← Spend the small cards on the divisor and the rest look after themselves · Build the Largest or Smallest Value from Digit Cards

Spend the small cards on the divisor and the rest look after themselves · 12 practice problems

6.NS.B.36.EE.B.8

Generated variants — 12

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

Variant 1 easy answer: 5.45

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 2 4 5 6
Show solution
1 · Understandwhat's really being asked

The cards 1, 2, 4, 5, 6 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 2, 4, 5, 6.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 2, and the smaller one goes in front to keep the divisor low.
1.21.2
The divisor is 1.2.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 4, 5 and 6 make the largest number when the biggest is in front.
6.546.54
The dividend is 6.54.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
6.54÷1.2=5.456.54 \div 1.2 = 5.45
The largest quotient is 5.45.
Answer: 5.45
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.19, so 5.45 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 2 easy answer: 2.25

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

3 4 5 6 7
Show solution
1 · Understandwhat's really being asked

The cards 3, 4, 5, 6, 7 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 3, 4, 5, 6, 7.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 3 and 4, and the smaller one goes in front to keep the divisor low.
3.43.4
The divisor is 3.4.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 5, 6 and 7 make the largest number when the biggest is in front.
7.657.65
The dividend is 7.65.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
7.65÷3.4=2.257.65 \div 3.4 = 2.25
The largest quotient is 2.25.
Answer: 2.25
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.45, so 2.25 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 3 easy answer: 7.3

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 2 6 7 8
Show solution
1 · Understandwhat's really being asked

The cards 1, 2, 6, 7, 8 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 2, 6, 7, 8.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 2, and the smaller one goes in front to keep the divisor low.
1.21.2
The divisor is 1.2.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 6, 7 and 8 make the largest number when the biggest is in front.
8.768.76
The dividend is 8.76.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
8.76÷1.2=7.38.76 \div 1.2 = 7.3
The largest quotient is 7.3.
Answer: 7.3
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.14, so 7.3 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 4 easy answer: 7.2

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 2 4 6 8
Show solution
1 · Understandwhat's really being asked

The cards 1, 2, 4, 6, 8 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 2, 4, 6, 8.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 2, and the smaller one goes in front to keep the divisor low.
1.21.2
The divisor is 1.2.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 4, 6 and 8 make the largest number when the biggest is in front.
8.648.64
The dividend is 8.64.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
8.64÷1.2=7.28.64 \div 1.2 = 7.2
The largest quotient is 7.2.
Answer: 7.2
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.14, so 7.2 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 5 medium answer: 7.275

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 2 3 7 8
Show solution
1 · Understandwhat's really being asked

The cards 1, 2, 3, 7, 8 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 2, 3, 7, 8.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 2, and the smaller one goes in front to keep the divisor low.
1.21.2
The divisor is 1.2.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 3, 7 and 8 make the largest number when the biggest is in front.
8.738.73
The dividend is 8.73.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
8.73÷1.2=7.2758.73 \div 1.2 = 7.275
The largest quotient is 7.275.
Answer: 7.275
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.14, so 7.275 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 6 medium answer: 6.25

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 4 5 7 8
Show solution
1 · Understandwhat's really being asked

The cards 1, 4, 5, 7, 8 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 4, 5, 7, 8.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 4, and the smaller one goes in front to keep the divisor low.
1.41.4
The divisor is 1.4.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 5, 7 and 8 make the largest number when the biggest is in front.
8.758.75
The dividend is 8.75.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
8.75÷1.4=6.258.75 \div 1.4 = 6.25
The largest quotient is 6.25.
Answer: 6.25
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.17, so 6.25 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 7 hard answer: 8.025

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 2 3 6 9
Show solution
1 · Understandwhat's really being asked

The cards 1, 2, 3, 6, 9 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 2, 3, 6, 9.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 2, and the smaller one goes in front to keep the divisor low.
1.21.2
The divisor is 1.2.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 3, 6 and 9 make the largest number when the biggest is in front.
9.639.63
The dividend is 9.63.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
9.63÷1.2=8.0259.63 \div 1.2 = 8.025
The largest quotient is 8.025.
Answer: 8.025
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.13, so 8.025 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 8 hard answer: 7.05

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 4 7 8 9
Show solution
1 · Understandwhat's really being asked

The cards 1, 4, 7, 8, 9 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 4, 7, 8, 9.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 4, and the smaller one goes in front to keep the divisor low.
1.41.4
The divisor is 1.4.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 7, 8 and 9 make the largest number when the biggest is in front.
9.879.87
The dividend is 9.87.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
9.87÷1.4=7.059.87 \div 1.4 = 7.05
The largest quotient is 7.05.
Answer: 7.05
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.15, so 7.05 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 9 medium answer: 7.5

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 3 5 7 9
Show solution
1 · Understandwhat's really being asked

The cards 1, 3, 5, 7, 9 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 3, 5, 7, 9.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 3, and the smaller one goes in front to keep the divisor low.
1.31.3
The divisor is 1.3.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 5, 7 and 9 make the largest number when the biggest is in front.
9.759.75
The dividend is 9.75.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
9.75÷1.3=7.59.75 \div 1.3 = 7.5
The largest quotient is 7.5.
Answer: 7.5
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.14, so 7.5 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 10 hard answer: 3.904

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

2 5 6 7 9
Show solution
1 · Understandwhat's really being asked

The cards 2, 5, 6, 7, 9 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 2, 5, 6, 7, 9.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 2 and 5, and the smaller one goes in front to keep the divisor low.
2.52.5
The divisor is 2.5.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 6, 7 and 9 make the largest number when the biggest is in front.
9.769.76
The dividend is 9.76.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
9.76÷2.5=3.9049.76 \div 2.5 = 3.904
The largest quotient is 3.904.
Answer: 3.904
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.26, so 3.904 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 11 medium answer: 3.944

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

2 5 6 8 9
Show solution
1 · Understandwhat's really being asked

The cards 2, 5, 6, 8, 9 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 2, 5, 6, 8, 9.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 2 and 5, and the smaller one goes in front to keep the divisor low.
2.52.5
The divisor is 2.5.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 6, 8 and 9 make the largest number when the biggest is in front.
9.869.86
The dividend is 9.86.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
9.86÷2.5=3.9449.86 \div 2.5 = 3.944
The largest quotient is 3.944.
Answer: 3.944
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.26, so 3.944 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.
Variant 12 hard answer: 8.125

Using each of the five number cards exactly once, make the division (a number with two decimal places) ÷\div (a number with one decimal place) whose quotient is as large as possible, and find that quotient.

1 2 5 7 9
Show solution
1 · Understandwhat's really being asked

The cards 1, 2, 5, 7, 9 fill five places: three in a number with two decimal places and two in a number with one. We want the largest quotient.

Givens
  • The cards are 1, 2, 5, 7, 9.
  • Each card is used exactly once.
  • The dividend has two decimal places and the divisor one.
Unknowns
  • The largest quotient that can be made.
Constraints
  • The five cards must fill all five places.
2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #8 Analyze the Units#2 Make a Systematic List

A big quotient wants a big dividend and a small divisor. Check whether those two aims compete: the divisor needs two cards, and spending the two smallest there leaves the three largest for the dividend -- so they do not.

3 · Execute4 carry out the plan

1Say what makes a quotient large

#8 Analyze the Units 6.NS.B.3
More on top and less underneath both push it up.
largesmall\frac{\text{large}}{\text{small}}
Two things to arrange.

2Give the divisor the smallest cards

#3 Eliminate Possibilities 6.EE.B.8
The two smallest are 1 and 2, and the smaller one goes in front to keep the divisor low.
1.21.2
The divisor is 1.2.

3Build the dividend from what is left

#2 Make a Systematic List 6.EE.B.8
The remaining cards 5, 7 and 9 make the largest number when the biggest is in front.
9.759.75
The dividend is 9.75.

4Divide

#8 Analyze the Units 6.NS.B.3
Both numbers are as favourable as the cards allow.
9.75÷1.2=8.1259.75 \div 1.2 = 8.125
The largest quotient is 8.125.
Answer: 8.125
4 · Reviewdoes it hold up?

The other extreme is far away: the smallest quotient these cards can make is about 0.13, so 8.125 really is at the top of the range.

Another way: Swapping any one card between the two numbers makes the dividend smaller or the divisor bigger, and either way the quotient drops -- which is the check that the two aims do not compete.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.
  • 6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Comparing arrangements to find the largest.
💡Takeaway. Ask what makes the answer big, then check whether the two ways of doing it get in each other's way.