← Place big digits in high places to maximize product · Build the Largest or Smallest Value from Digit Cards

Place big digits in high places to maximize product · 12 practice problems

5.NBT.A.35.NBT.B.7

Generated variants — 12

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

Variant 1 easy answer: 46.15

Use the four number cards 55, 11, 77, and 66, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 5, 1, 7, 6 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 5, 1, 7, 6, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 7 and 6, go there.
7.□×6.□7.\square \times 6.\square
7 and 6 are settled; 5 and 1 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 5 and 1 can go either way round.
7.5 \times 6.1$ or $7.1 \times 6.5
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
7.5×6.1=45.757.5 \times 6.1 = 45.75
45.75.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
7.1×6.5=46.157.1 \times 6.5 = 46.15
46.15.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
46.15>45.7546.15 > 45.75
The greatest product is 46.15.
Answer: 46.15
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 46.15 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 2 easy answer: 62.32

Use the four number cards 77, 22, 66, and 88, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 7, 2, 6, 8 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 7, 2, 6, 8, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 8 and 7, go there.
8.□×7.□8.\square \times 7.\square
8 and 7 are settled; 6 and 2 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 6 and 2 can go either way round.
8.6 \times 7.2$ or $8.2 \times 7.6
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
8.6×7.2=61.928.6 \times 7.2 = 61.92
61.92.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
8.2×7.6=62.328.2 \times 7.6 = 62.32
62.32.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
62.32>61.9262.32 > 61.92
The greatest product is 62.32.
Answer: 62.32
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 62.32 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 3 easy answer: 60.75

Use the four number cards 55, 88, 11, and 77, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 5, 8, 1, 7 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 5, 8, 1, 7, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 8 and 7, go there.
8.□×7.□8.\square \times 7.\square
8 and 7 are settled; 5 and 1 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 5 and 1 can go either way round.
8.5 \times 7.1$ or $8.1 \times 7.5
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
8.5×7.1=60.358.5 \times 7.1 = 60.35
60.35.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
8.1×7.5=60.758.1 \times 7.5 = 60.75
60.75.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
60.75>60.3560.75 > 60.35
The greatest product is 60.75.
Answer: 60.75
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 60.75 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 4 easy answer: 59.13

Use the four number cards 33, 77, 88, and 11, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 3, 7, 8, 1 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 3, 7, 8, 1, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 8 and 7, go there.
8.□×7.□8.\square \times 7.\square
8 and 7 are settled; 3 and 1 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 3 and 1 can go either way round.
8.3 \times 7.1$ or $8.1 \times 7.3
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
8.3×7.1=58.938.3 \times 7.1 = 58.93
58.93.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
8.1×7.3=59.138.1 \times 7.3 = 59.13
59.13.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
59.13>58.9359.13 > 58.93
The greatest product is 59.13.
Answer: 59.13
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 59.13 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 5 medium answer: 43.46

Use the four number cards 33, 55, 88, and 22, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 3, 5, 8, 2 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 3, 5, 8, 2, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 8 and 5, go there.
8.□×5.□8.\square \times 5.\square
8 and 5 are settled; 3 and 2 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 3 and 2 can go either way round.
8.3 \times 5.2$ or $8.2 \times 5.3
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
8.3×5.2=43.168.3 \times 5.2 = 43.16
43.16.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
8.2×5.3=43.468.2 \times 5.3 = 43.46
43.46.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
43.46>43.1643.46 > 43.16
The greatest product is 43.46.
Answer: 43.46
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 43.46 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 6 medium answer: 61.42

Use the four number cards 88, 33, 44, and 77, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 8, 3, 4, 7 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 8, 3, 4, 7, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 8 and 7, go there.
8.□×7.□8.\square \times 7.\square
8 and 7 are settled; 4 and 3 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 4 and 3 can go either way round.
8.4 \times 7.3$ or $8.3 \times 7.4
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
8.4×7.3=61.328.4 \times 7.3 = 61.32
61.32.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
8.3×7.4=61.428.3 \times 7.4 = 61.42
61.42.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
61.42>61.3261.42 > 61.32
The greatest product is 61.42.
Answer: 61.42
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 61.42 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 7 hard answer: 78.12

Use the four number cards 44, 88, 33, and 99, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 4, 8, 3, 9 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 4, 8, 3, 9, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 9 and 8, go there.
9.□×8.□9.\square \times 8.\square
9 and 8 are settled; 4 and 3 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 4 and 3 can go either way round.
9.4 \times 8.3$ or $9.3 \times 8.4
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
9.4×8.3=78.029.4 \times 8.3 = 78.02
78.02.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
9.3×8.4=78.129.3 \times 8.4 = 78.12
78.12.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
78.12>78.0278.12 > 78.02
The greatest product is 78.12.
Answer: 78.12
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 78.12 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 8 hard answer: 48.23

Use the four number cards 11, 55, 99, and 33, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 1, 5, 9, 3 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 1, 5, 9, 3, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 9 and 5, go there.
9.□×5.□9.\square \times 5.\square
9 and 5 are settled; 3 and 1 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 3 and 1 can go either way round.
9.3 \times 5.1$ or $9.1 \times 5.3
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
9.3×5.1=47.439.3 \times 5.1 = 47.43
47.43.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
9.1×5.3=48.239.1 \times 5.3 = 48.23
48.23.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
48.23>47.4348.23 > 47.43
The greatest product is 48.23.
Answer: 48.23
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 48.23 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 9 hard answer: 58.24

Use the four number cards 99, 11, 66, and 44, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 9, 1, 6, 4 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 9, 1, 6, 4, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 9 and 6, go there.
9.□×6.□9.\square \times 6.\square
9 and 6 are settled; 4 and 1 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 4 and 1 can go either way round.
9.4 \times 6.1$ or $9.1 \times 6.4
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
9.4×6.1=57.349.4 \times 6.1 = 57.34
57.34.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
9.1×6.4=58.249.1 \times 6.4 = 58.24
58.24.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
58.24>57.3458.24 > 57.34
The greatest product is 58.24.
Answer: 58.24
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 58.24 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 10 medium answer: 58.88

Use the four number cards 66, 44, 99, and 22, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 6, 4, 9, 2 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 6, 4, 9, 2, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 9 and 6, go there.
9.□×6.□9.\square \times 6.\square
9 and 6 are settled; 4 and 2 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 4 and 2 can go either way round.
9.4 \times 6.2$ or $9.2 \times 6.4
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
9.4×6.2=58.289.4 \times 6.2 = 58.28
58.28.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
9.2×6.4=58.889.2 \times 6.4 = 58.88
58.88.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
58.88>58.2858.88 > 58.28
The greatest product is 58.88.
Answer: 58.88
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 58.88 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 11 hard answer: 59.8

Use the four number cards 22, 66, 99, and 55, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 2, 6, 9, 5 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 2, 6, 9, 5, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 9 and 6, go there.
9.□×6.□9.\square \times 6.\square
9 and 6 are settled; 5 and 2 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 5 and 2 can go either way round.
9.5 \times 6.2$ or $9.2 \times 6.5
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
9.5×6.2=58.99.5 \times 6.2 = 58.9
58.9.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
9.2×6.5=59.89.2 \times 6.5 = 59.8
59.8.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
59.8>58.959.8 > 58.9
The greatest product is 59.8.
Answer: 59.8
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 59.8 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.
Variant 12 medium answer: 58.88

Use the four number cards 22, 99, 44, and 66, each exactly once, to build a multiplication of the form shown below.

0.0×0.0\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}

Find the greatest possible product.

Show solution
1 · Understandwhat's really being asked

The four cards 2, 9, 4, 6 fill two decimal numbers, one digit in each place. We want the arrangement whose product is largest.

Givens
  • The cards are 2, 9, 4, 6, used once each.
  • Each number has one digit before the point and one after.
Unknowns
  • The greatest possible product.
Constraints
  • Every card is used exactly once.
2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

A ones digit is worth ten times a tenths digit, so the two biggest cards belong in the ones places. That leaves only two ways to place the other pair -- few enough to try both.

3 · Execute5 carry out the plan

1See which place matters most

#5 Look for a Pattern 5.NBT.A.3
A digit in the ones place counts ten times what it would in the tenths place, so the two largest cards, 9 and 6, go there.
9.□×6.□9.\square \times 6.\square
9 and 6 are settled; 4 and 2 are not.

2List the two ways to place the leftover cards

#2 Make a Systematic List 5.NBT.A.3
The remaining 4 and 2 can go either way round.
9.4 \times 6.2$ or $9.2 \times 6.4
Only two candidates left to compare.

3Check the first arrangement

#6 Guess and Check 5.NBT.B.7
Multiply it out.
9.4×6.2=58.289.4 \times 6.2 = 58.28
58.28.

4Check the second arrangement

#6 Guess and Check 5.NBT.B.7
And the other way round.
9.2×6.4=58.889.2 \times 6.4 = 58.88
58.88.

5Pick the larger product

#6 Guess and Check 5.NBT.A.3
Compare the two results.
58.88>58.2858.88 > 58.28
The greatest product is 58.88.
Answer: 58.88
4 · Reviewdoes it hold up?

Trying all 4 factorial arrangements would give the same maximum, 58.88 -- the place-value argument just skipped most of them.

Another way: Pair the largest card with the smallest of the two leftovers: that keeps the two factors close in size, which is what makes a product large for a fixed digit sum.

Standardsmin grade 5
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Judging which place each card is worth most in, and comparing the two results.
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying each candidate arrangement out.
💡Takeaway. A ones digit is worth ten tenths digits, so the big cards go left -- and then there is almost nothing left to try.