Practice · The rate fixes the shape, the difference fixes the size · Sensim Math

Variant 1 easy

Find the ratio that satisfies both conditions.

The rate is $0.5$ .
The base and the compared quantity differ by $6$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.5, and its base and compared quantity are 6 apart. We want the ratio itself.

Givens

The rate is 0.5.
The two terms differ by 6.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.5 means the compared quantity is 1 whenever the base is 2.

0.5 = \frac{1}{2} \rightarrow 1 : 2

The shape is 1 to 2.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

2 - 1 = 1

Each scaling step widens the gap by 1.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 6, so divide it by 1.

6 \div 1 = 6

Scale everything by 6.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

1 \times 6 : 2 \times 6 = 6 : 12

The ratio is 6 to 12.

Answer: 6 : 12

4 · Reviewdoes it hold up?

Both conditions hold: 6 divided by 12 is 0.5, and the two differ by 6.

Another way: Listing ratios with that rate -- 1 : 2, 2 : 4, 3 : 6 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 2 easy

Find the ratio that satisfies both conditions.

The rate is $0.4$ .
The base and the compared quantity differ by $9$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.4, and its base and compared quantity are 9 apart. We want the ratio itself.

Givens

The rate is 0.4.
The two terms differ by 9.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.4 means the compared quantity is 2 whenever the base is 5.

0.4 = \frac{2}{5} \rightarrow 2 : 5

The shape is 2 to 5.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

5 - 2 = 3

Each scaling step widens the gap by 3.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 9, so divide it by 3.

9 \div 3 = 3

Scale everything by 3.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

2 \times 3 : 5 \times 3 = 6 : 15

The ratio is 6 to 15.

Answer: 6 : 15

4 · Reviewdoes it hold up?

Both conditions hold: 6 divided by 15 is 0.4, and the two differ by 9.

Another way: Listing ratios with that rate -- 2 : 5, 4 : 10, 6 : 15 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 3 easy

Find the ratio that satisfies both conditions.

The rate is $0.6$ .
The base and the compared quantity differ by $10$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.6, and its base and compared quantity are 10 apart. We want the ratio itself.

Givens

The rate is 0.6.
The two terms differ by 10.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.6 means the compared quantity is 3 whenever the base is 5.

0.6 = \frac{3}{5} \rightarrow 3 : 5

The shape is 3 to 5.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

5 - 3 = 2

Each scaling step widens the gap by 2.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 10, so divide it by 2.

10 \div 2 = 5

Scale everything by 5.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

3 \times 5 : 5 \times 5 = 15 : 25

The ratio is 15 to 25.

Answer: 15 : 25

4 · Reviewdoes it hold up?

Both conditions hold: 15 divided by 25 is 0.6, and the two differ by 10.

Another way: Listing ratios with that rate -- 3 : 5, 6 : 10, 9 : 15 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 4 easy

Find the ratio that satisfies both conditions.

The rate is $0.2$ .
The base and the compared quantity differ by $16$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.2, and its base and compared quantity are 16 apart. We want the ratio itself.

Givens

The rate is 0.2.
The two terms differ by 16.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.2 means the compared quantity is 1 whenever the base is 5.

0.2 = \frac{1}{5} \rightarrow 1 : 5

The shape is 1 to 5.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

5 - 1 = 4

Each scaling step widens the gap by 4.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 16, so divide it by 4.

16 \div 4 = 4

Scale everything by 4.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

1 \times 4 : 5 \times 4 = 4 : 20

The ratio is 4 to 20.

Answer: 4 : 20

4 · Reviewdoes it hold up?

Both conditions hold: 4 divided by 20 is 0.2, and the two differ by 16.

Another way: Listing ratios with that rate -- 1 : 5, 2 : 10, 3 : 15 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 5 medium

Find the ratio that satisfies both conditions.

The rate is $0.75$ .
The base and the compared quantity differ by $18$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.75, and its base and compared quantity are 18 apart. We want the ratio itself.

Givens

The rate is 0.75.
The two terms differ by 18.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.75 means the compared quantity is 3 whenever the base is 4.

0.75 = \frac{3}{4} \rightarrow 3 : 4

The shape is 3 to 4.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

4 - 3 = 1

Each scaling step widens the gap by 1.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 18, so divide it by 1.

18 \div 1 = 18

Scale everything by 18.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

3 \times 18 : 4 \times 18 = 54 : 72

The ratio is 54 to 72.

Answer: 54 : 72

4 · Reviewdoes it hold up?

Both conditions hold: 54 divided by 72 is 0.75, and the two differ by 18.

Another way: Listing ratios with that rate -- 3 : 4, 6 : 8, 9 : 12 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 6 medium

Find the ratio that satisfies both conditions.

The rate is $0.6$ .
The base and the compared quantity differ by $20$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.6, and its base and compared quantity are 20 apart. We want the ratio itself.

Givens

The rate is 0.6.
The two terms differ by 20.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.6 means the compared quantity is 3 whenever the base is 5.

0.6 = \frac{3}{5} \rightarrow 3 : 5

The shape is 3 to 5.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

5 - 3 = 2

Each scaling step widens the gap by 2.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 20, so divide it by 2.

20 \div 2 = 10

Scale everything by 10.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

3 \times 10 : 5 \times 10 = 30 : 50

The ratio is 30 to 50.

Answer: 30 : 50

4 · Reviewdoes it hold up?

Both conditions hold: 30 divided by 50 is 0.6, and the two differ by 20.

Another way: Listing ratios with that rate -- 3 : 5, 6 : 10, 9 : 15 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 7 medium

Find the ratio that satisfies both conditions.

The rate is $0.7$ .
The base and the compared quantity differ by $21$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.7, and its base and compared quantity are 21 apart. We want the ratio itself.

Givens

The rate is 0.7.
The two terms differ by 21.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.7 means the compared quantity is 7 whenever the base is 10.

0.7 = \frac{7}{10} \rightarrow 7 : 10

The shape is 7 to 10.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

10 - 7 = 3

Each scaling step widens the gap by 3.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 21, so divide it by 3.

21 \div 3 = 7

Scale everything by 7.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

7 \times 7 : 10 \times 7 = 49 : 70

The ratio is 49 to 70.

Answer: 49 : 70

4 · Reviewdoes it hold up?

Both conditions hold: 49 divided by 70 is 0.7, and the two differ by 21.

Another way: Listing ratios with that rate -- 7 : 10, 14 : 20, 21 : 30 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 8 medium

Find the ratio that satisfies both conditions.

The rate is $0.2$ .
The base and the compared quantity differ by $32$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.2, and its base and compared quantity are 32 apart. We want the ratio itself.

Givens

The rate is 0.2.
The two terms differ by 32.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.2 means the compared quantity is 1 whenever the base is 5.

0.2 = \frac{1}{5} \rightarrow 1 : 5

The shape is 1 to 5.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

5 - 1 = 4

Each scaling step widens the gap by 4.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 32, so divide it by 4.

32 \div 4 = 8

Scale everything by 8.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

1 \times 8 : 5 \times 8 = 8 : 40

The ratio is 8 to 40.

Answer: 8 : 40

4 · Reviewdoes it hold up?

Both conditions hold: 8 divided by 40 is 0.2, and the two differ by 32.

Another way: Listing ratios with that rate -- 1 : 5, 2 : 10, 3 : 15 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 9 hard

Find the ratio that satisfies both conditions.

The rate is $0.5$ .
The base and the compared quantity differ by $32$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.5, and its base and compared quantity are 32 apart. We want the ratio itself.

Givens

The rate is 0.5.
The two terms differ by 32.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.5 means the compared quantity is 1 whenever the base is 2.

0.5 = \frac{1}{2} \rightarrow 1 : 2

The shape is 1 to 2.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

2 - 1 = 1

Each scaling step widens the gap by 1.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 32, so divide it by 1.

32 \div 1 = 32

Scale everything by 32.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

1 \times 32 : 2 \times 32 = 32 : 64

The ratio is 32 to 64.

Answer: 32 : 64

4 · Reviewdoes it hold up?

Both conditions hold: 32 divided by 64 is 0.5, and the two differ by 32.

Another way: Listing ratios with that rate -- 1 : 2, 2 : 4, 3 : 6 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 10 hard

Find the ratio that satisfies both conditions.

The rate is $0.56$ .
The base and the compared quantity differ by $33$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.56, and its base and compared quantity are 33 apart. We want the ratio itself.

Givens

The rate is 0.56.
The two terms differ by 33.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.56 means the compared quantity is 14 whenever the base is 25.

0.56 = \frac{14}{25} \rightarrow 14 : 25

The shape is 14 to 25.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

25 - 14 = 11

Each scaling step widens the gap by 11.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 33, so divide it by 11.

33 \div 11 = 3

Scale everything by 3.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

14 \times 3 : 25 \times 3 = 42 : 75

The ratio is 42 to 75.

Answer: 42 : 75

4 · Reviewdoes it hold up?

Both conditions hold: 42 divided by 75 is 0.56, and the two differ by 33.

Another way: Listing ratios with that rate -- 14 : 25, 28 : 50, 42 : 75 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 11 hard

Find the ratio that satisfies both conditions.

The rate is $0.9$ .
The base and the compared quantity differ by $37$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.9, and its base and compared quantity are 37 apart. We want the ratio itself.

Givens

The rate is 0.9.
The two terms differ by 37.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.9 means the compared quantity is 9 whenever the base is 10.

0.9 = \frac{9}{10} \rightarrow 9 : 10

The shape is 9 to 10.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

10 - 9 = 1

Each scaling step widens the gap by 1.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 37, so divide it by 1.

37 \div 1 = 37

Scale everything by 37.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

9 \times 37 : 10 \times 37 = 333 : 370

The ratio is 333 to 370.

Answer: 333 : 370

4 · Reviewdoes it hold up?

Both conditions hold: 333 divided by 370 is 0.9, and the two differ by 37.

Another way: Listing ratios with that rate -- 9 : 10, 18 : 20, 27 : 30 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

Variant 12 hard

Find the ratio that satisfies both conditions.

The rate is $0.6$ .
The base and the compared quantity differ by $38$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A ratio has rate 0.6, and its base and compared quantity are 38 apart. We want the ratio itself.

Givens

The rate is 0.6.
The two terms differ by 38.

Unknowns

The ratio, written compared to base.

Constraints

The rate is the compared quantity divided by the base, so the base is the larger of the two.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #5 Look for a Pattern#7 Identify Subproblems

The rate alone allows infinitely many ratios -- it fixes their shape, not their size. Write the simplest ratio with that rate, see how far apart its terms are, and scale until the gap is the one asked for.

3 · Execute4 carry out the plan

1Write the rate as a simplest ratio

#5 Look for a Pattern 6.RP.A.1

A rate of 0.6 means the compared quantity is 3 whenever the base is 5.

0.6 = \frac{3}{5} \rightarrow 3 : 5

The shape is 3 to 5.

2See how far apart the simplest terms are

#7 Identify Subproblems 6.RP.A.3

That gap is what one unit of scaling is worth.

5 - 3 = 2

Each scaling step widens the gap by 2.

3Find the scale factor

#13 Convert to Algebra 6.EE.B.7

The real gap is 38, so divide it by 2.

38 \div 2 = 19

Scale everything by 19.

4Scale both terms

#5 Look for a Pattern 6.RP.A.1

Multiplying both terms by the same number leaves the rate unchanged.

3 \times 19 : 5 \times 19 = 57 : 95

The ratio is 57 to 95.

Answer: 57 : 95

4 · Reviewdoes it hold up?

Both conditions hold: 57 divided by 95 is 0.6, and the two differ by 38.

Another way: Listing ratios with that rate -- 3 : 5, 6 : 10, 9 : 15 -- and watching the gap grow reaches the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the rate as a ratio and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the gap between terms to the scale factor.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the scale factor from the stated difference.

💡Takeaway. A rate tells you the shape of a ratio. It takes one more fact -- a difference or a total -- to say how big it is.

The rate fixes the shape, the difference fixes the size · 12 practice problems

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