← 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

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Generated variants — 12

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

Variant 1 easy

Use the four number cards $5$ , $1$ , $7$ , and $6$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 45.75

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

Use the four number cards $7$ , $2$ , $6$ , and $8$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 61.92

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

Use the four number cards $5$ , $8$ , $1$ , and $7$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 60.35

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

Use the four number cards $3$ , $7$ , $8$ , and $1$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 58.93

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

Use the four number cards $3$ , $5$ , $8$ , and $2$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 43.16

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

Use the four number cards $8$ , $3$ , $4$ , and $7$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 61.32

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

Use the four number cards $4$ , $8$ , $3$ , and $9$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 78.02

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

Use the four number cards $1$ , $5$ , $9$ , and $3$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 47.43

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

Use the four number cards $9$ , $1$ , $6$ , and $4$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 57.34

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

Use the four number cards $6$ , $4$ , $9$ , and $2$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 58.28

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

Use the four number cards $2$ , $6$ , $9$ , and $5$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 58.9

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

Use the four number cards $2$ , $9$ , $4$ , and $6$ , each exactly once, to build a multiplication of the form shown below.

$\boxed{\phantom{0}}.\boxed{\phantom{0}} \times \boxed{\phantom{0}}.\boxed{\phantom{0}}$

Find the greatest possible product.

Your answer

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.\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 \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 \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 > 58.28

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.

Where next?

← Back to the problem: Place big digits in high places to maximize product