← A rate applies to the whole, so only the whole has to be measured · Ratio, Rate and Percent

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

6.RP.A.36.G.A.1

Generated variants — 12

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

Variant 1 easy answer: 32 cm²

In rectangle ABCD at the right, the shaded part's area is 410\frac{4}{10} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 8 cm 10 cm
Show solution
1 · Understandwhat's really being asked

A rectangle 8 cm by 10 cm has a shaded triangle covering 4 over 10 of it. We want the shaded area.

Givens
  • The rectangle is 8 cm wide and 10 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.
8×10=808 \times 10 = 80
The rectangle is 80 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.
410×80\frac{4}{10} \times 80
No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3
80 divided by 10 is 8, and 4 of those.
80÷10×4=3280 \div 10 \times 4 = 32
The shaded part is 32 cm2.
Answer: 32 cm²
4 · Reviewdoes it hold up?

32 out of 80 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 2 easy answer: 117 cm²

In rectangle ABCD at the right, the shaded part's area is 35\frac{3}{5} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 13 cm 15 cm
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.
13×15=19513 \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.
35×195\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÷5×3=117195 \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 answer: 90 cm²

In rectangle ABCD at the right, the shaded part's area is 511\frac{5}{11} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 18 cm 11 cm
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.
18×11=19818 \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.
511×198\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÷11×5=90198 \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 answer: 108 cm²

In rectangle ABCD at the right, the shaded part's area is 510\frac{5}{10} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 18 cm 12 cm
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.
18×12=21618 \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.
510×216\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÷10×5=108216 \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 answer: 72 cm²

In rectangle ABCD at the right, the shaded part's area is 35\frac{3}{5} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 20 cm 6 cm
Show solution
1 · Understandwhat's really being asked

A rectangle 20 cm by 6 cm has a shaded triangle covering 3 over 5 of it. We want the shaded area.

Givens
  • The rectangle is 20 cm wide and 6 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.
20×6=12020 \times 6 = 120
The rectangle is 120 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.
35×120\frac{3}{5} \times 120
No triangle measurements needed.

3Work it out

#8 Analyze the Units 6.RP.A.3
120 divided by 5 is 24, and 3 of those.
120÷5×3=72120 \div 5 \times 3 = 72
The shaded part is 72 cm2.
Answer: 72 cm²
4 · Reviewdoes it hold up?

72 out of 120 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 6 medium answer: 240 cm²

In rectangle ABCD at the right, the shaded part's area is 59\frac{5}{9} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 24 cm 18 cm
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.
24×18=43224 \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.
59×432\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÷9×5=240432 \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 answer: 156 cm²

In rectangle ABCD at the right, the shaded part's area is 1325\frac{13}{25} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 15 cm 20 cm
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.
15×20=30015 \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.
1325×300\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÷25×13=156300 \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 answer: 208 cm²

In rectangle ABCD at the right, the shaded part's area is 410\frac{4}{10} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 26 cm 20 cm
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.
26×20=52026 \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.
410×520\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÷10×4=208520 \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 answer: 171 cm²

In rectangle ABCD at the right, the shaded part's area is 1326\frac{13}{26} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 19 cm 18 cm
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.
19×18=34219 \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.
1326×342\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÷26×13=171342 \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 answer: 216 cm²

In rectangle ABCD at the right, the shaded part's area is 1530\frac{15}{30} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 27 cm 16 cm
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.
27×16=43227 \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.
1530×432\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÷30×15=216432 \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 answer: 56 cm²

In rectangle ABCD at the right, the shaded part's area is 1434\frac{14}{34} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 8 cm 17 cm
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.
8×17=1368 \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.
1434×136\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÷34×14=56136 \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 answer: 62 cm²

In rectangle ABCD at the right, the shaded part's area is 3163\frac{31}{63} of the whole area. What is the area of the shaded part, in cm2\text{cm}^2?

A D B C 14 cm 9 cm
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.
14×9=12614 \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.
3163×126\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÷63×31=62126 \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.