Practice · Clear denominators to reshape the condition · Sensim Math

Variant 1 easy

If $\frac{1}{a}-\frac{1}{b}=3$ , find the value of $\frac{4a-4b-2ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=3$ but not $a$ or $b$ themselves. Find the value of $\frac{4a-4b-2ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=3$

Unknowns

The value of $\frac{4a-4b-2ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=3

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=3ab \Rightarrow a-b=-3ab

a-b

is just

-3

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

4

out of the first two terms of the numerator.

4a-4b-2ab=4(a-b)-2ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-3ab

everywhere;

ab

cancels because it is not

0

.

\frac{4(-3ab)-2ab}{-3ab}=\frac{-14ab}{-3ab}=\frac{14}{3}

Only a number is left.

Answer:

\frac{14}{3}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{4}$ , $b=1$ : $4-1=3$ works. Then $4a-4b-2ab=1-4-\frac{1}{2}=-\frac{7}{2}$ and $a-b=-\frac{3}{4}$ , so the value is $\frac{14}{3}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{4\left(\frac{1}{b}-\frac{1}{a}\right)-2}{\frac{1}{b}-\frac{1}{a}}=\frac{4(-3)-2}{-3}=\frac{14}{3}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $4$ out of $4a-4b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 2 easy

If $\frac{1}{a}-\frac{1}{b}=4$ , find the value of $\frac{2a-2b-3ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=4$ but not $a$ or $b$ themselves. Find the value of $\frac{2a-2b-3ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=4$

Unknowns

The value of $\frac{2a-2b-3ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=4

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=4ab \Rightarrow a-b=-4ab

a-b

is just

-4

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

2

out of the first two terms of the numerator.

2a-2b-3ab=2(a-b)-3ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-4ab

everywhere;

ab

cancels because it is not

0

.

\frac{2(-4ab)-3ab}{-4ab}=\frac{-11ab}{-4ab}=\frac{11}{4}

Only a number is left.

Answer:

\frac{11}{4}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{5}$ , $b=1$ : $5-1=4$ works. Then $2a-2b-3ab=\frac{2}{5}-2-\frac{3}{5}=-\frac{11}{5}$ and $a-b=-\frac{4}{5}$ , so the value is $\frac{11}{4}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{2\left(\frac{1}{b}-\frac{1}{a}\right)-3}{\frac{1}{b}-\frac{1}{a}}=\frac{2(-4)-3}{-4}=\frac{11}{4}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $2$ out of $2a-2b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 3 easy

If $\frac{1}{a}-\frac{1}{b}=4$ , find the value of $\frac{3a-3b-2ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=4$ but not $a$ or $b$ themselves. Find the value of $\frac{3a-3b-2ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=4$

Unknowns

The value of $\frac{3a-3b-2ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=4

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=4ab \Rightarrow a-b=-4ab

a-b

is just

-4

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

3

out of the first two terms of the numerator.

3a-3b-2ab=3(a-b)-2ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-4ab

everywhere;

ab

cancels because it is not

0

.

\frac{3(-4ab)-2ab}{-4ab}=\frac{-14ab}{-4ab}=\frac{7}{2}

Only a number is left.

Answer:

\frac{7}{2}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{5}$ , $b=1$ : $5-1=4$ works. Then $3a-3b-2ab=\frac{3}{5}-3-\frac{2}{5}=-\frac{14}{5}$ and $a-b=-\frac{4}{5}$ , so the value is $\frac{14}{4}=\frac{7}{2}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{3\left(\frac{1}{b}-\frac{1}{a}\right)-2}{\frac{1}{b}-\frac{1}{a}}=\frac{3(-4)-2}{-4}=\frac{7}{2}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $3$ out of $3a-3b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 4 easy

If $\frac{1}{a}-\frac{1}{b}=5$ , find the value of $\frac{2a-2b-3ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=5$ but not $a$ or $b$ themselves. Find the value of $\frac{2a-2b-3ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=5$

Unknowns

The value of $\frac{2a-2b-3ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=5

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=5ab \Rightarrow a-b=-5ab

a-b

is just

-5

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

2

out of the first two terms of the numerator.

2a-2b-3ab=2(a-b)-3ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-5ab

everywhere;

ab

cancels because it is not

0

.

\frac{2(-5ab)-3ab}{-5ab}=\frac{-13ab}{-5ab}=\frac{13}{5}

Only a number is left.

Answer:

\frac{13}{5}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{6}$ , $b=1$ : $6-1=5$ works. Then $2a-2b-3ab=\frac{1}{3}-2-\frac{1}{2}=-\frac{13}{6}$ and $a-b=-\frac{5}{6}$ , so the value is $\frac{13}{5}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{2\left(\frac{1}{b}-\frac{1}{a}\right)-3}{\frac{1}{b}-\frac{1}{a}}=\frac{2(-5)-3}{-5}=\frac{13}{5}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $2$ out of $2a-2b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 5 medium

If $\frac{1}{a}-\frac{1}{b}=2$ , find the value of $\frac{5a-5b-3ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=2$ but not $a$ or $b$ themselves. Find the value of $\frac{5a-5b-3ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=2$

Unknowns

The value of $\frac{5a-5b-3ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=2

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=2ab \Rightarrow a-b=-2ab

a-b

is just

-2

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

5

out of the first two terms of the numerator.

5a-5b-3ab=5(a-b)-3ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-2ab

everywhere;

ab

cancels because it is not

0

.

\frac{5(-2ab)-3ab}{-2ab}=\frac{-13ab}{-2ab}=\frac{13}{2}

Only a number is left.

Answer:

\frac{13}{2}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{3}$ , $b=1$ : $3-1=2$ works. Then $5a-5b-3ab=\frac{5}{3}-5-1=-\frac{13}{3}$ and $a-b=-\frac{2}{3}$ , so the value is $\frac{13}{2}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{5\left(\frac{1}{b}-\frac{1}{a}\right)-3}{\frac{1}{b}-\frac{1}{a}}=\frac{5(-2)-3}{-2}=\frac{13}{2}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $5$ out of $5a-5b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 6 medium

If $\frac{1}{a}-\frac{1}{b}=3$ , find the value of $\frac{2a-2b-5ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=3$ but not $a$ or $b$ themselves. Find the value of $\frac{2a-2b-5ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=3$

Unknowns

The value of $\frac{2a-2b-5ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=3

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=3ab \Rightarrow a-b=-3ab

a-b

is just

-3

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

2

out of the first two terms of the numerator.

2a-2b-5ab=2(a-b)-5ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-3ab

everywhere;

ab

cancels because it is not

0

.

\frac{2(-3ab)-5ab}{-3ab}=\frac{-11ab}{-3ab}=\frac{11}{3}

Only a number is left.

Answer:

\frac{11}{3}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{4}$ , $b=1$ : $4-1=3$ works. Then $2a-2b-5ab=\frac{1}{2}-2-\frac{5}{4}=-\frac{11}{4}$ and $a-b=-\frac{3}{4}$ , so the value is $\frac{11}{3}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{2\left(\frac{1}{b}-\frac{1}{a}\right)-5}{\frac{1}{b}-\frac{1}{a}}=\frac{2(-3)-5}{-3}=\frac{11}{3}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $2$ out of $2a-2b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 7 medium

If $\frac{1}{a}-\frac{1}{b}=6$ , find the value of $\frac{3a-3b-2ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=6$ but not $a$ or $b$ themselves. Find the value of $\frac{3a-3b-2ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=6$

Unknowns

The value of $\frac{3a-3b-2ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=6

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=6ab \Rightarrow a-b=-6ab

a-b

is just

-6

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

3

out of the first two terms of the numerator.

3a-3b-2ab=3(a-b)-2ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-6ab

everywhere;

ab

cancels because it is not

0

.

\frac{3(-6ab)-2ab}{-6ab}=\frac{-20ab}{-6ab}=\frac{10}{3}

Only a number is left.

Answer:

\frac{10}{3}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{7}$ , $b=1$ : $7-1=6$ works. Then $3a-3b-2ab=\frac{3}{7}-3-\frac{2}{7}=-\frac{20}{7}$ and $a-b=-\frac{6}{7}$ , so the value is $\frac{20}{6}=\frac{10}{3}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{3\left(\frac{1}{b}-\frac{1}{a}\right)-2}{\frac{1}{b}-\frac{1}{a}}=\frac{3(-6)-2}{-6}=\frac{10}{3}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $3$ out of $3a-3b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 8 medium

If $\frac{1}{a}-\frac{1}{b}=6$ , find the value of $\frac{5a-5b-4ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=6$ but not $a$ or $b$ themselves. Find the value of $\frac{5a-5b-4ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=6$

Unknowns

The value of $\frac{5a-5b-4ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=6

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=6ab \Rightarrow a-b=-6ab

a-b

is just

-6

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

5

out of the first two terms of the numerator.

5a-5b-4ab=5(a-b)-4ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-6ab

everywhere;

ab

cancels because it is not

0

.

\frac{5(-6ab)-4ab}{-6ab}=\frac{-34ab}{-6ab}=\frac{17}{3}

Only a number is left.

Answer:

\frac{17}{3}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{7}$ , $b=1$ : $7-1=6$ works. Then $5a-5b-4ab=\frac{5}{7}-5-\frac{4}{7}=-\frac{34}{7}$ and $a-b=-\frac{6}{7}$ , so the value is $\frac{34}{6}=\frac{17}{3}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{5\left(\frac{1}{b}-\frac{1}{a}\right)-4}{\frac{1}{b}-\frac{1}{a}}=\frac{5(-6)-4}{-6}=\frac{17}{3}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $5$ out of $5a-5b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 9 hard

If $\frac{1}{a}-\frac{1}{b}=7$ , find the value of $\frac{3a-3b-4ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=7$ but not $a$ or $b$ themselves. Find the value of $\frac{3a-3b-4ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=7$

Unknowns

The value of $\frac{3a-3b-4ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=7

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=7ab \Rightarrow a-b=-7ab

a-b

is just

-7

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

3

out of the first two terms of the numerator.

3a-3b-4ab=3(a-b)-4ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-7ab

everywhere;

ab

cancels because it is not

0

.

\frac{3(-7ab)-4ab}{-7ab}=\frac{-25ab}{-7ab}=\frac{25}{7}

Only a number is left.

Answer:

\frac{25}{7}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{8}$ , $b=1$ : $8-1=7$ works. Then $3a-3b-4ab=\frac{3}{8}-3-\frac{1}{2}=-\frac{25}{8}$ and $a-b=-\frac{7}{8}$ , so the value is $\frac{25}{7}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{3\left(\frac{1}{b}-\frac{1}{a}\right)-4}{\frac{1}{b}-\frac{1}{a}}=\frac{3(-7)-4}{-7}=\frac{25}{7}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $3$ out of $3a-3b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 10 hard

If $\frac{1}{a}-\frac{1}{b}=8$ , find the value of $\frac{3a-3b-6ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=8$ but not $a$ or $b$ themselves. Find the value of $\frac{3a-3b-6ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=8$

Unknowns

The value of $\frac{3a-3b-6ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=8

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=8ab \Rightarrow a-b=-8ab

a-b

is just

-8

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

3

out of the first two terms of the numerator.

3a-3b-6ab=3(a-b)-6ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-8ab

everywhere;

ab

cancels because it is not

0

.

\frac{3(-8ab)-6ab}{-8ab}=\frac{-30ab}{-8ab}=\frac{15}{4}

Only a number is left.

Answer:

\frac{15}{4}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{9}$ , $b=1$ : $9-1=8$ works. Then $3a-3b-6ab=\frac{1}{3}-3-\frac{2}{3}=-\frac{10}{3}$ and $a-b=-\frac{8}{9}$ , so the value is $\frac{30}{8}=\frac{15}{4}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{3\left(\frac{1}{b}-\frac{1}{a}\right)-6}{\frac{1}{b}-\frac{1}{a}}=\frac{3(-8)-6}{-8}=\frac{15}{4}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $3$ out of $3a-3b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 11 hard

If $\frac{1}{a}-\frac{1}{b}=9$ , find the value of $\frac{2a-2b-6ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=9$ but not $a$ or $b$ themselves. Find the value of $\frac{2a-2b-6ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=9$

Unknowns

The value of $\frac{2a-2b-6ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=9

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=9ab \Rightarrow a-b=-9ab

a-b

is just

-9

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

2

out of the first two terms of the numerator.

2a-2b-6ab=2(a-b)-6ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-9ab

everywhere;

ab

cancels because it is not

0

.

\frac{2(-9ab)-6ab}{-9ab}=\frac{-24ab}{-9ab}=\frac{8}{3}

Only a number is left.

Answer:

\frac{8}{3}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{10}$ , $b=1$ : $10-1=9$ works. Then $2a-2b-6ab=\frac{1}{5}-2-\frac{3}{5}=-\frac{12}{5}$ and $a-b=-\frac{9}{10}$ , so the value is $\frac{24}{9}=\frac{8}{3}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{2\left(\frac{1}{b}-\frac{1}{a}\right)-6}{\frac{1}{b}-\frac{1}{a}}=\frac{2(-9)-6}{-9}=\frac{8}{3}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $2$ out of $2a-2b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Variant 12 hard

If $\frac{1}{a}-\frac{1}{b}=10$ , find the value of $\frac{4a-4b-5ab}{a-b}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

We know $\frac{1}{a}-\frac{1}{b}=10$ but not $a$ or $b$ themselves. Find the value of $\frac{4a-4b-5ab}{a-b}$ .

Givens

$\frac{1}{a}-\frac{1}{b}=10$

Unknowns

The value of $\frac{4a-4b-5ab}{a-b}$ .

Constraints

$a$ and $b$ are not $0$ (they are in denominators), and $a \neq b$ .

2 · Planchoose the strategy

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

Many pairs $(a, b)$ satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as $a-b$ in terms of $ab$ , regroup the target so $a-b$ shows, substitute, and let $ab$ cancel.

3 · Execute4 carry out the plan

1Combine the condition

#15 Organize Information in More Ways 7.EE.A.2

Subtract the fractions over the common denominator

ab

.

\frac{1}{a}-\frac{1}{b}=\frac{b-a}{ab}=10

The condition now involves

b-a

and

ab

.

2Clear the denominator

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

Multiply by

ab

, then flip the sign to get

a-b

.

b-a=10ab \Rightarrow a-b=-10ab

a-b

is just

-10

times

ab

.

3Let a-b show in the top

#7 Identify Subproblems 6.EE.A.3

Factor

4

out of the first two terms of the numerator.

4a-4b-5ab=4(a-b)-5ab

Now both top and bottom are written with

a-b

.

4Substitute and cancel

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

Replace

a-b

with

-10ab

everywhere;

ab

cancels because it is not

0

.

\frac{4(-10ab)-5ab}{-10ab}=\frac{-45ab}{-10ab}=\frac{9}{2}

Only a number is left.

Answer:

\frac{9}{2}

4 · Reviewdoes it hold up?

Take $a=\frac{1}{11}$ , $b=1$ : $11-1=10$ works. Then $4a-4b-5ab=\frac{4}{11}-4-\frac{5}{11}=-\frac{45}{11}$ and $a-b=-\frac{10}{11}$ , so the value is $\frac{45}{10}=\frac{9}{2}$ .

Another way: Divide top and bottom by $ab$ : the expression becomes $\frac{4\left(\frac{1}{b}-\frac{1}{a}\right)-5}{\frac{1}{b}-\frac{1}{a}}=\frac{4(-10)-5}{-10}=\frac{9}{2}$ .

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the condition and the target so the same block $a-b$ appears in both.
6.EE.A.3 Apply properties to generate equivalent expressions — Factoring $4$ out of $4a-4b$.

💡Takeaway. When the letters can't be pinned down, reshape the condition until it fits the expression you need.

Clear denominators to reshape the condition · 12 practice problems

Generated variants — 12

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