← Every ratio with a given value is the simplest one scaled up · Proportion and Proportional Division

Every ratio with a given value is the simplest one scaled up · 12 practice problems

6.RP.A.16.RP.A.3

0 / 12 solved

Show all answers (parent / teacher)

Generated variants — 12

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

Variant 1 easy

Among the ratios whose value is $\frac{3}{5}$ , find the one whose two terms add to $40$ .

Your answer (check it yourself)

Variant 2 easy

Among the ratios whose value is $\frac{2}{3}$ , find the one whose two terms add to $45$ .

Your answer (check it yourself)

Variant 3 easy

Among the ratios whose value is $\frac{6}{7}$ , find the one whose two terms add to $52$ .

Your answer (check it yourself)

Variant 4 easy

Among the ratios whose value is $\frac{7}{12}$ , find the one whose two terms add to $57$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Of all the ratios worth 7/12, we want the one whose two terms add to 57.

Givens

The value of the ratio is 7/12.
The two terms add to 57.

Unknowns

The two terms of the ratio.

Constraints

Infinitely many ratios share the value; only one has that sum.

2 · Planchoose the strategy

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

The value alone allows any number of ratios, because scaling both terms leaves it unchanged. Write the simplest one, see what its terms add to, and scale until that sum is the one asked for.

3 · Execute4 carry out the plan

1Write the simplest ratio

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

A value of 7/12 in lowest terms is 7 to 12.

\frac{7}{12} = \frac{7}{12} \rightarrow 7 : 12

The shape is 7 to 12.

2Add its terms

#7 Identify Subproblems 6.RP.A.3

That sum is what one unit of scaling is worth.

7 + 12 = 19

Each scaling step adds 19.

3Find the scale factor

#13 Convert to Algebra 6.RP.A.3

The real sum is 57, so divide by 19.

57 \div 19 = 3

Scale everything by 3.

4Scale both terms

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

Multiplying both by the same number leaves the value alone.

7 \times 3 : 12 \times 3 = 21 : 36

The ratio is 21 to 36.

Answer: 21 : 36

4 · Reviewdoes it hold up?

Both conditions hold: 21 plus 36 is 57, and 21 over 36 simplifies back to 7/12.

Another way: Listing 7 : 12, 14 : 24, 21 : 36 and watching the sums climb by 19 arrives at the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the value as a ratio in lowest terms and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the sum of the terms to the scale factor.

💡Takeaway. A ratio's value says its shape. One more fact -- a sum or a difference -- says how big it is.

Variant 5 medium

Among the ratios whose value is $\frac{7}{9}$ , find the one whose two terms add to $64$ .

Your answer (check it yourself)

Variant 6 medium

Among the ratios whose value is $\frac{4}{9}$ , find the one whose two terms add to $65$ .

Your answer (check it yourself)

Variant 7 medium

Among the ratios whose value is $\frac{4}{7}$ , find the one whose two terms add to $66$ .

Your answer (check it yourself)

Variant 8 medium

Among the ratios whose value is $\frac{8}{11}$ , find the one whose two terms add to $76$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Of all the ratios worth 8/11, we want the one whose two terms add to 76.

Givens

The value of the ratio is 8/11.
The two terms add to 76.

Unknowns

The two terms of the ratio.

Constraints

Infinitely many ratios share the value; only one has that sum.

2 · Planchoose the strategy

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

The value alone allows any number of ratios, because scaling both terms leaves it unchanged. Write the simplest one, see what its terms add to, and scale until that sum is the one asked for.

3 · Execute4 carry out the plan

1Write the simplest ratio

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

A value of 8/11 in lowest terms is 8 to 11.

\frac{8}{11} = \frac{8}{11} \rightarrow 8 : 11

The shape is 8 to 11.

2Add its terms

#7 Identify Subproblems 6.RP.A.3

That sum is what one unit of scaling is worth.

8 + 11 = 19

Each scaling step adds 19.

3Find the scale factor

#13 Convert to Algebra 6.RP.A.3

The real sum is 76, so divide by 19.

76 \div 19 = 4

Scale everything by 4.

4Scale both terms

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

Multiplying both by the same number leaves the value alone.

8 \times 4 : 11 \times 4 = 32 : 44

The ratio is 32 to 44.

Answer: 32 : 44

4 · Reviewdoes it hold up?

Both conditions hold: 32 plus 44 is 76, and 32 over 44 simplifies back to 8/11.

Another way: Listing 8 : 11, 16 : 22, 24 : 33 and watching the sums climb by 19 arrives at the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the value as a ratio in lowest terms and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the sum of the terms to the scale factor.

💡Takeaway. A ratio's value says its shape. One more fact -- a sum or a difference -- says how big it is.

Variant 9 hard

Among the ratios whose value is $\frac{5}{8}$ , find the one whose two terms add to $78$ .

Your answer (check it yourself)

Variant 10 hard

Among the ratios whose value is $\frac{3}{4}$ , find the one whose two terms add to $84$ .

Your answer (check it yourself)

Variant 11 hard

Among the ratios whose value is $\frac{5}{6}$ , find the one whose two terms add to $88$ .

Your answer (check it yourself)

Variant 12 hard

Among the ratios whose value is $\frac{9}{10}$ , find the one whose two terms add to $95$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Of all the ratios worth 9/10, we want the one whose two terms add to 95.

Givens

The value of the ratio is 9/10.
The two terms add to 95.

Unknowns

The two terms of the ratio.

Constraints

Infinitely many ratios share the value; only one has that sum.

2 · Planchoose the strategy

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

The value alone allows any number of ratios, because scaling both terms leaves it unchanged. Write the simplest one, see what its terms add to, and scale until that sum is the one asked for.

3 · Execute4 carry out the plan

1Write the simplest ratio

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

A value of 9/10 in lowest terms is 9 to 10.

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

The shape is 9 to 10.

2Add its terms

#7 Identify Subproblems 6.RP.A.3

That sum is what one unit of scaling is worth.

9 + 10 = 19

Each scaling step adds 19.

3Find the scale factor

#13 Convert to Algebra 6.RP.A.3

The real sum is 95, so divide by 19.

95 \div 19 = 5

Scale everything by 5.

4Scale both terms

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

Multiplying both by the same number leaves the value alone.

9 \times 5 : 10 \times 5 = 45 : 50

The ratio is 45 to 50.

Answer: 45 : 50

4 · Reviewdoes it hold up?

Both conditions hold: 45 plus 50 is 95, and 45 over 50 simplifies back to 9/10.

Another way: Listing 9 : 10, 18 : 20, 27 : 30 and watching the sums climb by 19 arrives at the same place, one step at a time.

Standardsmin grade 6

6.RP.A.1 Understand ratio concepts and use ratio language — Writing the value as a ratio in lowest terms and scaling it.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Relating the sum of the terms to the scale factor.

💡Takeaway. A ratio's value says its shape. One more fact -- a sum or a difference -- says how big it is.

Where next?

← Back to the problem: Every ratio with a given value is the simplest one scaled up