← Map length equals real distance times scale · Part-Whole Fraction Reasoning

Map length equals real distance times scale · 12 practice problems

5.MD.A.15.NF.B.65.NF.B.5

Generated variants — 12

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

Variant 1 medium answer: 21 inches

A real-world distance of $14$ miles is drawn on a map with a scale of $\dfrac{1}{42240}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 14 miles in the real world is drawn on a map at a scale of one to 42240. We need how long that distance is on the map, in inches.

Givens

The real distance is 14 miles.
The scale is $\frac{1}{42240}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 14 of them.

14 \times 63360 = 887040\ \text{in}

The real distance is 887040 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 42240 means the drawing is

\frac{1}{42240}

of the real thing, so divide by 42240.

887040 \times \frac{1}{42240} = 21\ \text{in}

On the map it measures 21 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 21 inches is far less than 887040.

21 \times 42240 = 887040

Scaling back up recovers the real distance, so the direction was right.

Answer: 21 inches

4 · Reviewdoes it hold up?

A map inch stands for 42240 real inches, so 21 map inches stand for 887040 real inches -- exactly the 14 miles we started with.

Another way: Since 42240 inches is 2/3 of a mile, each map inch is that much real distance; dividing 14 miles by it gives 21 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 14 miles into 887040 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 2 medium answer: 9 inches

A real-world distance of $6$ miles is drawn on a map with a scale of $\dfrac{1}{42240}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 6 miles in the real world is drawn on a map at a scale of one to 42240. We need how long that distance is on the map, in inches.

Givens

The real distance is 6 miles.
The scale is $\frac{1}{42240}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 6 of them.

6 \times 63360 = 380160\ \text{in}

The real distance is 380160 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 42240 means the drawing is

\frac{1}{42240}

of the real thing, so divide by 42240.

380160 \times \frac{1}{42240} = 9\ \text{in}

On the map it measures 9 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 9 inches is far less than 380160.

9 \times 42240 = 380160

Scaling back up recovers the real distance, so the direction was right.

Answer: 9 inches

4 · Reviewdoes it hold up?

A map inch stands for 42240 real inches, so 9 map inches stand for 380160 real inches -- exactly the 6 miles we started with.

Another way: Since 42240 inches is 2/3 of a mile, each map inch is that much real distance; dividing 6 miles by it gives 9 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 6 miles into 380160 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 3 easy answer: 6 inches

A real-world distance of $3$ miles is drawn on a map with a scale of $\dfrac{1}{31680}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 3 miles in the real world is drawn on a map at a scale of one to 31680. We need how long that distance is on the map, in inches.

Givens

The real distance is 3 miles.
The scale is $\frac{1}{31680}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 3 of them.

3 \times 63360 = 190080\ \text{in}

The real distance is 190080 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 31680 means the drawing is

\frac{1}{31680}

of the real thing, so divide by 31680.

190080 \times \frac{1}{31680} = 6\ \text{in}

On the map it measures 6 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 6 inches is far less than 190080.

6 \times 31680 = 190080

Scaling back up recovers the real distance, so the direction was right.

Answer: 6 inches

4 · Reviewdoes it hold up?

A map inch stands for 31680 real inches, so 6 map inches stand for 190080 real inches -- exactly the 3 miles we started with.

Another way: Since 31680 inches is 1/2 of a mile, each map inch is that much real distance; dividing 3 miles by it gives 6 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 3 miles into 190080 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 4 easy answer: 5 inches

A real-world distance of $5$ miles is drawn on a map with a scale of $\dfrac{1}{63360}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 5 miles in the real world is drawn on a map at a scale of one to 63360. We need how long that distance is on the map, in inches.

Givens

The real distance is 5 miles.
The scale is $\frac{1}{63360}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 5 of them.

5 \times 63360 = 316800\ \text{in}

The real distance is 316800 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 63360 means the drawing is

\frac{1}{63360}

of the real thing, so divide by 63360.

316800 \times \frac{1}{63360} = 5\ \text{in}

On the map it measures 5 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 5 inches is far less than 316800.

5 \times 63360 = 316800

Scaling back up recovers the real distance, so the direction was right.

Answer: 5 inches

4 · Reviewdoes it hold up?

A map inch stands for 63360 real inches, so 5 map inches stand for 316800 real inches -- exactly the 5 miles we started with.

Another way: Since 63360 inches is 1 of a mile, each map inch is that much real distance; dividing 5 miles by it gives 5 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 5 miles into 316800 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 5 medium answer: 9 inches

A real-world distance of $9$ miles is drawn on a map with a scale of $\dfrac{1}{63360}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 9 miles in the real world is drawn on a map at a scale of one to 63360. We need how long that distance is on the map, in inches.

Givens

The real distance is 9 miles.
The scale is $\frac{1}{63360}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 9 of them.

9 \times 63360 = 570240\ \text{in}

The real distance is 570240 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 63360 means the drawing is

\frac{1}{63360}

of the real thing, so divide by 63360.

570240 \times \frac{1}{63360} = 9\ \text{in}

On the map it measures 9 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 9 inches is far less than 570240.

9 \times 63360 = 570240

Scaling back up recovers the real distance, so the direction was right.

Answer: 9 inches

4 · Reviewdoes it hold up?

A map inch stands for 63360 real inches, so 9 map inches stand for 570240 real inches -- exactly the 9 miles we started with.

Another way: Since 63360 inches is 1 of a mile, each map inch is that much real distance; dividing 9 miles by it gives 9 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 9 miles into 570240 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 6 easy answer: 6 inches

A real-world distance of $2$ miles is drawn on a map with a scale of $\dfrac{1}{21120}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 2 miles in the real world is drawn on a map at a scale of one to 21120. We need how long that distance is on the map, in inches.

Givens

The real distance is 2 miles.
The scale is $\frac{1}{21120}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 2 of them.

2 \times 63360 = 126720\ \text{in}

The real distance is 126720 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 21120 means the drawing is

\frac{1}{21120}

of the real thing, so divide by 21120.

126720 \times \frac{1}{21120} = 6\ \text{in}

On the map it measures 6 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 6 inches is far less than 126720.

6 \times 21120 = 126720

Scaling back up recovers the real distance, so the direction was right.

Answer: 6 inches

4 · Reviewdoes it hold up?

A map inch stands for 21120 real inches, so 6 map inches stand for 126720 real inches -- exactly the 2 miles we started with.

Another way: Since 21120 inches is 1/3 of a mile, each map inch is that much real distance; dividing 2 miles by it gives 6 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 2 miles into 126720 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 7 easy answer: 12 inches

A real-world distance of $12$ miles is drawn on a map with a scale of $\dfrac{1}{63360}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 12 miles in the real world is drawn on a map at a scale of one to 63360. We need how long that distance is on the map, in inches.

Givens

The real distance is 12 miles.
The scale is $\frac{1}{63360}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 12 of them.

12 \times 63360 = 760320\ \text{in}

The real distance is 760320 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 63360 means the drawing is

\frac{1}{63360}

of the real thing, so divide by 63360.

760320 \times \frac{1}{63360} = 12\ \text{in}

On the map it measures 12 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 12 inches is far less than 760320.

12 \times 63360 = 760320

Scaling back up recovers the real distance, so the direction was right.

Answer: 12 inches

4 · Reviewdoes it hold up?

A map inch stands for 63360 real inches, so 12 map inches stand for 760320 real inches -- exactly the 12 miles we started with.

Another way: Since 63360 inches is 1 of a mile, each map inch is that much real distance; dividing 12 miles by it gives 12 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 12 miles into 760320 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 8 medium answer: 8 inches

A real-world distance of $4$ miles is drawn on a map with a scale of $\dfrac{1}{31680}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 4 miles in the real world is drawn on a map at a scale of one to 31680. We need how long that distance is on the map, in inches.

Givens

The real distance is 4 miles.
The scale is $\frac{1}{31680}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 4 of them.

4 \times 63360 = 253440\ \text{in}

The real distance is 253440 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 31680 means the drawing is

\frac{1}{31680}

of the real thing, so divide by 31680.

253440 \times \frac{1}{31680} = 8\ \text{in}

On the map it measures 8 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 8 inches is far less than 253440.

8 \times 31680 = 253440

Scaling back up recovers the real distance, so the direction was right.

Answer: 8 inches

4 · Reviewdoes it hold up?

A map inch stands for 31680 real inches, so 8 map inches stand for 253440 real inches -- exactly the 4 miles we started with.

Another way: Since 31680 inches is 1/2 of a mile, each map inch is that much real distance; dividing 4 miles by it gives 8 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 4 miles into 253440 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 9 hard answer: 6 inches

A real-world distance of $10$ miles is drawn on a map with a scale of $\dfrac{1}{105600}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 10 miles in the real world is drawn on a map at a scale of one to 105600. We need how long that distance is on the map, in inches.

Givens

The real distance is 10 miles.
The scale is $\frac{1}{105600}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 10 of them.

10 \times 63360 = 633600\ \text{in}

The real distance is 633600 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 105600 means the drawing is

\frac{1}{105600}

of the real thing, so divide by 105600.

633600 \times \frac{1}{105600} = 6\ \text{in}

On the map it measures 6 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 6 inches is far less than 633600.

6 \times 105600 = 633600

Scaling back up recovers the real distance, so the direction was right.

Answer: 6 inches

4 · Reviewdoes it hold up?

A map inch stands for 105600 real inches, so 6 map inches stand for 633600 real inches -- exactly the 10 miles we started with.

Another way: Since 105600 inches is 5/3 of a mile, each map inch is that much real distance; dividing 10 miles by it gives 6 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 10 miles into 633600 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 10 hard answer: 4 inches

A real-world distance of $8$ miles is drawn on a map with a scale of $\dfrac{1}{126720}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 8 miles in the real world is drawn on a map at a scale of one to 126720. We need how long that distance is on the map, in inches.

Givens

The real distance is 8 miles.
The scale is $\frac{1}{126720}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 8 of them.

8 \times 63360 = 506880\ \text{in}

The real distance is 506880 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 126720 means the drawing is

\frac{1}{126720}

of the real thing, so divide by 126720.

506880 \times \frac{1}{126720} = 4\ \text{in}

On the map it measures 4 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 4 inches is far less than 506880.

4 \times 126720 = 506880

Scaling back up recovers the real distance, so the direction was right.

Answer: 4 inches

4 · Reviewdoes it hold up?

A map inch stands for 126720 real inches, so 4 map inches stand for 506880 real inches -- exactly the 8 miles we started with.

Another way: Since 126720 inches is 2 of a mile, each map inch is that much real distance; dividing 8 miles by it gives 4 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 8 miles into 506880 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 11 hard answer: 10 inches

A real-world distance of $20$ miles is drawn on a map with a scale of $\dfrac{1}{126720}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 20 miles in the real world is drawn on a map at a scale of one to 126720. We need how long that distance is on the map, in inches.

Givens

The real distance is 20 miles.
The scale is $\frac{1}{126720}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 20 of them.

20 \times 63360 = 1267200\ \text{in}

The real distance is 1267200 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 126720 means the drawing is

\frac{1}{126720}

of the real thing, so divide by 126720.

1267200 \times \frac{1}{126720} = 10\ \text{in}

On the map it measures 10 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 10 inches is far less than 1267200.

10 \times 126720 = 1267200

Scaling back up recovers the real distance, so the direction was right.

Answer: 10 inches

4 · Reviewdoes it hold up?

A map inch stands for 126720 real inches, so 10 map inches stand for 1267200 real inches -- exactly the 20 miles we started with.

Another way: Since 126720 inches is 2 of a mile, each map inch is that much real distance; dividing 20 miles by it gives 10 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 20 miles into 1267200 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.

Variant 12 hard answer: 6 inches

A real-world distance of $15$ miles is drawn on a map with a scale of $\dfrac{1}{158400}$ .

(Note: $1$ mile $= 63360$ inches.)

How many inches long is this distance on the map?

Show solution

1 · Understandwhat's really being asked

A distance of 15 miles in the real world is drawn on a map at a scale of one to 158400. We need how long that distance is on the map, in inches.

Givens

The real distance is 15 miles.
The scale is $\frac{1}{158400}$ .
1 mile is 63360 inches.

Unknowns

The length on the map, in inches.

Constraints

A scale compares two lengths, so both must be in the same unit before it can be applied.

2 · Planchoose the strategy

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

The answer is wanted in inches and the scale is a bare ratio, so put the real distance into inches first. Then the scale is just a fraction to multiply by.

3 · Execute3 carry out the plan

1Convert the real distance to inches

#8 Analyze the Units 5.MD.A.1

Each mile is 63360 inches, and there are 15 of them.

15 \times 63360 = 950400\ \text{in}

The real distance is 950400 inches.

2Multiply the real length by the scale

#7 Identify Subproblems 5.NF.B.6

A scale of one to 158400 means the drawing is

\frac{1}{158400}

of the real thing, so divide by 158400.

950400 \times \frac{1}{158400} = 6\ \text{in}

On the map it measures 6 inches.

3Check the scaling makes sense

#8 Analyze the Units 5.NF.B.5

Multiplying by a fraction below 1 must shrink the length, and 6 inches is far less than 950400.

6 \times 158400 = 950400

Scaling back up recovers the real distance, so the direction was right.

Answer: 6 inches

4 · Reviewdoes it hold up?

A map inch stands for 158400 real inches, so 6 map inches stand for 950400 real inches -- exactly the 15 miles we started with.

Another way: Since 158400 inches is 5/2 of a mile, each map inch is that much real distance; dividing 15 miles by it gives 6 inches the same way.

Standardsmin grade 5

5.MD.A.1 Convert among different-sized standard measurement units within a given system — Turning 15 miles into 950400 inches so the scale applies.
5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers — Multiplying the real length by the scale fraction.
5.NF.B.5 Interpret multiplication as scaling or resizing — Reading the scale as a shrinking factor and checking the direction.

💡Takeaway. A scale is a ratio, so it only works when both lengths are in the same unit -- convert first, then multiply.