Practice · Chain two ratios, then evaluate by ratios · Sensim Math

Variant 1 easy

If $a:b=1:3$ and $b:c=2:1$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=1:3$
$b:c=2:1$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=1:3=2:6

and

b:c=2:1=6:3

.

a:b:c=2:6:3

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=2:6:3

.

\frac{a}{c}=\frac{2}{3},\ \frac{b}{a}=3,\ \frac{c}{b}=\frac{1}{2}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{25}{6}

, and then add

3

.

\frac{2}{3}+3+\frac{1}{2}+3=\frac{43}{6}

Just fraction addition now.

Answer:

\frac{43}{6}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=2$ , $b=6$ , $c=3$ : the top is $2 \times 30+6 \times 24+3 \times 18=258$ and $abc=36$ , and $\frac{258}{36}=\frac{43}{6}$ .

Another way: Set $a=2k$ , $b=6k$ , $c=3k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{43}{6}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 2 easy

If $a:b=2:1$ and $b:c=2:3$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=2:1$
$b:c=2:3$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=2:1=4:2

and

b:c=2:3

.

a:b:c=4:2:3

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=4:2:3

.

\frac{a}{c}=\frac{4}{3},\ \frac{b}{a}=\frac{1}{2},\ \frac{c}{b}=\frac{3}{2}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{10}{3}

, and then add

3

.

\frac{4}{3}+\frac{1}{2}+\frac{3}{2}+3=\frac{19}{3}

Just fraction addition now.

Answer:

\frac{19}{3}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=4$ , $b=2$ , $c=3$ : the top is $4 \times 14+2 \times 18+3 \times 20=152$ and $abc=24$ , and $\frac{152}{24}=\frac{19}{3}$ .

Another way: Set $a=4k$ , $b=2k$ , $c=3k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{19}{3}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 3 easy

If $a:b=1:2$ and $b:c=3:1$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=1:2$
$b:c=3:1$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=1:2=3:6

and

b:c=3:1=6:2

.

a:b:c=3:6:2

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=3:6:2

.

\frac{a}{c}=\frac{3}{2},\ \frac{b}{a}=2,\ \frac{c}{b}=\frac{1}{3}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{23}{6}

, and then add

3

.

\frac{3}{2}+2+\frac{1}{3}+3=\frac{41}{6}

Just fraction addition now.

Answer:

\frac{41}{6}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=3$ , $b=6$ , $c=2$ : the top is $3 \times 30+6 \times 18+2 \times 24=246$ and $abc=36$ , and $\frac{246}{36}=\frac{41}{6}$ .

Another way: Set $a=3k$ , $b=6k$ , $c=2k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{41}{6}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 4 easy

If $a:b=3:2$ and $b:c=3:4$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=3:2$
$b:c=3:4$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=3:2=9:6

and

b:c=3:4=6:8

.

a:b:c=9:6:8

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=9:6:8

.

\frac{a}{c}=\frac{9}{8},\ \frac{b}{a}=\frac{2}{3},\ \frac{c}{b}=\frac{4}{3}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{25}{8}

, and then add

3

.

\frac{9}{8}+\frac{2}{3}+\frac{4}{3}+3=\frac{49}{8}

Just fraction addition now.

Answer:

\frac{49}{8}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=9$ , $b=6$ , $c=8$ : the top is $9 \times 102+6 \times 120+8 \times 126=2646$ and $abc=432$ , and $\frac{2646}{432}=\frac{49}{8}$ .

Another way: Set $a=9k$ , $b=6k$ , $c=8k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{49}{8}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 5 medium

If $a:b=1:2$ and $b:c=3:4$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=1:2$
$b:c=3:4$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=1:2=3:6

and

b:c=3:4=6:8

.

a:b:c=3:6:8

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=3:6:8

.

\frac{a}{c}=\frac{3}{8},\ \frac{b}{a}=2,\ \frac{c}{b}=\frac{4}{3}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{89}{24}

, and then add

3

.

\frac{3}{8}+2+\frac{4}{3}+3=\frac{161}{24}

Just fraction addition now.

Answer:

\frac{161}{24}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=3$ , $b=6$ , $c=8$ : the top is $3 \times 66+6 \times 72+8 \times 42=966$ and $abc=144$ , and $\frac{966}{144}=\frac{161}{24}$ .

Another way: Set $a=3k$ , $b=6k$ , $c=8k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{161}{24}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 6 medium

If $a:b=4:3$ and $b:c=2:1$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=4:3$
$b:c=2:1$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=4:3=8:6

and

b:c=2:1=6:3

.

a:b:c=8:6:3

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=8:6:3

.

\frac{a}{c}=\frac{8}{3},\ \frac{b}{a}=\frac{3}{4},\ \frac{c}{b}=\frac{1}{2}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{47}{12}

, and then add

3

.

\frac{8}{3}+\frac{3}{4}+\frac{1}{2}+3=\frac{83}{12}

Just fraction addition now.

Answer:

\frac{83}{12}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=8$ , $b=6$ , $c=3$ : the top is $8 \times 66+6 \times 42+3 \times 72=996$ and $abc=144$ , and $\frac{996}{144}=\frac{83}{12}$ .

Another way: Set $a=8k$ , $b=6k$ , $c=3k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{83}{12}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 7 medium

If $a:b=4:1$ and $b:c=3:2$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=4:1$
$b:c=3:2$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=4:1=12:3

and

b:c=3:2

.

a:b:c=12:3:2

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=12:3:2

.

\frac{a}{c}=6,\ \frac{b}{a}=\frac{1}{4},\ \frac{c}{b}=\frac{2}{3}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{83}{12}

, and then add

3

.

6+\frac{1}{4}+\frac{2}{3}+3=\frac{119}{12}

Just fraction addition now.

Answer:

\frac{119}{12}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=12$ , $b=3$ , $c=2$ : the top is $12 \times 42+3 \times 30+2 \times 60=714$ and $abc=72$ , and $\frac{714}{72}=\frac{119}{12}$ .

Another way: Set $a=12k$ , $b=3k$ , $c=2k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{119}{12}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 8 medium

If $a:b=3:1$ and $b:c=2:5$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=3:1$
$b:c=2:5$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=3:1=6:2

and

b:c=2:5

.

a:b:c=6:2:5

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=6:2:5

.

\frac{a}{c}=\frac{6}{5},\ \frac{b}{a}=\frac{1}{3},\ \frac{c}{b}=\frac{5}{2}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{121}{30}

, and then add

3

.

\frac{6}{5}+\frac{1}{3}+\frac{5}{2}+3=\frac{211}{30}

Just fraction addition now.

Answer:

\frac{211}{30}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=6$ , $b=2$ , $c=5$ : the top is $6 \times 22+2 \times 40+5 \times 42=422$ and $abc=60$ , and $\frac{422}{60}=\frac{211}{30}$ .

Another way: Set $a=6k$ , $b=2k$ , $c=5k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{211}{30}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 9 hard

If $a:b=5:4$ and $b:c=2:3$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=5:4$
$b:c=2:3$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=5:4

and

b:c=2:3=4:6

.

a:b:c=5:4:6

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=5:4:6

.

\frac{a}{c}=\frac{5}{6},\ \frac{b}{a}=\frac{4}{5},\ \frac{c}{b}=\frac{3}{2}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{47}{15}

, and then add

3

.

\frac{5}{6}+\frac{4}{5}+\frac{3}{2}+3=\frac{92}{15}

Just fraction addition now.

Answer:

\frac{92}{15}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=5$ , $b=4$ , $c=6$ : the top is $5 \times 44+4 \times 54+6 \times 50=736$ and $abc=120$ , and $\frac{736}{120}=\frac{92}{15}$ .

Another way: Set $a=5k$ , $b=4k$ , $c=6k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{92}{15}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 10 hard

If $a:b=5:2$ and $b:c=3:2$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=5:2$
$b:c=3:2$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=5:2=15:6

and

b:c=3:2=6:4

.

a:b:c=15:6:4

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=15:6:4

.

\frac{a}{c}=\frac{15}{4},\ \frac{b}{a}=\frac{2}{5},\ \frac{c}{b}=\frac{2}{3}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{289}{60}

, and then add

3

.

\frac{15}{4}+\frac{2}{5}+\frac{2}{3}+3=\frac{469}{60}

Just fraction addition now.

Answer:

\frac{469}{60}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=15$ , $b=6$ , $c=4$ : the top is $15 \times 114+6 \times 84+4 \times 150=2814$ and $abc=360$ , and $\frac{2814}{360}=\frac{469}{60}$ .

Another way: Set $a=15k$ , $b=6k$ , $c=4k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{469}{60}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 11 hard

If $a:b=5:3$ and $b:c=2:1$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=5:3$
$b:c=2:1$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=5:3=10:6

and

b:c=2:1=6:3

.

a:b:c=10:6:3

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=10:6:3

.

\frac{a}{c}=\frac{10}{3},\ \frac{b}{a}=\frac{3}{5},\ \frac{c}{b}=\frac{1}{2}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{133}{30}

, and then add

3

.

\frac{10}{3}+\frac{3}{5}+\frac{1}{2}+3=\frac{223}{30}

Just fraction addition now.

Answer:

\frac{223}{30}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=10$ , $b=6$ , $c=3$ : the top is $10 \times 78+6 \times 48+3 \times 90=1338$ and $abc=180$ , and $\frac{1338}{180}=\frac{223}{30}$ .

Another way: Set $a=10k$ , $b=6k$ , $c=3k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{223}{30}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Variant 12 hard

If $a:b=2:3$ and $b:c=4:5$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two ratios link $a$ , $b$ , and $c$ . Find the value of a fraction built from $a$ , $b$ , $c$ .

Givens

$a:b=2:3$
$b:c=4:5$

Unknowns

The value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

Constraints

$a$ , $b$ , $c$ are not $0$ (they divide), and only their ratios are known.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#13 Convert to Algebra

Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus $3$ ; chaining the two given ratios supplies all three.

3 · Execute4 carry out the plan

1Chain the two ratios

#15 Organize Information in More Ways 6.RP.A.3

Make

b

the same in both:

a:b=2:3=8:12

and

b:c=4:5=12:15

.

a:b:c=8:12:15

One ratio for all three letters.

2Split the fraction

#7 Identify Subproblems 7.EE.A.2

Expand the top and divide each piece by

abc

.

\frac{a^2b+abc}{abc}+\frac{b^2c+abc}{abc}+\frac{c^2a+abc}{abc}=\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3

Each piece becomes one ratio plus

1

.

3Read off the ratios

#15 Organize Information in More Ways 6.RP.A.3

From

a:b:c=8:12:15

.

\frac{a}{c}=\frac{8}{15},\ \frac{b}{a}=\frac{3}{2},\ \frac{c}{b}=\frac{5}{4}

The size of

a

,

b

,

c

never matters, only their ratio.

4Add

#13 Convert to Algebra 7.EE.A.2

The three ratios add to

\frac{197}{60}

, and then add

3

.

\frac{8}{15}+\frac{3}{2}+\frac{5}{4}+3=\frac{377}{60}

Just fraction addition now.

Answer:

\frac{377}{60}

4 · Reviewdoes it hold up?

Use actual numbers in the ratio, $a=8$ , $b=12$ , $c=15$ : the top is $8 \times 276+12 \times 300+15 \times 216=9048$ and $abc=1440$ , and $\frac{9048}{1440}=\frac{377}{60}$ .

Another way: Set $a=8k$ , $b=12k$ , $c=15k$ with $k \neq 0$ . Every term on top has $k^3$ , as does $abc$ , so $k$ cancels and the same $\frac{377}{60}$ comes out.

Standardsmin grade 7

6.RP.A.3 Use ratio and rate reasoning to solve problems — Combining two ratios into one and reading off part-to-part ratios.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the fraction as a sum of ratios plus $3$.

💡Takeaway. If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Chain two ratios, then evaluate by ratios · 12 practice problems

Generated variants — 12

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