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Rewrite the division as one fraction and the placement becomes obvious · 12 practice problems

6.NS.A.16.EE.B.8

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Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 1, 2, 3, 4 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 1, 2, 3, 4.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 1 over 4 divided by 2 over 3.

\frac{1}{4} \div \frac{2}{3}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{1}{4} \times \frac{3}{2} = \frac{3}{8}

The smallest quotient is 3/8.

Answer:

\frac{3}{8}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 2 and 2/3, so 3/8 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 2 easy

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 2, 3, 4, 5 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 2, 3, 4, 5.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 2 over 5 divided by 3 over 4.

\frac{2}{5} \div \frac{3}{4}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}

The smallest quotient is 8/15.

Answer:

\frac{8}{15}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 1 and 7/8, so 8/15 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 3 easy

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 3, 4, 5, 6 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 3, 4, 5, 6.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 3 over 6 divided by 4 over 5.

\frac{3}{6} \div \frac{4}{5}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{3}{6} \times \frac{5}{4} = \frac{5}{8}

The smallest quotient is 5/8.

Answer:

\frac{5}{8}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 1 and 3/5, so 5/8 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 4 medium

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 4, 5, 6, 7 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 4, 5, 6, 7.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 4 over 7 divided by 5 over 6.

\frac{4}{7} \div \frac{5}{6}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{4}{7} \times \frac{6}{5} = \frac{24}{35}

The smallest quotient is 24/35.

Answer:

\frac{24}{35}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 1 and 11/24, so 24/35 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 5 easy

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 2, 4, 5, 7 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 2, 4, 5, 7.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 2 over 7 divided by 4 over 5.

\frac{2}{7} \div \frac{4}{5}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{2}{7} \times \frac{5}{4} = \frac{5}{14}

The smallest quotient is 5/14.

Answer:

\frac{5}{14}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 2 and 4/5, so 5/14 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 6 medium

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 1, 3, 5, 7 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 1, 3, 5, 7.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 1 over 7 divided by 3 over 5.

\frac{1}{7} \div \frac{3}{5}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{1}{7} \times \frac{5}{3} = \frac{5}{21}

The smallest quotient is 5/21.

Answer:

\frac{5}{21}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 4 and 1/5, so 5/21 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 7 hard

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 5, 6, 7, 8 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 5, 6, 7, 8.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 5 over 8 divided by 6 over 7.

\frac{5}{8} \div \frac{6}{7}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{5}{8} \times \frac{7}{6} = \frac{35}{48}

The smallest quotient is 35/48.

Answer:

\frac{35}{48}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 1 and 13/35, so 35/48 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 8 medium

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 2, 3, 5, 8 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 2, 3, 5, 8.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 2 over 8 divided by 3 over 5.

\frac{2}{8} \div \frac{3}{5}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{2}{8} \times \frac{5}{3} = \frac{5}{12}

The smallest quotient is 5/12.

Answer:

\frac{5}{12}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 2 and 2/5, so 5/12 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 9 medium

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 3, 5, 6, 8 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 3, 5, 6, 8.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 3 over 8 divided by 5 over 6.

\frac{3}{8} \div \frac{5}{6}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{3}{8} \times \frac{6}{5} = \frac{9}{20}

The smallest quotient is 9/20.

Answer:

\frac{9}{20}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 2 and 2/9, so 9/20 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 10 hard

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 2, 6, 7, 9 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 2, 6, 7, 9.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 2 over 9 divided by 6 over 7.

\frac{2}{9} \div \frac{6}{7}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{2}{9} \times \frac{7}{6} = \frac{7}{27}

The smallest quotient is 7/27.

Answer:

\frac{7}{27}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 3 and 6/7, so 7/27 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 11 hard

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 1, 4, 6, 9 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 1, 4, 6, 9.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 1 over 9 divided by 4 over 6.

\frac{1}{9} \div \frac{4}{6}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{1}{9} \times \frac{6}{4} = \frac{1}{6}

The smallest quotient is 1/6.

Answer:

\frac{1}{6}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 6, so 1/6 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Variant 12 hard

Using each of the four number cards exactly once, make the division (proper fraction) $\div$ (proper fraction) whose result is as small as possible. Find that quotient.

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 3, 4, 7, 9 each go into one of four places in (proper fraction) divided by (proper fraction). We want the smallest quotient.

Givens

The cards are 3, 4, 7, 9.
Each card is used exactly once.
Both fractions must be proper.

Unknowns

The smallest quotient that can be made.

Constraints

A proper fraction has its numerator smaller than its denominator.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

Rewrite the division as a single fraction first. Then two cards are on top and two underneath, and smallest means small cards up, large cards down -- with the proper-fraction rule deciding the ties.

3 · Execute4 carry out the plan

1Write the division as one fraction

#13 Convert to Algebra 6.NS.A.1

Dividing by a fraction multiplies by its reciprocal, so the second fraction turns upside down.

\frac{\square}{\square} \div \frac{\square}{\square} = \frac{\square \times \square}{\square \times \square}

Two cards on top, two underneath.

2Say what makes it small

#3 Eliminate Possibilities 6.EE.B.8

A fraction is smallest when its top is small and its bottom large, so the two smallest cards want to be on top.

\text{small} \div \text{large}

But both fractions still have to be proper.

3Work through the placings the rule allows

#2 Make a Systematic List 6.EE.B.8

There are 6 arrangements where both fractions are proper; the best of them puts 3 over 9 divided by 4 over 7.

\frac{3}{9} \div \frac{4}{7}

One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1

Flip and multiply.

\frac{3}{9} \times \frac{7}{4} = \frac{7}{12}

The smallest quotient is 7/12.

Answer:

\frac{7}{12}

4 · Reviewdoes it hold up?

The largest quotient these cards can make is 1 and 5/7, so 7/12 really is at the small end of what is possible.

Another way: Listing all 6 allowed arrangements and comparing them gives the same winner; the top-and-bottom rule is what makes it unnecessary to compute all of them.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Turning a division of fractions into one fraction.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing the arrangements the proper-fraction rule allows.

💡Takeaway. Turn the division into one fraction. Then making it small is just deciding which cards go on top.

Where next?

← Back to the problem: Rewrite the division as one fraction and the placement becomes obvious