Practice · A rate applies to the whole, so only the whole has to be measured · Sensim Math

Variant 1 easy

In rectangle ABCD at the right, the shaded part's area is $\frac{4}{10}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Variant 2 easy

In rectangle ABCD at the right, the shaded part's area is $\frac{3}{5}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 13 cm by 15 cm has a shaded triangle covering 3 over 5 of it. We want the shaded area.

Givens

The rectangle is 13 cm wide and 15 cm tall.
The shaded part is 3/5 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

13 \times 15 = 195

The rectangle is 195 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

3 over 5 of the whole means the whole divided by 5, taken 3 times.

\frac{3}{5} \times 195

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

195 divided by 5 is 39, and 3 of those.

195 \div 5 \times 3 = 117

The shaded part is 117 cm2.

Answer: 117 cm²

4 · Reviewdoes it hold up?

117 out of 195 is a bit over half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 3 easy

In rectangle ABCD at the right, the shaded part's area is $\frac{5}{11}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 18 cm by 11 cm has a shaded triangle covering 5 over 11 of it. We want the shaded area.

Givens

The rectangle is 18 cm wide and 11 cm tall.
The shaded part is 5/11 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

18 \times 11 = 198

The rectangle is 198 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

5 over 11 of the whole means the whole divided by 11, taken 5 times.

\frac{5}{11} \times 198

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

198 divided by 11 is 18, and 5 of those.

198 \div 11 \times 5 = 90

The shaded part is 90 cm2.

Answer: 90 cm²

4 · Reviewdoes it hold up?

90 out of 198 is a bit under half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 4 easy

In rectangle ABCD at the right, the shaded part's area is $\frac{5}{10}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 18 cm by 12 cm has a shaded triangle covering 5 over 10 of it. We want the shaded area.

Givens

The rectangle is 18 cm wide and 12 cm tall.
The shaded part is 5/10 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

18 \times 12 = 216

The rectangle is 216 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

5 over 10 of the whole means the whole divided by 10, taken 5 times.

\frac{5}{10} \times 216

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

216 divided by 10 is 21, and 5 of those.

216 \div 10 \times 5 = 108

The shaded part is 108 cm2.

Answer: 108 cm²

4 · Reviewdoes it hold up?

108 out of 216 is a bit over half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 5 medium

In rectangle ABCD at the right, the shaded part's area is $\frac{3}{5}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Variant 6 medium

In rectangle ABCD at the right, the shaded part's area is $\frac{5}{9}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 24 cm by 18 cm has a shaded triangle covering 5 over 9 of it. We want the shaded area.

Givens

The rectangle is 24 cm wide and 18 cm tall.
The shaded part is 5/9 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

24 \times 18 = 432

The rectangle is 432 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

5 over 9 of the whole means the whole divided by 9, taken 5 times.

\frac{5}{9} \times 432

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

432 divided by 9 is 48, and 5 of those.

432 \div 9 \times 5 = 240

The shaded part is 240 cm2.

Answer: 240 cm²

4 · Reviewdoes it hold up?

240 out of 432 is a bit over half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 7 medium

In rectangle ABCD at the right, the shaded part's area is $\frac{13}{25}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 15 cm by 20 cm has a shaded triangle covering 13 over 25 of it. We want the shaded area.

Givens

The rectangle is 15 cm wide and 20 cm tall.
The shaded part is 13/25 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

15 \times 20 = 300

The rectangle is 300 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

13 over 25 of the whole means the whole divided by 25, taken 13 times.

\frac{13}{25} \times 300

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

300 divided by 25 is 12, and 13 of those.

300 \div 25 \times 13 = 156

The shaded part is 156 cm2.

Answer: 156 cm²

4 · Reviewdoes it hold up?

156 out of 300 is a bit over half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 8 medium

In rectangle ABCD at the right, the shaded part's area is $\frac{4}{10}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 26 cm by 20 cm has a shaded triangle covering 4 over 10 of it. We want the shaded area.

Givens

The rectangle is 26 cm wide and 20 cm tall.
The shaded part is 4/10 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

26 \times 20 = 520

The rectangle is 520 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

4 over 10 of the whole means the whole divided by 10, taken 4 times.

\frac{4}{10} \times 520

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

520 divided by 10 is 52, and 4 of those.

520 \div 10 \times 4 = 208

The shaded part is 208 cm2.

Answer: 208 cm²

4 · Reviewdoes it hold up?

208 out of 520 is a bit under half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 9 hard

In rectangle ABCD at the right, the shaded part's area is $\frac{13}{26}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 19 cm by 18 cm has a shaded triangle covering 13 over 26 of it. We want the shaded area.

Givens

The rectangle is 19 cm wide and 18 cm tall.
The shaded part is 13/26 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

19 \times 18 = 342

The rectangle is 342 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

13 over 26 of the whole means the whole divided by 26, taken 13 times.

\frac{13}{26} \times 342

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

342 divided by 26 is 13, and 13 of those.

342 \div 26 \times 13 = 171

The shaded part is 171 cm2.

Answer: 171 cm²

4 · Reviewdoes it hold up?

171 out of 342 is a bit over half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 10 hard

In rectangle ABCD at the right, the shaded part's area is $\frac{15}{30}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 27 cm by 16 cm has a shaded triangle covering 15 over 30 of it. We want the shaded area.

Givens

The rectangle is 27 cm wide and 16 cm tall.
The shaded part is 15/30 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

27 \times 16 = 432

The rectangle is 432 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

15 over 30 of the whole means the whole divided by 30, taken 15 times.

\frac{15}{30} \times 432

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

432 divided by 30 is 14, and 15 of those.

432 \div 30 \times 15 = 216

The shaded part is 216 cm2.

Answer: 216 cm²

4 · Reviewdoes it hold up?

216 out of 432 is a bit over half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 11 hard

In rectangle ABCD at the right, the shaded part's area is $\frac{14}{34}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 8 cm by 17 cm has a shaded triangle covering 14 over 34 of it. We want the shaded area.

Givens

The rectangle is 8 cm wide and 17 cm tall.
The shaded part is 14/34 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

8 \times 17 = 136

The rectangle is 136 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

14 over 34 of the whole means the whole divided by 34, taken 14 times.

\frac{14}{34} \times 136

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

136 divided by 34 is 4, and 14 of those.

136 \div 34 \times 14 = 56

The shaded part is 56 cm2.

Answer: 56 cm²

4 · Reviewdoes it hold up?

56 out of 136 is a bit under half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

Variant 12 hard

In rectangle ABCD at the right, the shaded part's area is $\frac{31}{63}$ of the whole area. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle 14 cm by 9 cm has a shaded triangle covering 31 over 63 of it. We want the shaded area.

Givens

The rectangle is 14 cm wide and 9 cm tall.
The shaded part is 31/63 of the whole rectangle.

Unknowns

The area of the shaded part.

Constraints

The triangle's own base and height are not given.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#16 Count the Complement

The triangle cannot be measured -- the point on the bottom side is not located. It does not need to be: a rate is an instruction to take a share of the whole, so measure the rectangle and take the share.

3 · Execute3 carry out the plan

1Find the whole area

#7 Identify Subproblems 6.G.A.1

The rectangle's two sides multiplied.

14 \times 9 = 126

The rectangle is 126 cm2.

2Read the rate as an instruction

#8 Analyze the Units 6.RP.A.3

31 over 63 of the whole means the whole divided by 63, taken 31 times.

\frac{31}{63} \times 126

No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3

126 divided by 63 is 2, and 31 of those.

126 \div 63 \times 31 = 62

The shaded part is 62 cm2.

Answer: 62 cm²

4 · Reviewdoes it hold up?

62 out of 126 is a bit under half the rectangle, which matches a triangle that reaches the full width and the full height.

Another way: Measuring the triangle would need the point on BC to be located, and the problem never says where it is -- so the rate is not a shortcut here, it is the only route.

Standardsmin grade 6

6.RP.A.3 Use ratio and rate reasoning to solve problems — Taking a stated share of a whole.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Finding the rectangle's area from its sides.

💡Takeaway. When you are told what share a part is, measure the whole and take that share. The part does not need measuring at all.

A rate applies to the whole, so only the whole has to be measured · 12 practice problems

Generated variants — 12

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