← Find a rectangle side via one unknown · Find Two Unknowns from Sum and Difference

Find a rectangle side via one unknown · 12 practice problems

3.MD.D.83.OA.C.73.OA.A.3

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Generated variants — 12

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

Variant 1 easy

Olivia used a piece of wire $40$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $4$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 40-inch wire is bent into a rectangle whose long side is 4 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 40 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 4 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 40 inches (all wire used).
Long side = 4 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (4 shorts), and again, giving 10 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 4 shorts, so the perimeter is 1 + 4 + 1 + 4 = 10 short-side lengths.

1 + 4 + 1 + 4 = 10 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 10 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 40-inch wire is divided into 10 equal short-side pieces.

40 \div 10 = 4 \text{ inches}

Sharing the total length equally among the 10 units gives one short side.

3Find the long side

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

The long side is 4 times the short side, so multiply 4 by 4.

4 \times 4 = 16 \text{ inches}

The long side is 4 short pieces joined, so scale the short side by 4.

Answer: 16 inches

4 · Reviewdoes it hold up?

Check: short 4, long 16, perimeter = 2 times (4 + 16) = 2 times 20 = 40 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 4s) = 10s = 40, so s = 4 and long side = 4s = 16.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 40-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 40 by 10 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 4 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 10 short sides, so divide, then multiply for the long side!

Variant 2 easy

Mia used a piece of wire $48$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $3$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 48-inch wire is bent into a rectangle whose long side is 3 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 48 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 3 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 48 inches (all wire used).
Long side = 3 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (3 shorts), and again, giving 8 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 3 shorts, so the perimeter is 1 + 3 + 1 + 3 = 8 short-side lengths.

1 + 3 + 1 + 3 = 8 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 8 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 48-inch wire is divided into 8 equal short-side pieces.

48 \div 8 = 6 \text{ inches}

Sharing the total length equally among the 8 units gives one short side.

3Find the long side

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

The long side is 3 times the short side, so multiply 6 by 3.

6 \times 3 = 18 \text{ inches}

The long side is 3 short pieces joined, so scale the short side by 3.

Answer: 18 inches

4 · Reviewdoes it hold up?

Check: short 6, long 18, perimeter = 2 times (6 + 18) = 2 times 24 = 48 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 3s) = 8s = 48, so s = 6 and long side = 3s = 18.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 48-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 48 by 8 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 3 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 8 short sides, so divide, then multiply for the long side!

Variant 3 easy

Sophia used a piece of wire $50$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $4$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 50-inch wire is bent into a rectangle whose long side is 4 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 50 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 4 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 50 inches (all wire used).
Long side = 4 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (4 shorts), and again, giving 10 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 4 shorts, so the perimeter is 1 + 4 + 1 + 4 = 10 short-side lengths.

1 + 4 + 1 + 4 = 10 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 10 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 50-inch wire is divided into 10 equal short-side pieces.

50 \div 10 = 5 \text{ inches}

Sharing the total length equally among the 10 units gives one short side.

3Find the long side

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

The long side is 4 times the short side, so multiply 5 by 4.

5 \times 4 = 20 \text{ inches}

The long side is 4 short pieces joined, so scale the short side by 4.

Answer: 20 inches

4 · Reviewdoes it hold up?

Check: short 5, long 20, perimeter = 2 times (5 + 20) = 2 times 25 = 50 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 4s) = 10s = 50, so s = 5 and long side = 4s = 20.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 50-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 50 by 10 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 4 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 10 short sides, so divide, then multiply for the long side!

Variant 4 easy

Lucas used a piece of wire $54$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $2$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 54-inch wire is bent into a rectangle whose long side is 2 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 54 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 2 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 54 inches (all wire used).
Long side = 2 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (2 shorts), and again, giving 6 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 2 shorts, so the perimeter is 1 + 2 + 1 + 2 = 6 short-side lengths.

1 + 2 + 1 + 2 = 6 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 6 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 54-inch wire is divided into 6 equal short-side pieces.

54 \div 6 = 9 \text{ inches}

Sharing the total length equally among the 6 units gives one short side.

3Find the long side

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

The long side is 2 times the short side, so multiply 9 by 2.

9 \times 2 = 18 \text{ inches}

The long side is 2 short pieces joined, so scale the short side by 2.

Answer: 18 inches

4 · Reviewdoes it hold up?

Check: short 9, long 18, perimeter = 2 times (9 + 18) = 2 times 27 = 54 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 2s) = 6s = 54, so s = 9 and long side = 2s = 18.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 54-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 54 by 6 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 2 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 6 short sides, so divide, then multiply for the long side!

Variant 5 medium

Noah used a piece of wire $60$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $5$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 60-inch wire is bent into a rectangle whose long side is 5 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 60 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 5 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 60 inches (all wire used).
Long side = 5 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (5 shorts), and again, giving 12 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 5 shorts, so the perimeter is 1 + 5 + 1 + 5 = 12 short-side lengths.

1 + 5 + 1 + 5 = 12 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 12 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 60-inch wire is divided into 12 equal short-side pieces.

60 \div 12 = 5 \text{ inches}

Sharing the total length equally among the 12 units gives one short side.

3Find the long side

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

The long side is 5 times the short side, so multiply 5 by 5.

5 \times 5 = 25 \text{ inches}

The long side is 5 short pieces joined, so scale the short side by 5.

Answer: 25 inches

4 · Reviewdoes it hold up?

Check: short 5, long 25, perimeter = 2 times (5 + 25) = 2 times 30 = 60 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 5s) = 12s = 60, so s = 5 and long side = 5s = 25.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 60-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 60 by 12 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 5 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 12 short sides, so divide, then multiply for the long side!

Variant 6 medium

Mason used a piece of wire $64$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $7$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 64-inch wire is bent into a rectangle whose long side is 7 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 64 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 7 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 64 inches (all wire used).
Long side = 7 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (7 shorts), and again, giving 16 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 7 shorts, so the perimeter is 1 + 7 + 1 + 7 = 16 short-side lengths.

1 + 7 + 1 + 7 = 16 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 16 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 64-inch wire is divided into 16 equal short-side pieces.

64 \div 16 = 4 \text{ inches}

Sharing the total length equally among the 16 units gives one short side.

3Find the long side

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

The long side is 7 times the short side, so multiply 4 by 7.

4 \times 7 = 28 \text{ inches}

The long side is 7 short pieces joined, so scale the short side by 7.

Answer: 28 inches

4 · Reviewdoes it hold up?

Check: short 4, long 28, perimeter = 2 times (4 + 28) = 2 times 32 = 64 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 7s) = 16s = 64, so s = 4 and long side = 7s = 28.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 64-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 64 by 16 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 7 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 16 short sides, so divide, then multiply for the long side!

Variant 7 medium

Chloe used a piece of wire $70$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $4$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 70-inch wire is bent into a rectangle whose long side is 4 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 70 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 4 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 70 inches (all wire used).
Long side = 4 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (4 shorts), and again, giving 10 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 4 shorts, so the perimeter is 1 + 4 + 1 + 4 = 10 short-side lengths.

1 + 4 + 1 + 4 = 10 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 10 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 70-inch wire is divided into 10 equal short-side pieces.

70 \div 10 = 7 \text{ inches}

Sharing the total length equally among the 10 units gives one short side.

3Find the long side

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

The long side is 4 times the short side, so multiply 7 by 4.

7 \times 4 = 28 \text{ inches}

The long side is 4 short pieces joined, so scale the short side by 4.

Answer: 28 inches

4 · Reviewdoes it hold up?

Check: short 7, long 28, perimeter = 2 times (7 + 28) = 2 times 35 = 70 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 4s) = 10s = 70, so s = 7 and long side = 4s = 28.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 70-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 70 by 10 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 4 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 10 short sides, so divide, then multiply for the long side!

Variant 8 medium

Emma used a piece of wire $72$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $2$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 72-inch wire is bent into a rectangle whose long side is 2 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 72 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 2 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 72 inches (all wire used).
Long side = 2 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (2 shorts), and again, giving 6 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 2 shorts, so the perimeter is 1 + 2 + 1 + 2 = 6 short-side lengths.

1 + 2 + 1 + 2 = 6 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 6 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 72-inch wire is divided into 6 equal short-side pieces.

72 \div 6 = 12 \text{ inches}

Sharing the total length equally among the 6 units gives one short side.

3Find the long side

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

The long side is 2 times the short side, so multiply 12 by 2.

12 \times 2 = 24 \text{ inches}

The long side is 2 short pieces joined, so scale the short side by 2.

Answer: 24 inches

4 · Reviewdoes it hold up?

Check: short 12, long 24, perimeter = 2 times (12 + 24) = 2 times 36 = 72 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 2s) = 6s = 72, so s = 12 and long side = 2s = 24.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 72-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 72 by 6 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 2 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 6 short sides, so divide, then multiply for the long side!

Variant 9 hard

Ethan used a piece of wire $84$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $5$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 84-inch wire is bent into a rectangle whose long side is 5 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 84 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 5 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 84 inches (all wire used).
Long side = 5 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (5 shorts), and again, giving 12 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 5 shorts, so the perimeter is 1 + 5 + 1 + 5 = 12 short-side lengths.

1 + 5 + 1 + 5 = 12 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 12 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 84-inch wire is divided into 12 equal short-side pieces.

84 \div 12 = 7 \text{ inches}

Sharing the total length equally among the 12 units gives one short side.

3Find the long side

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

The long side is 5 times the short side, so multiply 7 by 5.

7 \times 5 = 35 \text{ inches}

The long side is 5 short pieces joined, so scale the short side by 5.

Answer: 35 inches

4 · Reviewdoes it hold up?

Check: short 7, long 35, perimeter = 2 times (7 + 35) = 2 times 42 = 84 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 5s) = 12s = 84, so s = 7 and long side = 5s = 35.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 84-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 84 by 12 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 5 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 12 short sides, so divide, then multiply for the long side!

Variant 10 hard

Jackson used a piece of wire $84$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $2$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 84-inch wire is bent into a rectangle whose long side is 2 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 84 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 2 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 84 inches (all wire used).
Long side = 2 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (2 shorts), and again, giving 6 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 2 shorts, so the perimeter is 1 + 2 + 1 + 2 = 6 short-side lengths.

1 + 2 + 1 + 2 = 6 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 6 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 84-inch wire is divided into 6 equal short-side pieces.

84 \div 6 = 14 \text{ inches}

Sharing the total length equally among the 6 units gives one short side.

3Find the long side

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

The long side is 2 times the short side, so multiply 14 by 2.

14 \times 2 = 28 \text{ inches}

The long side is 2 short pieces joined, so scale the short side by 2.

Answer: 28 inches

4 · Reviewdoes it hold up?

Check: short 14, long 28, perimeter = 2 times (14 + 28) = 2 times 42 = 84 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 2s) = 6s = 84, so s = 14 and long side = 2s = 28.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 84-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 84 by 6 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 2 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 6 short sides, so divide, then multiply for the long side!

Variant 11 hard

Liam used a piece of wire $88$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $3$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 88-inch wire is bent into a rectangle whose long side is 3 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 88 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 3 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 88 inches (all wire used).
Long side = 3 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (3 shorts), and again, giving 8 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 3 shorts, so the perimeter is 1 + 3 + 1 + 3 = 8 short-side lengths.

1 + 3 + 1 + 3 = 8 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 8 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 88-inch wire is divided into 8 equal short-side pieces.

88 \div 8 = 11 \text{ inches}

Sharing the total length equally among the 8 units gives one short side.

3Find the long side

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

The long side is 3 times the short side, so multiply 11 by 3.

11 \times 3 = 33 \text{ inches}

The long side is 3 short pieces joined, so scale the short side by 3.

Answer: 33 inches

4 · Reviewdoes it hold up?

Check: short 11, long 33, perimeter = 2 times (11 + 33) = 2 times 44 = 88 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 3s) = 8s = 88, so s = 11 and long side = 3s = 33.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 88-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 88 by 8 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 3 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 8 short sides, so divide, then multiply for the long side!

Variant 12 hard

Ava used a piece of wire $96$ inches long, using all of it, to bend a rectangle. The long side of the rectangle is $3$ times the length of the short side. How many inches long is the long side?

Your answer

Show solution

1 · Understandwhat's really being asked

An 96-inch wire is bent into a rectangle whose long side is 3 times the short side, using all the wire. We need the length of the long side.

Givens

The wire is 96 inches long and all of it is used.
It forms a rectangle (the wire is the perimeter).
The long side is 3 times the short side.

Unknowns

The length of the long side in inches.

Constraints

The perimeter equals 96 inches (all wire used).
Long side = 3 times short side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #6 Guess and Check

Sketch the rectangle and count the short-side units around the perimeter: a short, a long (3 shorts), and again, giving 8 equal short-side pieces; divide the wire among them.

3 · Execute3 carry out the plan

1Count short-side units in the perimeter

#1 Draw a Diagram 3.MD.D.8

Going around: short + long + short + long. Each long is 3 shorts, so the perimeter is 1 + 3 + 1 + 3 = 8 short-side lengths.

1 + 3 + 1 + 3 = 8 \text{ short sides}

Measuring the whole border in short-side units turns the wire into 8 equal pieces.

2Find the short side

#6 Guess and Check 3.OA.C.7

The 96-inch wire is divided into 8 equal short-side pieces.

96 \div 8 = 12 \text{ inches}

Sharing the total length equally among the 8 units gives one short side.

3Find the long side

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

The long side is 3 times the short side, so multiply 12 by 3.

12 \times 3 = 36 \text{ inches}

The long side is 3 short pieces joined, so scale the short side by 3.

Answer: 36 inches

4 · Reviewdoes it hold up?

Check: short 12, long 36, perimeter = 2 times (12 + 36) = 2 times 48 = 96 inches, matching the wire exactly. The long side being longer than the short fits the picture.

Another way: Convert to algebra (tool 13): with short side s, perimeter 2(s + 3s) = 8s = 96, so s = 12 and long side = 3s = 36.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the 96-inch perimeter as a sum of short-side units.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 96 by 8 to find the short side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Scaling the short side by 3 to get the long side.

💡Takeaway. This only needs Grade 3 perimeter sense: the border is 8 short sides, so divide, then multiply for the long side!

Where next?

← Back to the problem: Find a rectangle side via one unknown