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

Variant 1 easy answer:

\frac{1}{40}

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

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

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 answer:

\frac{1}{16}

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

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

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 answer:

\frac{1}{18}

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

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

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 answer:

\frac{1}{27}

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

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

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 answer:

\frac{1}{40}

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

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

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 answer:

\frac{1}{18}

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

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

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 answer:

\frac{1}{24}

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

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

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 answer:

\frac{1}{6}

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

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

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 answer:

\frac{1}{75}

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

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

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 answer:

\frac{1}{12}

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

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

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 answer:

\frac{1}{100}

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

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

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 answer:

\frac{1}{18}

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

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

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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