Practice · Smaller divisor makes a bigger quotient · Sensim Math

Variant 1 medium

Using the number cards $3$ , $5$ , $7$ , $2$ , $8$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 3, 5, 7, 2, 8 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 3, 5, 7, 2, 8 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 23.

\text{smallest divisor} = 23

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 875, putting the biggest digit first.

\text{largest dividend} = 875

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 875 divided by 23. Since 23 x 38 = 874, the quotient is 38 with a remainder of 1.

875 \div 23 = 38 \text{ remainder } 1 \quad (23 \times 38 = 874,\; 875 - 874 = 1)

Find how many 23s fit in 875; 38 of them reach 874, leaving 1 left over.

Answer: 875 / 23 = 38 remainder 1 (quotient 38)

4 · Reviewdoes it hold up?

Any divisor bigger than 23 lowers the quotient, and 875 is the biggest dividend left, so 875 / 23 gives the largest quotient. Check: 23 x 38 + 1 = 875.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 875 by 23 to get quotient 38 and remainder 1.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 2 easy

Using the number cards $5$ , $4$ , $1$ , $8$ , $2$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 5, 4, 1, 8, 2 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 5, 4, 1, 8, 2 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 12.

\text{smallest divisor} = 12

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 854, putting the biggest digit first.

\text{largest dividend} = 854

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 854 divided by 12. Since 12 x 71 = 852, the quotient is 71 with a remainder of 2.

854 \div 12 = 71 \text{ remainder } 2 \quad (12 \times 71 = 852,\; 854 - 852 = 2)

Find how many 12s fit in 854; 71 of them reach 852, leaving 2 left over.

Answer: 854 / 12 = 71 remainder 2 (quotient 71)

4 · Reviewdoes it hold up?

Any divisor bigger than 12 lowers the quotient, and 854 is the biggest dividend left, so 854 / 12 gives the largest quotient. Check: 12 x 71 + 2 = 854.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 854 by 12 to get quotient 71 and remainder 2.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 3 easy

Using the number cards $7$ , $2$ , $5$ , $3$ , $8$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 7, 2, 5, 3, 8 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 7, 2, 5, 3, 8 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 23.

\text{smallest divisor} = 23

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 875, putting the biggest digit first.

\text{largest dividend} = 875

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 875 divided by 23. Since 23 x 38 = 874, the quotient is 38 with a remainder of 1.

875 \div 23 = 38 \text{ remainder } 1 \quad (23 \times 38 = 874,\; 875 - 874 = 1)

Find how many 23s fit in 875; 38 of them reach 874, leaving 1 left over.

Answer: 875 / 23 = 38 remainder 1 (quotient 38)

4 · Reviewdoes it hold up?

Any divisor bigger than 23 lowers the quotient, and 875 is the biggest dividend left, so 875 / 23 gives the largest quotient. Check: 23 x 38 + 1 = 875.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 875 by 23 to get quotient 38 and remainder 1.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 4 easy

Using the number cards $2$ , $5$ , $8$ , $3$ , $6$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 2, 5, 8, 3, 6 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 2, 5, 8, 3, 6 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 23.

\text{smallest divisor} = 23

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 865, putting the biggest digit first.

\text{largest dividend} = 865

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 865 divided by 23. Since 23 x 37 = 851, the quotient is 37 with a remainder of 14.

865 \div 23 = 37 \text{ remainder } 14 \quad (23 \times 37 = 851,\; 865 - 851 = 14)

Find how many 23s fit in 865; 37 of them reach 851, leaving 14 left over.

Answer: 865 / 23 = 37 remainder 14 (quotient 37)

4 · Reviewdoes it hold up?

Any divisor bigger than 23 lowers the quotient, and 865 is the biggest dividend left, so 865 / 23 gives the largest quotient. Check: 23 x 37 + 14 = 865.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 865 by 23 to get quotient 37 and remainder 14.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 5 hard

Using the number cards $9$ , $8$ , $1$ , $4$ , $5$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 9, 8, 1, 4, 5 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 9, 8, 1, 4, 5 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 14.

\text{smallest divisor} = 14

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 985, putting the biggest digit first.

\text{largest dividend} = 985

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 985 divided by 14. Since 14 x 70 = 980, the quotient is 70 with a remainder of 5.

985 \div 14 = 70 \text{ remainder } 5 \quad (14 \times 70 = 980,\; 985 - 980 = 5)

Find how many 14s fit in 985; 70 of them reach 980, leaving 5 left over.

Answer: 985 / 14 = 70 remainder 5 (quotient 70)

4 · Reviewdoes it hold up?

Any divisor bigger than 14 lowers the quotient, and 985 is the biggest dividend left, so 985 / 14 gives the largest quotient. Check: 14 x 70 + 5 = 985.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 985 by 14 to get quotient 70 and remainder 5.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 6 medium

Using the number cards $4$ , $8$ , $1$ , $9$ , $3$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 4, 8, 1, 9, 3 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 4, 8, 1, 9, 3 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 13.

\text{smallest divisor} = 13

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 984, putting the biggest digit first.

\text{largest dividend} = 984

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 984 divided by 13. Since 13 x 75 = 975, the quotient is 75 with a remainder of 9.

984 \div 13 = 75 \text{ remainder } 9 \quad (13 \times 75 = 975,\; 984 - 975 = 9)

Find how many 13s fit in 984; 75 of them reach 975, leaving 9 left over.

Answer: 984 / 13 = 75 remainder 9 (quotient 75)

4 · Reviewdoes it hold up?

Any divisor bigger than 13 lowers the quotient, and 984 is the biggest dividend left, so 984 / 13 gives the largest quotient. Check: 13 x 75 + 9 = 984.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 984 by 13 to get quotient 75 and remainder 9.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 7 hard

Using the number cards $2$ , $7$ , $4$ , $9$ , $3$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 2, 7, 4, 9, 3 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 2, 7, 4, 9, 3 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 23.

\text{smallest divisor} = 23

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 974, putting the biggest digit first.

\text{largest dividend} = 974

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 974 divided by 23. Since 23 x 42 = 966, the quotient is 42 with a remainder of 8.

974 \div 23 = 42 \text{ remainder } 8 \quad (23 \times 42 = 966,\; 974 - 966 = 8)

Find how many 23s fit in 974; 42 of them reach 966, leaving 8 left over.

Answer: 974 / 23 = 42 remainder 8 (quotient 42)

4 · Reviewdoes it hold up?

Any divisor bigger than 23 lowers the quotient, and 974 is the biggest dividend left, so 974 / 23 gives the largest quotient. Check: 23 x 42 + 8 = 974.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 974 by 23 to get quotient 42 and remainder 8.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 8 medium

Using the number cards $3$ , $6$ , $9$ , $1$ , $7$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 3, 6, 9, 1, 7 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 3, 6, 9, 1, 7 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 13.

\text{smallest divisor} = 13

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 976, putting the biggest digit first.

\text{largest dividend} = 976

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 976 divided by 13. Since 13 x 75 = 975, the quotient is 75 with a remainder of 1.

976 \div 13 = 75 \text{ remainder } 1 \quad (13 \times 75 = 975,\; 976 - 975 = 1)

Find how many 13s fit in 976; 75 of them reach 975, leaving 1 left over.

Answer: 976 / 13 = 75 remainder 1 (quotient 75)

4 · Reviewdoes it hold up?

Any divisor bigger than 13 lowers the quotient, and 976 is the biggest dividend left, so 976 / 13 gives the largest quotient. Check: 13 x 75 + 1 = 976.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 976 by 13 to get quotient 75 and remainder 1.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 9 hard

Using the number cards $6$ , $1$ , $7$ , $2$ , $9$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 6, 1, 7, 2, 9 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 6, 1, 7, 2, 9 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 12.

\text{smallest divisor} = 12

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 976, putting the biggest digit first.

\text{largest dividend} = 976

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 976 divided by 12. Since 12 x 81 = 972, the quotient is 81 with a remainder of 4.

976 \div 12 = 81 \text{ remainder } 4 \quad (12 \times 81 = 972,\; 976 - 972 = 4)

Find how many 12s fit in 976; 81 of them reach 972, leaving 4 left over.

Answer: 976 / 12 = 81 remainder 4 (quotient 81)

4 · Reviewdoes it hold up?

Any divisor bigger than 12 lowers the quotient, and 976 is the biggest dividend left, so 976 / 12 gives the largest quotient. Check: 12 x 81 + 4 = 976.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 976 by 12 to get quotient 81 and remainder 4.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Variant 10 medium

Using the number cards $9$ , $4$ , $1$ , $6$ , $7$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Use the five digit cards 9, 4, 1, 6, 7 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 9, 4, 1, 6, 7 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

3 · Execute3 carry out the plan

1Make the divisor as small as possible

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards is 14.

\text{smallest divisor} = 14

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

2Make the dividend as large as possible

#6 Guess and Check 4.NBT.A.2

From the remaining cards the largest 3-digit number is 976, putting the biggest digit first.

\text{largest dividend} = 976

Place the biggest digit in the hundreds spot to make the number as large as it can be.

3Divide

#6 Guess and Check 4.NBT.B.6

Compute 976 divided by 14. Since 14 x 69 = 966, the quotient is 69 with a remainder of 10.

976 \div 14 = 69 \text{ remainder } 10 \quad (14 \times 69 = 966,\; 976 - 966 = 10)

Find how many 14s fit in 976; 69 of them reach 966, leaving 10 left over.

Answer: 976 / 14 = 69 remainder 10 (quotient 69)

4 · Reviewdoes it hold up?

Any divisor bigger than 14 lowers the quotient, and 976 is the biggest dividend left, so 976 / 14 gives the largest quotient. Check: 14 x 69 + 10 = 976.

Another way: Make a systematic list (tool 2): pair each small divisor with the largest leftover 3-digit number and divide; the smallest divisor with the largest dividend wins, confirming the answer.

Standardsmin grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 976 by 14 to get quotient 69 and remainder 10.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡Takeaway. Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!

Smaller divisor makes a bigger quotient · 10 practice problems

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