← The rate fixes the shape, the difference fixes the size · Ratio, Rate and Percent

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

6.RP.A.16.RP.A.36.EE.B.7

Generated variants — 12

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

Variant 1 easy answer: 6 : 12

Find the ratio that satisfies both conditions.

  • The rate is 0.50.5.
  • The base and the compared quantity differ by 66.
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=12→1:20.5 = \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=12 - 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÷1=66 \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.
1×6:2×6=6:121 \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 answer: 6 : 15

Find the ratio that satisfies both conditions.

  • The rate is 0.40.4.
  • The base and the compared quantity differ by 99.
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=25→2:50.4 = \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=35 - 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÷3=39 \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.
2×3:5×3=6:152 \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 answer: 15 : 25

Find the ratio that satisfies both conditions.

  • The rate is 0.60.6.
  • The base and the compared quantity differ by 1010.
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=35→3:50.6 = \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=25 - 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÷2=510 \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.
3×5:5×5=15:253 \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 answer: 4 : 20

Find the ratio that satisfies both conditions.

  • The rate is 0.20.2.
  • The base and the compared quantity differ by 1616.
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=15→1:50.2 = \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=45 - 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÷4=416 \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.
1×4:5×4=4:201 \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 answer: 54 : 72

Find the ratio that satisfies both conditions.

  • The rate is 0.750.75.
  • The base and the compared quantity differ by 1818.
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=34→3:40.75 = \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=14 - 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÷1=1818 \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.
3×18:4×18=54:723 \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 answer: 30 : 50

Find the ratio that satisfies both conditions.

  • The rate is 0.60.6.
  • The base and the compared quantity differ by 2020.
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=35→3:50.6 = \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=25 - 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÷2=1020 \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.
3×10:5×10=30:503 \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 answer: 49 : 70

Find the ratio that satisfies both conditions.

  • The rate is 0.70.7.
  • The base and the compared quantity differ by 2121.
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=710→7:100.7 = \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=310 - 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÷3=721 \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.
7×7:10×7=49:707 \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 answer: 8 : 40

Find the ratio that satisfies both conditions.

  • The rate is 0.20.2.
  • The base and the compared quantity differ by 3232.
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=15→1:50.2 = \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=45 - 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÷4=832 \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.
1×8:5×8=8:401 \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 answer: 32 : 64

Find the ratio that satisfies both conditions.

  • The rate is 0.50.5.
  • The base and the compared quantity differ by 3232.
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=12→1:20.5 = \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=12 - 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÷1=3232 \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.
1×32:2×32=32:641 \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 answer: 42 : 75

Find the ratio that satisfies both conditions.

  • The rate is 0.560.56.
  • The base and the compared quantity differ by 3333.
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=1425→14:250.56 = \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=1125 - 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÷11=333 \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.
14×3:25×3=42:7514 \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 answer: 333 : 370

Find the ratio that satisfies both conditions.

  • The rate is 0.90.9.
  • The base and the compared quantity differ by 3737.
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=910→9:100.9 = \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=110 - 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÷1=3737 \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.
9×37:10×37=333:3709 \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 answer: 57 : 95

Find the ratio that satisfies both conditions.

  • The rate is 0.60.6.
  • The base and the compared quantity differ by 3838.
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=35→3:50.6 = \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=25 - 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÷2=1938 \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.
3×19:5×19=57:953 \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.