Practice · Equal side products cancel the shared corner · Sensim Math

Variant 1 easy

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $-2$ , $6$ , $x$
Right side: $-2$ , $4.5$ , $y$
Bottom side: $x$ , $-\frac{3}{4}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $-2$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

-2

.

(-2) \times 6 \times x = (-2) \times 4.5 \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

-2

, which is not

0

.

6 \times x = 4.5 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

6 \times y

and write

4.5

as

\frac{9}{2}

.

\frac{x}{y} = \frac{9}{2} \div 6 = \frac{3}{4}

So

\frac{x}{y} = \frac{3}{4}

.

Answer:

\frac{3}{4}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-12 = \left(-\frac{3}{4}\right) \times y$ , so $y = 16$ and then $x = \frac{3}{4} \times 16 = 12$ . Each side's product is then $-144$ : for example $(-2) \times 6 \times 12 = -144$ and $12 \times \left(-\frac{3}{4}\right) \times 16 = -144$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=16$ ), then for $x$ from the left and right sides ( $x=12$ ), and divide: $12 \div 16 = \frac{3}{4}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 2 easy

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $-\frac{2}{5}$ , $9$ , $x$
Right side: $-\frac{2}{5}$ , $1.5$ , $y$
Bottom side: $x$ , $3$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $-\frac{2}{5}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

-\frac{2}{5}

.

\left(-\frac{2}{5}\right) \times 9 \times x = \left(-\frac{2}{5}\right) \times 1.5 \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

-\frac{2}{5}

, which is not

0

.

9 \times x = 1.5 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

9 \times y

and write

1.5

as

\frac{3}{2}

.

\frac{x}{y} = \frac{3}{2} \div 9 = \frac{1}{6}

So

\frac{x}{y} = \frac{1}{6}

.

Answer:

\frac{1}{6}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-\frac{18}{5} = 3 \times y$ , so $y = -\frac{6}{5}$ and then $x = \frac{1}{6} \times \left(-\frac{6}{5}\right) = -\frac{1}{5}$ . Each side's product is then $\frac{18}{25}$ : for example $\left(-\frac{2}{5}\right) \times 9 \times \left(-\frac{1}{5}\right) = \frac{18}{25}$ and $-\frac{1}{5} \times 3 \times \left(-\frac{6}{5}\right) = \frac{18}{25}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=-\frac{6}{5}$ ), then for $x$ from the left and right sides ( $x=-\frac{1}{5}$ ), and divide: $-\frac{1}{5} \div \left(-\frac{6}{5}\right) = \frac{1}{6}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 3 easy

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $-\frac{3}{2}$ , $4$ , $x$
Right side: $-\frac{3}{2}$ , $3.25$ , $y$
Bottom side: $x$ , $-\frac{2}{3}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $-\frac{3}{2}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

-\frac{3}{2}

.

\left(-\frac{3}{2}\right) \times 4 \times x = \left(-\frac{3}{2}\right) \times 3.25 \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

-\frac{3}{2}

, which is not

0

.

4 \times x = 3.25 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

4 \times y

and write

3.25

as

\frac{13}{4}

.

\frac{x}{y} = \frac{13}{4} \div 4 = \frac{13}{16}

So

\frac{x}{y} = \frac{13}{16}

.

Answer:

\frac{13}{16}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-6 = \left(-\frac{2}{3}\right) \times y$ , so $y = 9$ and then $x = \frac{13}{16} \times 9 = \frac{117}{16}$ . Each side's product is then $-\frac{351}{8}$ : for example $\left(-\frac{3}{2}\right) \times 4 \times \frac{117}{16} = -\frac{351}{8}$ and $\frac{117}{16} \times \left(-\frac{2}{3}\right) \times 9 = -\frac{351}{8}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=9$ ), then for $x$ from the left and right sides ( $x=\frac{117}{16}$ ), and divide: $\frac{117}{16} \div 9 = \frac{13}{16}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 4 medium

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $\frac{4}{5}$ , $-2$ , $x$
Right side: $\frac{4}{5}$ , $-\frac{3}{5}$ , $y$
Bottom side: $x$ , $-4$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $\frac{4}{5}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

\frac{4}{5}

.

\frac{4}{5} \times (-2) \times x = \frac{4}{5} \times \left(-\frac{3}{5}\right) \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

\frac{4}{5}

, which is not

0

.

-2 \times x = -\frac{3}{5} \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

(-2) \times y

.

\frac{x}{y} = -\frac{3}{5} \div (-2) = \frac{3}{10}

So

\frac{x}{y} = \frac{3}{10}

.

Answer:

\frac{3}{10}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-\frac{8}{5} = (-4) \times y$ , so $y = \frac{2}{5}$ and then $x = \frac{3}{10} \times \frac{2}{5} = \frac{3}{25}$ . Each side's product is then $-\frac{24}{125}$ : for example $\frac{4}{5} \times (-2) \times \frac{3}{25} = -\frac{24}{125}$ and $\frac{3}{25} \times (-4) \times \frac{2}{5} = -\frac{24}{125}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=\frac{2}{5}$ ), then for $x$ from the left and right sides ( $x=\frac{3}{25}$ ), and divide: $\frac{3}{25} \div \frac{2}{5} = \frac{3}{10}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 5 medium

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $-\frac{4}{3}$ , $5$ , $x$
Right side: $-\frac{4}{3}$ , $1.25$ , $y$
Bottom side: $x$ , $\frac{2}{5}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $-\frac{4}{3}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

-\frac{4}{3}

.

\left(-\frac{4}{3}\right) \times 5 \times x = \left(-\frac{4}{3}\right) \times 1.25 \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

-\frac{4}{3}

, which is not

0

.

5 \times x = 1.25 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

5 \times y

and write

1.25

as

\frac{5}{4}

.

\frac{x}{y} = \frac{5}{4} \div 5 = \frac{1}{4}

So

\frac{x}{y} = \frac{1}{4}

.

Answer:

\frac{1}{4}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-\frac{20}{3} = \frac{2}{5} \times y$ , so $y = -\frac{50}{3}$ and then $x = \frac{1}{4} \times \left(-\frac{50}{3}\right) = -\frac{25}{6}$ . Each side's product is then $\frac{250}{9}$ : for example $\left(-\frac{4}{3}\right) \times 5 \times \left(-\frac{25}{6}\right) = \frac{250}{9}$ and $-\frac{25}{6} \times \frac{2}{5} \times \left(-\frac{50}{3}\right) = \frac{250}{9}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=-\frac{50}{3}$ ), then for $x$ from the left and right sides ( $x=-\frac{25}{6}$ ), and divide: $-\frac{25}{6} \div \left(-\frac{50}{3}\right) = \frac{1}{4}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 6 medium

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $\frac{3}{4}$ , $2$ , $x$
Right side: $\frac{3}{4}$ , $3.5$ , $y$
Bottom side: $x$ , $-\frac{5}{2}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $\frac{3}{4}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

\frac{3}{4}

.

\frac{3}{4} \times 2 \times x = \frac{3}{4} \times 3.5 \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

\frac{3}{4}

, which is not

0

.

2 \times x = 3.5 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

2 \times y

and write

3.5

as

\frac{7}{2}

.

\frac{x}{y} = \frac{7}{2} \div 2 = \frac{7}{4}

So

\frac{x}{y} = 1\frac{3}{4}

.

Answer:

1\frac{3}{4}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $\frac{3}{2} = \left(-\frac{5}{2}\right) \times y$ , so $y = -\frac{3}{5}$ and then $x = \frac{7}{4} \times \left(-\frac{3}{5}\right) = -\frac{21}{20}$ . Each side's product is then $-\frac{63}{40}$ : for example $\frac{3}{4} \times 2 \times \left(-\frac{21}{20}\right) = -\frac{63}{40}$ and $-\frac{21}{20} \times \left(-\frac{5}{2}\right) \times \left(-\frac{3}{5}\right) = -\frac{63}{40}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=-\frac{3}{5}$ ), then for $x$ from the left and right sides ( $x=-\frac{21}{20}$ ), and divide: $-\frac{21}{20} \div \left(-\frac{3}{5}\right) = \frac{7}{4}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 7 easy

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $-\frac{5}{2}$ , $3$ , $x$
Right side: $-\frac{5}{2}$ , $2.5$ , $y$
Bottom side: $x$ , $-\frac{4}{3}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $-\frac{5}{2}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

-\frac{5}{2}

.

\left(-\frac{5}{2}\right) \times 3 \times x = \left(-\frac{5}{2}\right) \times 2.5 \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

-\frac{5}{2}

, which is not

0

.

3 \times x = 2.5 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

3 \times y

and write

2.5

as

\frac{5}{2}

.

\frac{x}{y} = \frac{5}{2} \div 3 = \frac{5}{6}

So

\frac{x}{y} = \frac{5}{6}

.

Answer:

\frac{5}{6}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-\frac{15}{2} = \left(-\frac{4}{3}\right) \times y$ , so $y = \frac{45}{8}$ and then $x = \frac{5}{6} \times \frac{45}{8} = \frac{75}{16}$ . Each side's product is then $-\frac{1125}{32}$ : for example $\left(-\frac{5}{2}\right) \times 3 \times \frac{75}{16} = -\frac{1125}{32}$ and $\frac{75}{16} \times \left(-\frac{4}{3}\right) \times \frac{45}{8} = -\frac{1125}{32}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=\frac{45}{8}$ ), then for $x$ from the left and right sides ( $x=\frac{75}{16}$ ), and divide: $\frac{75}{16} \div \frac{45}{8} = \frac{5}{6}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 8 medium

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $\frac{2}{3}$ , $-6$ , $x$
Right side: $\frac{2}{3}$ , $-1.5$ , $y$
Bottom side: $x$ , $\frac{3}{4}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $\frac{2}{3}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

\frac{2}{3}

.

\frac{2}{3} \times (-6) \times x = \frac{2}{3} \times (-1.5) \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

\frac{2}{3}

, which is not

0

.

-6 \times x = -1.5 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

(-6) \times y

and write

-1.5

as

-\frac{3}{2}

.

\frac{x}{y} = -\frac{3}{2} \div (-6) = \frac{1}{4}

So

\frac{x}{y} = \frac{1}{4}

.

Answer:

\frac{1}{4}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-4 = \frac{3}{4} \times y$ , so $y = -\frac{16}{3}$ and then $x = \frac{1}{4} \times \left(-\frac{16}{3}\right) = -\frac{4}{3}$ . Each side's product is then $\frac{16}{3}$ : for example $\frac{2}{3} \times (-6) \times \left(-\frac{4}{3}\right) = \frac{16}{3}$ and $-\frac{4}{3} \times \frac{3}{4} \times \left(-\frac{16}{3}\right) = \frac{16}{3}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=-\frac{16}{3}$ ), then for $x$ from the left and right sides ( $x=-\frac{4}{3}$ ), and divide: $-\frac{4}{3} \div \left(-\frac{16}{3}\right) = \frac{1}{4}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 9 hard

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $\frac{5}{2}$ , $-3$ , $x$
Right side: $\frac{5}{2}$ , $-0.75$ , $y$
Bottom side: $x$ , $\frac{6}{5}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $\frac{5}{2}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

\frac{5}{2}

.

\frac{5}{2} \times (-3) \times x = \frac{5}{2} \times (-0.75) \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

\frac{5}{2}

, which is not

0

.

-3 \times x = -0.75 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

(-3) \times y

and write

-0.75

as

-\frac{3}{4}

.

\frac{x}{y} = -\frac{3}{4} \div (-3) = \frac{1}{4}

So

\frac{x}{y} = \frac{1}{4}

.

Answer:

\frac{1}{4}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-\frac{15}{2} = \frac{6}{5} \times y$ , so $y = -\frac{25}{4}$ and then $x = \frac{1}{4} \times \left(-\frac{25}{4}\right) = -\frac{25}{16}$ . Each side's product is then $\frac{375}{32}$ : for example $\frac{5}{2} \times (-3) \times \left(-\frac{25}{16}\right) = \frac{375}{32}$ and $-\frac{25}{16} \times \frac{6}{5} \times \left(-\frac{25}{4}\right) = \frac{375}{32}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=-\frac{25}{4}$ ), then for $x$ from the left and right sides ( $x=-\frac{25}{16}$ ), and divide: $-\frac{25}{16} \div \left(-\frac{25}{4}\right) = \frac{1}{4}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 10 hard

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $\frac{3}{2}$ , $-5$ , $x$
Right side: $\frac{3}{2}$ , $-2.4$ , $y$
Bottom side: $x$ , $-\frac{5}{6}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $\frac{3}{2}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

\frac{3}{2}

.

\frac{3}{2} \times (-5) \times x = \frac{3}{2} \times (-2.4) \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

\frac{3}{2}

, which is not

0

.

-5 \times x = -2.4 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

(-5) \times y

and write

-2.4

as

-\frac{12}{5}

.

\frac{x}{y} = -\frac{12}{5} \div (-5) = \frac{12}{25}

So

\frac{x}{y} = \frac{12}{25}

.

Answer:

\frac{12}{25}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-\frac{15}{2} = \left(-\frac{5}{6}\right) \times y$ , so $y = 9$ and then $x = \frac{12}{25} \times 9 = \frac{108}{25}$ . Each side's product is then $-\frac{162}{5}$ : for example $\frac{3}{2} \times (-5) \times \frac{108}{25} = -\frac{162}{5}$ and $\frac{108}{25} \times \left(-\frac{5}{6}\right) \times 9 = -\frac{162}{5}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=9$ ), then for $x$ from the left and right sides ( $x=\frac{108}{25}$ ), and divide: $\frac{108}{25} \div 9 = \frac{12}{25}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 11 hard

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $-\frac{3}{4}$ , $8$ , $x$
Right side: $-\frac{3}{4}$ , $2.25$ , $y$
Bottom side: $x$ , $-\frac{3}{2}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $-\frac{3}{4}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

-\frac{3}{4}

.

\left(-\frac{3}{4}\right) \times 8 \times x = \left(-\frac{3}{4}\right) \times 2.25 \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

-\frac{3}{4}

, which is not

0

.

8 \times x = 2.25 \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

8 \times y

and write

2.25

as

\frac{9}{4}

.

\frac{x}{y} = \frac{9}{4} \div 8 = \frac{9}{32}

So

\frac{x}{y} = \frac{9}{32}

.

Answer:

\frac{9}{32}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-6 = \left(-\frac{3}{2}\right) \times y$ , so $y = 4$ and then $x = \frac{9}{32} \times 4 = \frac{9}{8}$ . Each side's product is then $-\frac{27}{4}$ : for example $\left(-\frac{3}{4}\right) \times 8 \times \frac{9}{8} = -\frac{27}{4}$ and $\frac{9}{8} \times \left(-\frac{3}{2}\right) \times 4 = -\frac{27}{4}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=4$ ), then for $x$ from the left and right sides ( $x=\frac{9}{8}$ ), and divide: $\frac{9}{8} \div 4 = \frac{9}{32}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Variant 12 hard

In the figure below, the product of the three numbers on each side of the triangle is the same for all three sides. Find the value of $\frac{x}{y}$ . (Assume $x \times y \neq 0$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Six numbers sit on a triangle, three on each side, and all three sides have the same product. Two corner values $x$ and $y$ are unknown; we need the ratio $\frac{x}{y}$ .

Givens

Left side: $-\frac{7}{2}$ , $3$ , $x$
Right side: $-\frac{7}{2}$ , $\frac{5}{4}$ , $y$
Bottom side: $x$ , $\frac{2}{3}$ , $y$
All three side products are equal, and $x \times y \neq 0$ .

Unknowns

$\frac{x}{y}$

Constraints

$x$ and $y$ are not $0$ , so we may divide by them.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #16 Count the Complement

We only need the ratio, not $x$ and $y$ themselves. The left and right sides share the top corner $-\frac{7}{2}$ , so set their products equal and divide that shared factor away: what is left is a direct link between $x$ and $y$ .

3 · Execute3 carry out the plan

1Set the two slanted sides equal

#13 Convert to Algebra 7.EE.B.4

Both slanted sides contain the top number

-\frac{7}{2}

.

\left(-\frac{7}{2}\right) \times 3 \times x = \left(-\frac{7}{2}\right) \times \frac{5}{4} \times y

One equation that involves only

x

and

y

.

2Cancel the shared corner

#16 Count the Complement 7.NS.A.2

Divide both sides by

-\frac{7}{2}

, which is not

0

.

3 \times x = \frac{5}{4} \times y

The shared corner drops out completely.

3Read off the ratio

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

Divide both sides by

3 \times y

.

\frac{x}{y} = \frac{5}{4} \div 3 = \frac{5}{12}

So

\frac{x}{y} = \frac{5}{12}

.

Answer:

\frac{5}{12}

4 · Reviewdoes it hold up?

Find the actual values. Setting the left side equal to the bottom side and dividing by $x$ gives $-\frac{21}{2} = \frac{2}{3} \times y$ , so $y = -\frac{63}{4}$ and then $x = \frac{5}{12} \times \left(-\frac{63}{4}\right) = -\frac{105}{16}$ . Each side's product is then $\frac{2205}{32}$ : for example $\left(-\frac{7}{2}\right) \times 3 \times \left(-\frac{105}{16}\right) = \frac{2205}{32}$ and $-\frac{105}{16} \times \frac{2}{3} \times \left(-\frac{63}{4}\right) = \frac{2205}{32}$ .

Another way: Solve for $y$ first from the left and bottom sides ( $y=-\frac{63}{4}$ ), then for $x$ from the left and right sides ( $x=-\frac{105}{16}$ ), and divide: $-\frac{105}{16} \div \left(-\frac{63}{4}\right) = \frac{5}{12}$ .

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Multiplying and dividing signed fractions and dividing out a common factor.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the equal-products condition as an equation in $x$ and $y$.

💡Takeaway. When two products share a factor, cancel it first and the rest falls out.

Equal side products cancel the shared corner · 12 practice problems

Generated variants — 12

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