← Rewrite the division as one fraction and the placement becomes obvious · Build the Largest or Smallest Value from Digit Cards

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

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

Generated variants — 12

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

Variant 1 easy answer: 38\frac{3}{8}

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.

1 2 3 4
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.
small÷large\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.
14÷23\frac{1}{4} \div \frac{2}{3}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
14×32=38\frac{1}{4} \times \frac{3}{2} = \frac{3}{8}
The smallest quotient is 3/8.
Answer: 38\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 answer: 815\frac{8}{15}

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.

2 3 4 5
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.
small÷large\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.
25÷34\frac{2}{5} \div \frac{3}{4}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
25×43=815\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}
The smallest quotient is 8/15.
Answer: 815\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 answer: 58\frac{5}{8}

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.

3 4 5 6
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.
small÷large\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.
36÷45\frac{3}{6} \div \frac{4}{5}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
36×54=58\frac{3}{6} \times \frac{5}{4} = \frac{5}{8}
The smallest quotient is 5/8.
Answer: 58\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 answer: 2435\frac{24}{35}

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.

4 5 6 7
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.
small÷large\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.
47÷56\frac{4}{7} \div \frac{5}{6}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
47×65=2435\frac{4}{7} \times \frac{6}{5} = \frac{24}{35}
The smallest quotient is 24/35.
Answer: 2435\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 answer: 514\frac{5}{14}

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.

2 4 5 7
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.
small÷large\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.
27÷45\frac{2}{7} \div \frac{4}{5}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
27×54=514\frac{2}{7} \times \frac{5}{4} = \frac{5}{14}
The smallest quotient is 5/14.
Answer: 514\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 answer: 521\frac{5}{21}

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.

1 3 5 7
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.
small÷large\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.
17÷35\frac{1}{7} \div \frac{3}{5}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
17×53=521\frac{1}{7} \times \frac{5}{3} = \frac{5}{21}
The smallest quotient is 5/21.
Answer: 521\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 answer: 3548\frac{35}{48}

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.

5 6 7 8
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.
small÷large\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.
58÷67\frac{5}{8} \div \frac{6}{7}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
58×76=3548\frac{5}{8} \times \frac{7}{6} = \frac{35}{48}
The smallest quotient is 35/48.
Answer: 3548\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 answer: 512\frac{5}{12}

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.

2 3 5 8
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.
small÷large\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.
28÷35\frac{2}{8} \div \frac{3}{5}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
28×53=512\frac{2}{8} \times \frac{5}{3} = \frac{5}{12}
The smallest quotient is 5/12.
Answer: 512\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 answer: 920\frac{9}{20}

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.

3 5 6 8
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.
small÷large\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.
38÷56\frac{3}{8} \div \frac{5}{6}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
38×65=920\frac{3}{8} \times \frac{6}{5} = \frac{9}{20}
The smallest quotient is 9/20.
Answer: 920\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 answer: 727\frac{7}{27}

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.

2 6 7 9
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.
small÷large\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.
29÷67\frac{2}{9} \div \frac{6}{7}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
29×76=727\frac{2}{9} \times \frac{7}{6} = \frac{7}{27}
The smallest quotient is 7/27.
Answer: 727\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 answer: 16\frac{1}{6}

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.

1 4 6 9
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.
small÷large\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.
19÷46\frac{1}{9} \div \frac{4}{6}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
19×64=16\frac{1}{9} \times \frac{6}{4} = \frac{1}{6}
The smallest quotient is 1/6.
Answer: 16\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 answer: 712\frac{7}{12}

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.

3 4 7 9
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.
small÷large\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.
39÷47\frac{3}{9} \div \frac{4}{7}
One arrangement wins.

4Work it out

#13 Convert to Algebra 6.NS.A.1
Flip and multiply.
39×74=712\frac{3}{9} \times \frac{7}{4} = \frac{7}{12}
The smallest quotient is 7/12.
Answer: 712\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.