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← 4-1 · Smaller divisor makes a bigger quotient · Build the Largest or Smallest Value from Digit Cards

Smaller divisor makes a bigger quotient · 10 practice problems

4.NBT.B.6

Generated variants — 10

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

Variant 1 answer: 985 / 14 = 70 remainder 5 (quotient 70)

Using the number cards 99, 88, 11, 44, 55 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=14\text{smallest divisor} = 14
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 985, putting the biggest digit first.
largest dividend=985\text{largest dividend} = 985
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷14=70 remainder 5(14×70=980,  985−980=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 2 answer: 984 / 13 = 75 remainder 9 (quotient 75)

Using the number cards 44, 88, 11, 99, 33 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=13\text{smallest divisor} = 13
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 984, putting the biggest digit first.
largest dividend=984\text{largest dividend} = 984
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷13=75 remainder 9(13×75=975,  984−975=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 3 answer: 875 / 23 = 38 remainder 1 (quotient 38)

Using the number cards 33, 55, 77, 22, 88 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=23\text{smallest divisor} = 23
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 875, putting the biggest digit first.
largest dividend=875\text{largest dividend} = 875
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷23=38 remainder 1(23×38=874,  875−874=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 4 answer: 854 / 12 = 71 remainder 2 (quotient 71)

Using the number cards 55, 44, 11, 88, 22 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=12\text{smallest divisor} = 12
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 854, putting the biggest digit first.
largest dividend=854\text{largest dividend} = 854
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷12=71 remainder 2(12×71=852,  854−852=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 5 answer: 875 / 23 = 38 remainder 1 (quotient 38)

Using the number cards 77, 22, 55, 33, 88 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=23\text{smallest divisor} = 23
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 875, putting the biggest digit first.
largest dividend=875\text{largest dividend} = 875
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷23=38 remainder 1(23×38=874,  875−874=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 6 answer: 974 / 23 = 42 remainder 8 (quotient 42)

Using the number cards 22, 77, 44, 99, 33 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=23\text{smallest divisor} = 23
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 974, putting the biggest digit first.
largest dividend=974\text{largest dividend} = 974
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷23=42 remainder 8(23×42=966,  974−966=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 7 answer: 976 / 13 = 75 remainder 1 (quotient 75)

Using the number cards 33, 66, 99, 11, 77 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=13\text{smallest divisor} = 13
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 976, putting the biggest digit first.
largest dividend=976\text{largest dividend} = 976
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷13=75 remainder 1(13×75=975,  976−975=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 8 answer: 976 / 12 = 81 remainder 4 (quotient 81)

Using the number cards 66, 11, 77, 22, 99 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=12\text{smallest divisor} = 12
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 976, putting the biggest digit first.
largest dividend=976\text{largest dividend} = 976
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷12=81 remainder 4(12×81=972,  976−972=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 9 answer: 865 / 23 = 37 remainder 14 (quotient 37)

Using the number cards 22, 55, 88, 33, 66 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=23\text{smallest divisor} = 23
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 865, putting the biggest digit first.
largest dividend=865\text{largest dividend} = 865
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷23=37 remainder 14(23×37=851,  865−851=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)

Review

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.

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.

Standards · min 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.
💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!
Variant 10 answer: 976 / 14 = 69 remainder 10 (quotient 69)

Using the number cards 99, 44, 11, 66, 77 each exactly once, form a (3-digit number) ÷\div (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

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.

Plan

#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.

Execute

#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.
smallest divisor=14\text{smallest divisor} = 14
Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.
#6 Guess and Check 4.NBT.A.2
From the remaining cards the largest 3-digit number is 976, putting the biggest digit first.
largest dividend=976\text{largest dividend} = 976
Place the biggest digit in the hundreds spot to make the number as large as it can be.
#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÷14=69 remainder 10(14×69=966,  976−966=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)

Review

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.

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.

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