Practice · One equation fixes the ratio, not either unknown · Sensim Math

Variant 1 easy

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 2 \times 5 \qquad \triangle \div 4 \div 4$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 2 then times 5 equals triangle divided by 4 then divided by 4. We want circle divided by triangle.

Givens

The left expression is the circle divided by 2 and multiplied by 5.
The right expression is the triangle divided by 4 and then by 4.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 2 and multiplying by 5 is one multiplication.

\bigcirc \div 2 \times 5 = \bigcirc \times 2\frac{1}{2}

The circle is scaled by 2 and 1/2.

2Collapse the right side

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

Two divisions in a row divide by the product, 16.

\triangle \div 4 \div 4 = \triangle \times \frac{1}{16}

The triangle is scaled by one 16th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 2\frac{1}{2} = \triangle \times \frac{1}{16}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{16} \div 2\frac{1}{2} = \frac{1}{40}

The circle is 1/40 of the triangle.

Answer:

\frac{1}{40}

4 · Reviewdoes it hold up?

Try triangle = 32: then the right side is 2, and a circle of $\frac{4}{5}$ gives the same on the left. Their quotient is $\frac{1}{40}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 2 easy

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 3 \times 6 \qquad \triangle \div 2 \div 4$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 3 then times 6 equals triangle divided by 2 then divided by 4. We want circle divided by triangle.

Givens

The left expression is the circle divided by 3 and multiplied by 6.
The right expression is the triangle divided by 2 and then by 4.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 3 and multiplying by 6 is one multiplication.

\bigcirc \div 3 \times 6 = \bigcirc \times 2

The circle is scaled by 2.

2Collapse the right side

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

Two divisions in a row divide by the product, 8.

\triangle \div 2 \div 4 = \triangle \times \frac{1}{8}

The triangle is scaled by one 8th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 2 = \triangle \times \frac{1}{8}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{8} \div 2 = \frac{1}{16}

The circle is 1/16 of the triangle.

Answer:

\frac{1}{16}

4 · Reviewdoes it hold up?

Try triangle = 24: then the right side is 3, and a circle of $1\frac{1}{2}$ gives the same on the left. Their quotient is $\frac{1}{16}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 3 easy

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 4 \times 6 \qquad \triangle \div 2 \div 6$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 4 then times 6 equals triangle divided by 2 then divided by 6. We want circle divided by triangle.

Givens

The left expression is the circle divided by 4 and multiplied by 6.
The right expression is the triangle divided by 2 and then by 6.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 4 and multiplying by 6 is one multiplication.

\bigcirc \div 4 \times 6 = \bigcirc \times 1\frac{1}{2}

The circle is scaled by 1 and 1/2.

2Collapse the right side

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

Two divisions in a row divide by the product, 12.

\triangle \div 2 \div 6 = \triangle \times \frac{1}{12}

The triangle is scaled by one 12th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 1\frac{1}{2} = \triangle \times \frac{1}{12}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{12} \div 1\frac{1}{2} = \frac{1}{18}

The circle is 1/18 of the triangle.

Answer:

\frac{1}{18}

4 · Reviewdoes it hold up?

Try triangle = 48: then the right side is 4, and a circle of $2\frac{2}{3}$ gives the same on the left. Their quotient is $\frac{1}{18}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 4 easy

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 2 \times 6 \qquad \triangle \div 3 \div 3$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 2 then times 6 equals triangle divided by 3 then divided by 3. We want circle divided by triangle.

Givens

The left expression is the circle divided by 2 and multiplied by 6.
The right expression is the triangle divided by 3 and then by 3.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 2 and multiplying by 6 is one multiplication.

\bigcirc \div 2 \times 6 = \bigcirc \times 3

The circle is scaled by 3.

2Collapse the right side

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

Two divisions in a row divide by the product, 9.

\triangle \div 3 \div 3 = \triangle \times \frac{1}{9}

The triangle is scaled by one 9th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 3 = \triangle \times \frac{1}{9}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{9} \div 3 = \frac{1}{27}

The circle is 1/27 of the triangle.

Answer:

\frac{1}{27}

4 · Reviewdoes it hold up?

Try triangle = 18: then the right side is 2, and a circle of $\frac{2}{3}$ gives the same on the left. Their quotient is $\frac{1}{27}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 5 medium

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 2 \times 8 \qquad \triangle \div 5 \div 2$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 2 then times 8 equals triangle divided by 5 then divided by 2. We want circle divided by triangle.

Givens

The left expression is the circle divided by 2 and multiplied by 8.
The right expression is the triangle divided by 5 and then by 2.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 2 and multiplying by 8 is one multiplication.

\bigcirc \div 2 \times 8 = \bigcirc \times 4

The circle is scaled by 4.

2Collapse the right side

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

Two divisions in a row divide by the product, 10.

\triangle \div 5 \div 2 = \triangle \times \frac{1}{10}

The triangle is scaled by one 10th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 4 = \triangle \times \frac{1}{10}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{10} \div 4 = \frac{1}{40}

The circle is 1/40 of the triangle.

Answer:

\frac{1}{40}

4 · Reviewdoes it hold up?

Try triangle = 20: then the right side is 2, and a circle of $\frac{1}{2}$ gives the same on the left. Their quotient is $\frac{1}{40}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 6 medium

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 4 \times 8 \qquad \triangle \div 3 \div 3$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 4 then times 8 equals triangle divided by 3 then divided by 3. We want circle divided by triangle.

Givens

The left expression is the circle divided by 4 and multiplied by 8.
The right expression is the triangle divided by 3 and then by 3.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 4 and multiplying by 8 is one multiplication.

\bigcirc \div 4 \times 8 = \bigcirc \times 2

The circle is scaled by 2.

2Collapse the right side

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

Two divisions in a row divide by the product, 9.

\triangle \div 3 \div 3 = \triangle \times \frac{1}{9}

The triangle is scaled by one 9th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 2 = \triangle \times \frac{1}{9}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{9} \div 2 = \frac{1}{18}

The circle is 1/18 of the triangle.

Answer:

\frac{1}{18}

4 · Reviewdoes it hold up?

Try triangle = 36: then the right side is 4, and a circle of $2$ gives the same on the left. Their quotient is $\frac{1}{18}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 7 medium

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 3 \times 9 \qquad \triangle \div 4 \div 2$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 3 then times 9 equals triangle divided by 4 then divided by 2. We want circle divided by triangle.

Givens

The left expression is the circle divided by 3 and multiplied by 9.
The right expression is the triangle divided by 4 and then by 2.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 3 and multiplying by 9 is one multiplication.

\bigcirc \div 3 \times 9 = \bigcirc \times 3

The circle is scaled by 3.

2Collapse the right side

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

Two divisions in a row divide by the product, 8.

\triangle \div 4 \div 2 = \triangle \times \frac{1}{8}

The triangle is scaled by one 8th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 3 = \triangle \times \frac{1}{8}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{8} \div 3 = \frac{1}{24}

The circle is 1/24 of the triangle.

Answer:

\frac{1}{24}

4 · Reviewdoes it hold up?

Try triangle = 24: then the right side is 3, and a circle of $1$ gives the same on the left. Their quotient is $\frac{1}{24}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 8 medium

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 6 \times 9 \qquad \triangle \div 2 \div 2$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 6 then times 9 equals triangle divided by 2 then divided by 2. We want circle divided by triangle.

Givens

The left expression is the circle divided by 6 and multiplied by 9.
The right expression is the triangle divided by 2 and then by 2.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 6 and multiplying by 9 is one multiplication.

\bigcirc \div 6 \times 9 = \bigcirc \times 1\frac{1}{2}

The circle is scaled by 1 and 1/2.

2Collapse the right side

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

Two divisions in a row divide by the product, 4.

\triangle \div 2 \div 2 = \triangle \times \frac{1}{4}

The triangle is scaled by one 4th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 1\frac{1}{2} = \triangle \times \frac{1}{4}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{4} \div 1\frac{1}{2} = \frac{1}{6}

The circle is 1/6 of the triangle.

Answer:

\frac{1}{6}

4 · Reviewdoes it hold up?

Try triangle = 24: then the right side is 6, and a circle of $4$ gives the same on the left. Their quotient is $\frac{1}{6}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 9 hard

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 2 \times 10 \qquad \triangle \div 3 \div 5$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 2 then times 10 equals triangle divided by 3 then divided by 5. We want circle divided by triangle.

Givens

The left expression is the circle divided by 2 and multiplied by 10.
The right expression is the triangle divided by 3 and then by 5.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 2 and multiplying by 10 is one multiplication.

\bigcirc \div 2 \times 10 = \bigcirc \times 5

The circle is scaled by 5.

2Collapse the right side

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

Two divisions in a row divide by the product, 15.

\triangle \div 3 \div 5 = \triangle \times \frac{1}{15}

The triangle is scaled by one 15th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 5 = \triangle \times \frac{1}{15}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{15} \div 5 = \frac{1}{75}

The circle is 1/75 of the triangle.

Answer:

\frac{1}{75}

4 · Reviewdoes it hold up?

Try triangle = 30: then the right side is 2, and a circle of $\frac{2}{5}$ gives the same on the left. Their quotient is $\frac{1}{75}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 10 hard

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 5 \times 10 \qquad \triangle \div 2 \div 3$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 5 then times 10 equals triangle divided by 2 then divided by 3. We want circle divided by triangle.

Givens

The left expression is the circle divided by 5 and multiplied by 10.
The right expression is the triangle divided by 2 and then by 3.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 5 and multiplying by 10 is one multiplication.

\bigcirc \div 5 \times 10 = \bigcirc \times 2

The circle is scaled by 2.

2Collapse the right side

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

Two divisions in a row divide by the product, 6.

\triangle \div 2 \div 3 = \triangle \times \frac{1}{6}

The triangle is scaled by one 6th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 2 = \triangle \times \frac{1}{6}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{6} \div 2 = \frac{1}{12}

The circle is 1/12 of the triangle.

Answer:

\frac{1}{12}

4 · Reviewdoes it hold up?

Try triangle = 30: then the right side is 5, and a circle of $2\frac{1}{2}$ gives the same on the left. Their quotient is $\frac{1}{12}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 11 hard

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 3 \times 12 \qquad \triangle \div 5 \div 5$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 3 then times 12 equals triangle divided by 5 then divided by 5. We want circle divided by triangle.

Givens

The left expression is the circle divided by 3 and multiplied by 12.
The right expression is the triangle divided by 5 and then by 5.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 3 and multiplying by 12 is one multiplication.

\bigcirc \div 3 \times 12 = \bigcirc \times 4

The circle is scaled by 4.

2Collapse the right side

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

Two divisions in a row divide by the product, 25.

\triangle \div 5 \div 5 = \triangle \times \frac{1}{25}

The triangle is scaled by one 25th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 4 = \triangle \times \frac{1}{25}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{25} \div 4 = \frac{1}{100}

The circle is 1/100 of the triangle.

Answer:

\frac{1}{100}

4 · Reviewdoes it hold up?

Try triangle = 75: then the right side is 3, and a circle of $\frac{3}{4}$ gives the same on the left. Their quotient is $\frac{1}{100}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

Variant 12 hard

The two expressions below give the same result. Find the value of $\bigcirc \div \triangle$ . (Neither $\bigcirc$ nor $\triangle$ is $0$ .)

$\bigcirc \div 5 \times 15 \qquad \triangle \div 3 \div 2$

Your answer

Show solution

1 · Understandwhat's really being asked

Circle divided by 5 then times 15 equals triangle divided by 3 then divided by 2. We want circle divided by triangle.

Givens

The left expression is the circle divided by 5 and multiplied by 15.
The right expression is the triangle divided by 3 and then by 2.
The two expressions are equal.

Unknowns

The value of circle divided by triangle.

Constraints

Neither symbol stands for 0, so dividing by them is allowed.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

One equation cannot pin down two unknowns, so do not chase either symbol. Write each side as its symbol times a single number; the equation between those two numbers is the ratio the question wants.

3 · Execute4 carry out the plan

1Collapse the left side

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

Dividing by 5 and multiplying by 15 is one multiplication.

\bigcirc \div 5 \times 15 = \bigcirc \times 3

The circle is scaled by 3.

2Collapse the right side

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

Two divisions in a row divide by the product, 6.

\triangle \div 3 \div 2 = \triangle \times \frac{1}{6}

The triangle is scaled by one 6th.

3Set the two equal

#13 Convert to Algebra 6.EE.B.6

The problem says the results match, so the two scaled symbols are the same number.

\bigcirc \times 3 = \triangle \times \frac{1}{6}

One equation, two unknowns -- but a fixed ratio.

4Read off the ratio

#7 Identify Subproblems 6.EE.B.5

Divide both sides by the triangle and by the number beside the circle. Neither symbol is known, and neither needs to be.

\bigcirc \div \triangle = \frac{1}{6} \div 3 = \frac{1}{18}

The circle is 1/18 of the triangle.

Answer:

\frac{1}{18}

4 · Reviewdoes it hold up?

Try triangle = 30: then the right side is 5, and a circle of $1\frac{2}{3}$ gives the same on the left. Their quotient is $\frac{1}{18}$ , whatever pair we pick.

Another way: Guessing a value for one symbol and solving for the other also works, and it shows the point: every guess gives a different pair but always the same quotient.

Standardsmin grade 6

6.EE.A.3 Apply properties to generate equivalent expressions — Rewriting each side as its symbol times one number.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two shapes as variables that stand for numbers.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Reading off what the equation does fix -- the ratio.

💡Takeaway. When one equation holds two unknowns, it will not tell you either one. It will tell you how they compare.

One equation fixes the ratio, not either unknown · 12 practice problems

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