Practice · Find a missing side, then sum sides as fractions · Sensim Math

Variant 1 easy

The width of a rectangle is $3\dfrac{1}{6}$ cm, and its height is $2\dfrac{5}{6}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 3 $\frac{1}{6}$ cm. Its height is 2 $\frac{5}{6}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 3 $\frac{1}{6}$ cm.
Height = width + 2 $\frac{5}{6}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 6; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 6.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 2

\frac{5}{6}

= 3

\frac{1}{6}

+ 2

\frac{5}{6}

. Whole: 3+2 = 5. Fractions:

\frac{1}{6}

+

\frac{5}{6}

=

\frac{6}{6}

. So height = 6 cm.

3\tfrac{1}{6}+2\tfrac{5}{6}=6

Same-denominator add, then regroup if the fraction part reaches 6/6.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 3

\frac{1}{6}

+ 6 = 9

\frac{1}{6}

cm. This is half the perimeter.

3\tfrac{1}{6}+6=9\tfrac{1}{6}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 9

\frac{1}{6}

= 18

\frac{2}{6}

= 18

\frac{1}{3}

cm.

2\times 9\tfrac{1}{6}=18\tfrac{2}{6}=18\tfrac{1}{3}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

18\frac{1}{3}

cm

4 · Reviewdoes it hold up?

Width is about 3.17 cm and height about 6.00 cm, so perimeter is about 2 x 9.17 = 18.33 cm, matching 18 $\frac{1}{3}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 3 $\frac{1}{6}$ + 3 $\frac{1}{6}$ + 6 + 6 = 18 $\frac{2}{6}$ = 18 $\frac{1}{3}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 6 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 2 easy

The width of a rectangle is $2\dfrac{4}{7}$ cm, and its height is $1\dfrac{5}{7}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 2 $\frac{4}{7}$ cm. Its height is 1 $\frac{5}{7}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 2 $\frac{4}{7}$ cm.
Height = width + 1 $\frac{5}{7}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 7; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 7.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 1

\frac{5}{7}

= 2

\frac{4}{7}

+ 1

\frac{5}{7}

. Whole: 2+1 = 3. Fractions:

\frac{4}{7}

+

\frac{5}{7}

=

\frac{9}{7}

. So height = 4

\frac{2}{7}

cm.

2\tfrac{4}{7}+1\tfrac{5}{7}=4\tfrac{2}{7}

Same-denominator add, then regroup if the fraction part reaches 7/7.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 2

\frac{4}{7}

+ 4

\frac{2}{7}

= 6

\frac{6}{7}

cm. This is half the perimeter.

2\tfrac{4}{7}+4\tfrac{2}{7}=6\tfrac{6}{7}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 6

\frac{6}{7}

= 13

\frac{5}{7}

= 13

\frac{5}{7}

cm.

2\times 6\tfrac{6}{7}=13\tfrac{5}{7}=13\tfrac{5}{7}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

13\frac{5}{7}

cm

4 · Reviewdoes it hold up?

Width is about 2.57 cm and height about 4.29 cm, so perimeter is about 2 x 6.86 = 13.71 cm, matching 13 $\frac{5}{7}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 2 $\frac{4}{7}$ + 2 $\frac{4}{7}$ + 4 $\frac{2}{7}$ + 4 $\frac{2}{7}$ = 13 $\frac{5}{7}$ = 13 $\frac{5}{7}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 7 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 3 easy

The width of a rectangle is $2\dfrac{3}{8}$ cm, and its height is $1\dfrac{5}{8}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 2 $\frac{3}{8}$ cm. Its height is 1 $\frac{5}{8}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 2 $\frac{3}{8}$ cm.
Height = width + 1 $\frac{5}{8}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 8; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 8.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 1

\frac{5}{8}

= 2

\frac{3}{8}

+ 1

\frac{5}{8}

. Whole: 2+1 = 3. Fractions:

\frac{3}{8}

+

\frac{5}{8}

=

\frac{8}{8}

. So height = 4 cm.

2\tfrac{3}{8}+1\tfrac{5}{8}=4

Same-denominator add, then regroup if the fraction part reaches 8/8.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 2

\frac{3}{8}

+ 4 = 6

\frac{3}{8}

cm. This is half the perimeter.

2\tfrac{3}{8}+4=6\tfrac{3}{8}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 6

\frac{3}{8}

= 12

\frac{6}{8}

= 12

\frac{3}{4}

cm.

2\times 6\tfrac{3}{8}=12\tfrac{6}{8}=12\tfrac{3}{4}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

12\frac{3}{4}

cm

4 · Reviewdoes it hold up?

Width is about 2.38 cm and height about 4.00 cm, so perimeter is about 2 x 6.38 = 12.75 cm, matching 12 $\frac{3}{4}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 2 $\frac{3}{8}$ + 2 $\frac{3}{8}$ + 4 + 4 = 12 $\frac{6}{8}$ = 12 $\frac{3}{4}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 8 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 4 medium

The width of a rectangle is $4\dfrac{2}{9}$ cm, and its height is $2\dfrac{4}{9}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 4 $\frac{2}{9}$ cm. Its height is 2 $\frac{4}{9}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 4 $\frac{2}{9}$ cm.
Height = width + 2 $\frac{4}{9}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 9; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 9.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 2

\frac{4}{9}

= 4

\frac{2}{9}

+ 2

\frac{4}{9}

. Whole: 4+2 = 6. Fractions:

\frac{2}{9}

+

\frac{4}{9}

=

\frac{6}{9}

. So height = 6

\frac{6}{9}

cm.

4\tfrac{2}{9}+2\tfrac{4}{9}=6\tfrac{6}{9}

Same-denominator add, then regroup if the fraction part reaches 9/9.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 4

\frac{2}{9}

+ 6

\frac{6}{9}

= 10

\frac{8}{9}

cm. This is half the perimeter.

4\tfrac{2}{9}+6\tfrac{6}{9}=10\tfrac{8}{9}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 10

\frac{8}{9}

= 21

\frac{7}{9}

= 21

\frac{7}{9}

cm.

2\times 10\tfrac{8}{9}=21\tfrac{7}{9}=21\tfrac{7}{9}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

21\frac{7}{9}

cm

4 · Reviewdoes it hold up?

Width is about 4.22 cm and height about 6.67 cm, so perimeter is about 2 x 10.89 = 21.78 cm, matching 21 $\frac{7}{9}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 4 $\frac{2}{9}$ + 4 $\frac{2}{9}$ + 6 $\frac{6}{9}$ + 6 $\frac{6}{9}$ = 21 $\frac{7}{9}$ = 21 $\frac{7}{9}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 9 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 5 medium

The width of a rectangle is $5\dfrac{3}{10}$ cm, and its height is $1\dfrac{7}{10}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 5 $\frac{3}{10}$ cm. Its height is 1 $\frac{7}{10}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 5 $\frac{3}{10}$ cm.
Height = width + 1 $\frac{7}{10}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 10; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 10.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 1

\frac{7}{10}

= 5

\frac{3}{10}

+ 1

\frac{7}{10}

. Whole: 5+1 = 6. Fractions:

\frac{3}{10}

+

\frac{7}{10}

=

\frac{10}{10}

. So height = 7 cm.

5\tfrac{3}{10}+1\tfrac{7}{10}=7

Same-denominator add, then regroup if the fraction part reaches 10/10.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 5

\frac{3}{10}

+ 7 = 12

\frac{3}{10}

cm. This is half the perimeter.

5\tfrac{3}{10}+7=12\tfrac{3}{10}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 12

\frac{3}{10}

= 24

\frac{6}{10}

= 24

\frac{3}{5}

cm.

2\times 12\tfrac{3}{10}=24\tfrac{6}{10}=24\tfrac{3}{5}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

24\frac{3}{5}

cm

4 · Reviewdoes it hold up?

Width is about 5.30 cm and height about 7.00 cm, so perimeter is about 2 x 12.30 = 24.60 cm, matching 24 $\frac{3}{5}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 5 $\frac{3}{10}$ + 5 $\frac{3}{10}$ + 7 + 7 = 24 $\frac{6}{10}$ = 24 $\frac{3}{5}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 10 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 6 medium

The width of a rectangle is $4\dfrac{6}{11}$ cm, and its height is $1\dfrac{8}{11}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 4 $\frac{6}{11}$ cm. Its height is 1 $\frac{8}{11}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 4 $\frac{6}{11}$ cm.
Height = width + 1 $\frac{8}{11}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 11; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 11.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 1

\frac{8}{11}

= 4

\frac{6}{11}

+ 1

\frac{8}{11}

. Whole: 4+1 = 5. Fractions:

\frac{6}{11}

+

\frac{8}{11}

=

\frac{14}{11}

. So height = 6

\frac{3}{11}

cm.

4\tfrac{6}{11}+1\tfrac{8}{11}=6\tfrac{3}{11}

Same-denominator add, then regroup if the fraction part reaches 11/11.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 4

\frac{6}{11}

+ 6

\frac{3}{11}

= 10

\frac{9}{11}

cm. This is half the perimeter.

4\tfrac{6}{11}+6\tfrac{3}{11}=10\tfrac{9}{11}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 10

\frac{9}{11}

= 21

\frac{7}{11}

= 21

\frac{7}{11}

cm.

2\times 10\tfrac{9}{11}=21\tfrac{7}{11}=21\tfrac{7}{11}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

21\frac{7}{11}

cm

4 · Reviewdoes it hold up?

Width is about 4.55 cm and height about 6.27 cm, so perimeter is about 2 x 10.82 = 21.64 cm, matching 21 $\frac{7}{11}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 4 $\frac{6}{11}$ + 4 $\frac{6}{11}$ + 6 $\frac{3}{11}$ + 6 $\frac{3}{11}$ = 21 $\frac{7}{11}$ = 21 $\frac{7}{11}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 11 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 7 medium

The width of a rectangle is $1\dfrac{7}{12}$ cm, and its height is $3\dfrac{9}{12}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 1 $\frac{7}{12}$ cm. Its height is 3 $\frac{9}{12}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 1 $\frac{7}{12}$ cm.
Height = width + 3 $\frac{9}{12}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 12; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 12.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 3

\frac{9}{12}

= 1

\frac{7}{12}

+ 3

\frac{9}{12}

. Whole: 1+3 = 4. Fractions:

\frac{7}{12}

+

\frac{9}{12}

=

\frac{16}{12}

. So height = 5

\frac{4}{12}

cm.

1\tfrac{7}{12}+3\tfrac{9}{12}=5\tfrac{4}{12}

Same-denominator add, then regroup if the fraction part reaches 12/12.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 1

\frac{7}{12}

+ 5

\frac{4}{12}

= 6

\frac{11}{12}

cm. This is half the perimeter.

1\tfrac{7}{12}+5\tfrac{4}{12}=6\tfrac{11}{12}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 6

\frac{11}{12}

= 13

\frac{10}{12}

= 13

\frac{5}{6}

cm.

2\times 6\tfrac{11}{12}=13\tfrac{10}{12}=13\tfrac{5}{6}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

13\frac{5}{6}

cm

4 · Reviewdoes it hold up?

Width is about 1.58 cm and height about 5.33 cm, so perimeter is about 2 x 6.92 = 13.83 cm, matching 13 $\frac{5}{6}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 1 $\frac{7}{12}$ + 1 $\frac{7}{12}$ + 5 $\frac{4}{12}$ + 5 $\frac{4}{12}$ = 13 $\frac{10}{12}$ = 13 $\frac{5}{6}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 12 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 8 hard

The width of a rectangle is $2\dfrac{5}{14}$ cm, and its height is $3\dfrac{11}{14}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 2 $\frac{5}{14}$ cm. Its height is 3 $\frac{11}{14}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 2 $\frac{5}{14}$ cm.
Height = width + 3 $\frac{11}{14}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 14; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 14.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 3

\frac{11}{14}

= 2

\frac{5}{14}

+ 3

\frac{11}{14}

. Whole: 2+3 = 5. Fractions:

\frac{5}{14}

+

\frac{11}{14}

=

\frac{16}{14}

. So height = 6

\frac{2}{14}

cm.

2\tfrac{5}{14}+3\tfrac{11}{14}=6\tfrac{2}{14}

Same-denominator add, then regroup if the fraction part reaches 14/14.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 2

\frac{5}{14}

+ 6

\frac{2}{14}

= 8

\frac{7}{14}

cm. This is half the perimeter.

2\tfrac{5}{14}+6\tfrac{2}{14}=8\tfrac{7}{14}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 8

\frac{7}{14}

= 17 = 17 cm.

2\times 8\tfrac{7}{14}=17=17

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

17

cm

4 · Reviewdoes it hold up?

Width is about 2.36 cm and height about 6.14 cm, so perimeter is about 2 x 8.50 = 17.00 cm, matching 17 cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 2 $\frac{5}{14}$ + 2 $\frac{5}{14}$ + 6 $\frac{2}{14}$ + 6 $\frac{2}{14}$ = 17 = 17 cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 14 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 9 hard

The width of a rectangle is $3\dfrac{5}{15}$ cm, and its height is $1\dfrac{11}{15}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 3 $\frac{5}{15}$ cm. Its height is 1 $\frac{11}{15}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 3 $\frac{5}{15}$ cm.
Height = width + 1 $\frac{11}{15}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 15; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 15.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 1

\frac{11}{15}

= 3

\frac{5}{15}

+ 1

\frac{11}{15}

. Whole: 3+1 = 4. Fractions:

\frac{5}{15}

+

\frac{11}{15}

=

\frac{16}{15}

. So height = 5

\frac{1}{15}

cm.

3\tfrac{5}{15}+1\tfrac{11}{15}=5\tfrac{1}{15}

Same-denominator add, then regroup if the fraction part reaches 15/15.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 3

\frac{5}{15}

+ 5

\frac{1}{15}

= 8

\frac{6}{15}

cm. This is half the perimeter.

3\tfrac{5}{15}+5\tfrac{1}{15}=8\tfrac{6}{15}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 8

\frac{6}{15}

= 16

\frac{12}{15}

= 16

\frac{4}{5}

cm.

2\times 8\tfrac{6}{15}=16\tfrac{12}{15}=16\tfrac{4}{5}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

16\frac{4}{5}

cm

4 · Reviewdoes it hold up?

Width is about 3.33 cm and height about 5.07 cm, so perimeter is about 2 x 8.40 = 16.80 cm, matching 16 $\frac{4}{5}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 3 $\frac{5}{15}$ + 3 $\frac{5}{15}$ + 5 $\frac{1}{15}$ + 5 $\frac{1}{15}$ = 16 $\frac{12}{15}$ = 16 $\frac{4}{5}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 15 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Variant 10 hard

The width of a rectangle is $1\dfrac{9}{20}$ cm, and its height is $2\dfrac{15}{20}$ cm longer than the width. Find the sum of the lengths of the four sides of this rectangle, in cm.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has width 1 $\frac{9}{20}$ cm. Its height is 2 $\frac{15}{20}$ cm longer than the width. I must find the perimeter (the sum of all four side lengths) in cm.

Givens

Width = 1 $\frac{9}{20}$ cm.
Height = width + 2 $\frac{15}{20}$ cm.
A rectangle has two widths and two heights.

Unknowns

The perimeter (sum of the four sides) of the rectangle.

Constraints

Fraction parts share denominator 20; perimeter = 2 x (width + height).

2 · Planchoose the strategy

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

First find the height by adding to the width, then apply the rectangle perimeter formula P = 2 x (width + height). Two clean subproblems, both with denominator 20.

3 · Execute3 carry out the plan

1Find the height

#7 Identify Subproblems 4.NF.B.3

Height = width + 2

\frac{15}{20}

= 1

\frac{9}{20}

+ 2

\frac{15}{20}

. Whole: 1+2 = 3. Fractions:

\frac{9}{20}

+

\frac{15}{20}

=

\frac{24}{20}

. So height = 4

\frac{4}{20}

cm.

1\tfrac{9}{20}+2\tfrac{15}{20}=4\tfrac{4}{20}

Same-denominator add, then regroup if the fraction part reaches 20/20.

2Add width and height

#7 Identify Subproblems 4.NF.B.3

width + height = 1

\frac{9}{20}

+ 4

\frac{4}{20}

= 5

\frac{13}{20}

cm. This is half the perimeter.

1\tfrac{9}{20}+4\tfrac{4}{20}=5\tfrac{13}{20}

Adding one width and one height gives half the perimeter.

3Double to get the perimeter

#7 Identify Subproblems 4.MD.A.3

Perimeter = 2 x (width + height) = 2 x 5

\frac{13}{20}

= 11

\frac{6}{20}

= 11

\frac{3}{10}

cm.

2\times 5\tfrac{13}{20}=11\tfrac{6}{20}=11\tfrac{3}{10}

A rectangle has two equal widths and two equal heights, so doubling the half-perimeter gives the whole.

Answer:

11\frac{3}{10}

cm

4 · Reviewdoes it hold up?

Width is about 1.45 cm and height about 4.20 cm, so perimeter is about 2 x 5.65 = 11.30 cm, matching 11 $\frac{3}{10}$ cm. Units stay in cm.

Another way: Add all four sides separately (tool 1): 1 $\frac{9}{20}$ + 1 $\frac{9}{20}$ + 4 $\frac{4}{20}$ + 4 $\frac{4}{20}$ = 11 $\frac{6}{20}$ = 11 $\frac{3}{10}$ cm, the same result.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the mixed-number side lengths with denominator 20 and regrouping improper fractions.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using perimeter = 2 x (width + height) to total the four sides.

💡Takeaway. This only needs Grade 4 fraction adding plus the rectangle perimeter formula — two widths plus two heights!

Find a missing side, then sum sides as fractions · 10 practice problems

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