (its other side).

2 \times 3 = 6

Stacking 2 squares each 3 cm tall gives a height of 6 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 15 cm by 6 cm. Add the four sides, or double the sum of width and height.

2 \times (15 + 6) = 2 \times 21 = 42

A rectangle's perimeter is twice the length plus twice the width.

Answer: 42 cm

Review

The rectangle is 15 cm by 6 cm. Its area is 15 times 6 equals 90 square cm, and 10 squares of area 3 times 3 equals 9 each also total 90 square cm, so the pieces fit exactly. The perimeter 15 + 6 + 15 + 6 = 42 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 3 that makes 5 squares per row times the rows equal 10; only 2 rows (6 cm) works, giving the same 42 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 15 cm by 6 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!

Variant 6 answer: 70 cm

A rectangular sheet of paper is $25\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $5\ \text{cm}$ on a side, giving $10$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 25 cm wide. It is cut with no leftover into 5 cm by 5 cm squares, making 10 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 25 cm wide (its top edge is 25 cm).
Each cut square is 5 cm by 5 cm.
Cutting the whole rectangle gives exactly 10 squares with no paper left over.
The figure shows the 25 cm width on top and one 5 cm by 5 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 5 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 5 cm squares fit across the 25 cm width (one row), then use the total of 10 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 5 cm wide and the rectangle is 25 cm wide, so divide to find how many squares fit in one row across the top.

25 \div 5 = 5

How many equal 5 cm pieces fit in 25 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 10 squares in all and 5 squares per row, so divide to find how many rows of squares there are.

10 \div 5 = 2

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 5 cm tall and there are 2 rows, so multiply to get the rectangle's height (its other side).

2 \times 5 = 10

Stacking 2 squares each 5 cm tall gives a height of 10 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 25 cm by 10 cm. Add the four sides, or double the sum of width and height.

2 \times (25 + 10) = 2 \times 35 = 70

A rectangle's perimeter is twice the length plus twice the width.

Answer: 70 cm

Review

The rectangle is 25 cm by 10 cm. Its area is 25 times 10 equals 250 square cm, and 10 squares of area 5 times 5 equals 25 each also total 250 square cm, so the pieces fit exactly. The perimeter 25 + 10 + 25 + 10 = 70 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 5 that makes 5 squares per row times the rows equal 10; only 2 rows (10 cm) works, giving the same 70 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 25 cm by 10 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!

Variant 7 answer: 54 cm

A rectangular sheet of paper is $18\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $3\ \text{cm}$ on a side, giving $18$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 18 cm wide. It is cut with no leftover into 3 cm by 3 cm squares, making 18 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 18 cm wide (its top edge is 18 cm).
Each cut square is 3 cm by 3 cm.
Cutting the whole rectangle gives exactly 18 squares with no paper left over.
The figure shows the 18 cm width on top and one 3 cm by 3 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 3 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 3 cm squares fit across the 18 cm width (one row), then use the total of 18 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 3 cm wide and the rectangle is 18 cm wide, so divide to find how many squares fit in one row across the top.

18 \div 3 = 6

How many equal 3 cm pieces fit in 18 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 18 squares in all and 6 squares per row, so divide to find how many rows of squares there are.

18 \div 6 = 3

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 3 cm tall and there are 3 rows, so multiply to get the rectangle's height (its other side).

3 \times 3 = 9

Stacking 3 squares each 3 cm tall gives a height of 9 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 18 cm by 9 cm. Add the four sides, or double the sum of width and height.

2 \times (18 + 9) = 2 \times 27 = 54

A rectangle's perimeter is twice the length plus twice the width.

Answer: 54 cm

Review

The rectangle is 18 cm by 9 cm. Its area is 18 times 9 equals 162 square cm, and 18 squares of area 3 times 3 equals 9 each also total 162 square cm, so the pieces fit exactly. The perimeter 18 + 9 + 18 + 9 = 54 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 3 that makes 6 squares per row times the rows equal 18; only 3 rows (9 cm) works, giving the same 54 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 18 cm by 9 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!

Variant 8 answer: 56 cm

A rectangular sheet of paper is $16\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $4\ \text{cm}$ on a side, giving $12$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 16 cm wide. It is cut with no leftover into 4 cm by 4 cm squares, making 12 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 16 cm wide (its top edge is 16 cm).
Each cut square is 4 cm by 4 cm.
Cutting the whole rectangle gives exactly 12 squares with no paper left over.
The figure shows the 16 cm width on top and one 4 cm by 4 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 4 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 4 cm squares fit across the 16 cm width (one row), then use the total of 12 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 4 cm wide and the rectangle is 16 cm wide, so divide to find how many squares fit in one row across the top.

16 \div 4 = 4

How many equal 4 cm pieces fit in 16 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 12 squares in all and 4 squares per row, so divide to find how many rows of squares there are.

12 \div 4 = 3

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 4 cm tall and there are 3 rows, so multiply to get the rectangle's height (its other side).

3 \times 4 = 12

Stacking 3 squares each 4 cm tall gives a height of 12 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 16 cm by 12 cm. Add the four sides, or double the sum of width and height.

2 \times (16 + 12) = 2 \times 28 = 56

A rectangle's perimeter is twice the length plus twice the width.

Answer: 56 cm

Review

The rectangle is 16 cm by 12 cm. Its area is 16 times 12 equals 192 square cm, and 12 squares of area 4 times 4 equals 16 each also total 192 square cm, so the pieces fit exactly. The perimeter 16 + 12 + 16 + 12 = 56 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 4 that makes 4 squares per row times the rows equal 12; only 3 rows (12 cm) works, giving the same 56 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 16 cm by 12 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!

Variant 9 answer: 32 cm

A rectangular sheet of paper is $10\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $2\ \text{cm}$ on a side, giving $15$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

Show solution

Understand

A rectangular sheet of paper is 10 cm wide. It is cut with no leftover into 2 cm by 2 cm squares, making 15 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 10 cm wide (its top edge is 10 cm).
Each cut square is 2 cm by 2 cm.
Cutting the whole rectangle gives exactly 15 squares with no paper left over.
The figure shows the 10 cm width on top and one 2 cm by 2 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 2 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 2 cm squares fit across the 10 cm width (one row), then use the total of 15 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 2 cm wide and the rectangle is 10 cm wide, so divide to find how many squares fit in one row across the top.

10 \div 2 = 5

How many equal 2 cm pieces fit in 10 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 15 squares in all and 5 squares per row, so divide to find how many rows of squares there are.

15 \div 5 = 3

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 2 cm tall and there are 3 rows, so multiply to get the rectangle's height (its other side).

3 \times 2 = 6

Stacking 3 squares each 2 cm tall gives a height of 6 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 10 cm by 6 cm. Add the four sides, or double the sum of width and height.

2 \times (10 + 6) = 2 \times 16 = 32

A rectangle's perimeter is twice the length plus twice the width.

Answer: 32 cm

Review

The rectangle is 10 cm by 6 cm. Its area is 10 times 6 equals 60 square cm, and 15 squares of area 2 times 2 equals 4 each also total 60 square cm, so the pieces fit exactly. The perimeter 10 + 6 + 10 + 6 = 32 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 2 that makes 5 squares per row times the rows equal 15; only 3 rows (6 cm) works, giving the same 32 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 10 cm by 6 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!