← Build the decimal closest to a target from cards · Get Closest to a Target Value

Build the decimal closest to a target from cards · 10 practice problems

5.NBT.B.74.NF.C.7

Generated variants — 10

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

Variant 1 easy answer: 23.96

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $25$ .

Number cards: $3$ , $9$ , $6$ , $2$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 3, 9, 6, 2 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 25.

Givens

The four cards are 3, 9, 6, 2.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 39.62.
The target value is 25.

Unknowns

The number, built from all four cards, that is closest to 25.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 25 means the whole-number part must be just below or just above 25. So we only need to check those few candidates and compare how far each is from 25.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 25, choose a tens digit so the whole part is just below or just above 25. The card pairs that bracket 25 are the ones to test.

\text{whole part} \approx 25

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 25 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 25. The two strongest candidates are 23.96 and 23.69.

23.96 \quad\text{and}\quad 23.69

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 25

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 25 by subtracting. The gaps are 1.04 and 1.31; the smaller gap, 1.04, belongs to 23.96.

|25-23.96|=1.04,\ \ |25-23.69|=1.31

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 23.96

4 · Reviewdoes it hold up?

Both candidates use the cards 3, 9, 6, 2 once each. 23.96 sits 1.04 from 25, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 25, which is again 23.96.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 25.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 25.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 25 as possible.

💡Takeaway. Closest to 25 means smallest gap - so just check the numbers right below and right above 25 and subtract!

Variant 2 easy answer: 32.47

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $30$ .

Number cards: $3$ , $2$ , $4$ , $7$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 3, 2, 4, 7 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 30.

Givens

The four cards are 3, 2, 4, 7.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 32.47.
The target value is 30.

Unknowns

The number, built from all four cards, that is closest to 30.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 30 means the whole-number part must be just below or just above 30. So we only need to check those few candidates and compare how far each is from 30.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 30, choose a tens digit so the whole part is just below or just above 30. The card pairs that bracket 30 are the ones to test.

\text{whole part} \approx 30

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 30 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 30. The two strongest candidates are 32.47 and 27.43.

32.47 \quad\text{and}\quad 27.43

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 30

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 30 by subtracting. The gaps are 2.47 and 2.57; the smaller gap, 2.47, belongs to 32.47.

|30-32.47|=2.47,\ \ |30-27.43|=2.57

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 32.47

4 · Reviewdoes it hold up?

Both candidates use the cards 3, 2, 4, 7 once each. 32.47 sits 2.47 from 30, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 30, which is again 32.47.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 30.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 30.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 30 as possible.

💡Takeaway. Closest to 30 means smallest gap - so just check the numbers right below and right above 30 and subtract!

Variant 3 easy answer: 29.54

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $35$ .

Number cards: $2$ , $5$ , $9$ , $4$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 2, 5, 9, 4 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 35.

Givens

The four cards are 2, 5, 9, 4.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 25.94.
The target value is 35.

Unknowns

The number, built from all four cards, that is closest to 35.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 35 means the whole-number part must be just below or just above 35. So we only need to check those few candidates and compare how far each is from 35.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 35, choose a tens digit so the whole part is just below or just above 35. The card pairs that bracket 35 are the ones to test.

\text{whole part} \approx 35

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 35 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 35. The two strongest candidates are 29.54 and 29.45.

29.54 \quad\text{and}\quad 29.45

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 35

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 35 by subtracting. The gaps are 5.46 and 5.55; the smaller gap, 5.46, belongs to 29.54.

|35-29.54|=5.46,\ \ |35-29.45|=5.55

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 29.54

4 · Reviewdoes it hold up?

Both candidates use the cards 2, 5, 9, 4 once each. 29.54 sits 5.46 from 35, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 35, which is again 29.54.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 35.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 35.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 35 as possible.

💡Takeaway. Closest to 35 means smallest gap - so just check the numbers right below and right above 35 and subtract!

Variant 4 medium answer: 51.68

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $40$ .

Number cards: $1$ , $5$ , $8$ , $6$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 1, 5, 8, 6 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 40.

Givens

The four cards are 1, 5, 8, 6.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 15.86.
The target value is 40.

Unknowns

The number, built from all four cards, that is closest to 40.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 40 means the whole-number part must be just below or just above 40. So we only need to check those few candidates and compare how far each is from 40.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 40, choose a tens digit so the whole part is just below or just above 40. The card pairs that bracket 40 are the ones to test.

\text{whole part} \approx 40

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 40 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 40. The two strongest candidates are 51.68 and 51.86.

51.68 \quad\text{and}\quad 51.86

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 40

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 40 by subtracting. The gaps are 11.68 and 11.86; the smaller gap, 11.68, belongs to 51.68.

|40-51.68|=11.68,\ \ |40-51.86|=11.86

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 51.68

4 · Reviewdoes it hold up?

Both candidates use the cards 1, 5, 8, 6 once each. 51.68 sits 11.68 from 40, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 40, which is again 51.68.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 40.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 40.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 40 as possible.

💡Takeaway. Closest to 40 means smallest gap - so just check the numbers right below and right above 40 and subtract!

Variant 5 medium answer: 38.71

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $45$ .

Number cards: $8$ , $1$ , $3$ , $7$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 8, 1, 3, 7 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 45.

Givens

The four cards are 8, 1, 3, 7.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 81.37.
The target value is 45.

Unknowns

The number, built from all four cards, that is closest to 45.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 45 means the whole-number part must be just below or just above 45. So we only need to check those few candidates and compare how far each is from 45.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 45, choose a tens digit so the whole part is just below or just above 45. The card pairs that bracket 45 are the ones to test.

\text{whole part} \approx 45

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 45 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 45. The two strongest candidates are 38.71 and 38.17.

38.71 \quad\text{and}\quad 38.17

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 45

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 45 by subtracting. The gaps are 6.29 and 6.83; the smaller gap, 6.29, belongs to 38.71.

|45-38.71|=6.29,\ \ |45-38.17|=6.83

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 38.71

4 · Reviewdoes it hold up?

Both candidates use the cards 8, 1, 3, 7 once each. 38.71 sits 6.29 from 45, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 45, which is again 38.71.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 45.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 45.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 45 as possible.

💡Takeaway. Closest to 45 means smallest gap - so just check the numbers right below and right above 45 and subtract!

Variant 6 medium answer: 49.21

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $50$ .

Number cards: $9$ , $2$ , $4$ , $1$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 9, 2, 4, 1 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 50.

Givens

The four cards are 9, 2, 4, 1.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 92.41.
The target value is 50.

Unknowns

The number, built from all four cards, that is closest to 50.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 50 means the whole-number part must be just below or just above 50. So we only need to check those few candidates and compare how far each is from 50.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 50, choose a tens digit so the whole part is just below or just above 50. The card pairs that bracket 50 are the ones to test.

\text{whole part} \approx 50

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 50 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 50. The two strongest candidates are 49.21 and 49.12.

49.21 \quad\text{and}\quad 49.12

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 50

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 50 by subtracting. The gaps are 0.79 and 0.88; the smaller gap, 0.79, belongs to 49.21.

|50-49.21|=0.79,\ \ |50-49.12|=0.88

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 49.21

4 · Reviewdoes it hold up?

Both candidates use the cards 9, 2, 4, 1 once each. 49.21 sits 0.79 from 50, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 50, which is again 49.21.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 50.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 50.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 50 as possible.

💡Takeaway. Closest to 50 means smallest gap - so just check the numbers right below and right above 50 and subtract!

Variant 7 medium answer: 49.62

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $55$ .

Number cards: $6$ , $4$ , $2$ , $9$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 6, 4, 2, 9 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 55.

Givens

The four cards are 6, 4, 2, 9.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 64.29.
The target value is 55.

Unknowns

The number, built from all four cards, that is closest to 55.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 55 means the whole-number part must be just below or just above 55. So we only need to check those few candidates and compare how far each is from 55.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 55, choose a tens digit so the whole part is just below or just above 55. The card pairs that bracket 55 are the ones to test.

\text{whole part} \approx 55

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 55 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 55. The two strongest candidates are 49.62 and 49.26.

49.62 \quad\text{and}\quad 49.26

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 55

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 55 by subtracting. The gaps are 5.38 and 5.74; the smaller gap, 5.38, belongs to 49.62.

|55-49.62|=5.38,\ \ |55-49.26|=5.74

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 49.62

4 · Reviewdoes it hold up?

Both candidates use the cards 6, 4, 2, 9 once each. 49.62 sits 5.38 from 55, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 55, which is again 49.62.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 55.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 55.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 55 as possible.

💡Takeaway. Closest to 55 means smallest gap - so just check the numbers right below and right above 55 and subtract!

Variant 8 hard answer: 63.78

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $60$ .

Number cards: $7$ , $3$ , $6$ , $8$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 7, 3, 6, 8 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 60.

Givens

The four cards are 7, 3, 6, 8.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 73.68.
The target value is 60.

Unknowns

The number, built from all four cards, that is closest to 60.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 60 means the whole-number part must be just below or just above 60. So we only need to check those few candidates and compare how far each is from 60.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 60, choose a tens digit so the whole part is just below or just above 60. The card pairs that bracket 60 are the ones to test.

\text{whole part} \approx 60

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 60 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 60. The two strongest candidates are 63.78 and 63.87.

63.78 \quad\text{and}\quad 63.87

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 60

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 60 by subtracting. The gaps are 3.78 and 3.87; the smaller gap, 3.78, belongs to 63.78.

|60-63.78|=3.78,\ \ |60-63.87|=3.87

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 63.78

4 · Reviewdoes it hold up?

Both candidates use the cards 7, 3, 6, 8 once each. 63.78 sits 3.78 from 60, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 60, which is again 63.78.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 60.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 60.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 60 as possible.

💡Takeaway. Closest to 60 means smallest gap - so just check the numbers right below and right above 60 and subtract!

Variant 9 hard answer: 58.41

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $65$ .

Number cards: $4$ , $8$ , $5$ , $1$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 4, 8, 5, 1 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 65.

Givens

The four cards are 4, 8, 5, 1.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 48.51.
The target value is 65.

Unknowns

The number, built from all four cards, that is closest to 65.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 65 means the whole-number part must be just below or just above 65. So we only need to check those few candidates and compare how far each is from 65.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 65, choose a tens digit so the whole part is just below or just above 65. The card pairs that bracket 65 are the ones to test.

\text{whole part} \approx 65

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 65 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 65. The two strongest candidates are 58.41 and 58.14.

58.41 \quad\text{and}\quad 58.14

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 65

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 65 by subtracting. The gaps are 6.59 and 6.86; the smaller gap, 6.59, belongs to 58.41.

|65-58.41|=6.59,\ \ |65-58.14|=6.86

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 58.41

4 · Reviewdoes it hold up?

Both candidates use the cards 4, 8, 5, 1 once each. 58.41 sits 6.59 from 65, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 65, which is again 58.41.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 65.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 65.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 65 as possible.

💡Takeaway. Closest to 65 means smallest gap - so just check the numbers right below and right above 65 and subtract!

Variant 10 hard answer: 71.58

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $70$ .

Number cards: $5$ , $7$ , $1$ , $8$

Show solution

1 · Understandwhat's really being asked

Use each of the cards 5, 7, 1, 8 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 70.

Givens

The four cards are 5, 7, 1, 8.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 57.18.
The target value is 70.

Unknowns

The number, built from all four cards, that is closest to 70.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 70 means the whole-number part must be just below or just above 70. So we only need to check those few candidates and compare how far each is from 70.

3 · Execute3 carry out the plan

1Decide the two whole-number digits

#2 Make a Systematic List 5.NBT.A.1

To land near 70, choose a tens digit so the whole part is just below or just above 70. The card pairs that bracket 70 are the ones to test.

\text{whole part} \approx 70

Reading the tens place tells you the rough size, just like ordering whole numbers.

2Make each candidate as close to 70 as possible

#6 Guess and Check 4.NF.C.7

For each whole part, arrange the remaining two cards to push the value toward 70. The two strongest candidates are 71.58 and 71.85.

71.58 \quad\text{and}\quad 71.85

Pick the digit arrangement that nudges each candidate toward the target.

3Compare the distances to 70

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 70 by subtracting. The gaps are 1.58 and 1.85; the smaller gap, 1.58, belongs to 71.58.

|70-71.58|=1.58,\ \ |70-71.85|=1.85

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 71.58

4 · Reviewdoes it hold up?

Both candidates use the cards 5, 7, 1, 8 once each. 71.58 sits 1.58 from 70, the smallest gap of any arrangement, so it is the closest.

Another way: Make a Systematic List: list every closest-of-its-block candidate and read off the one with the smallest gap to 70, which is again 71.58.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 70.
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (0) to make the whole part near 70.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 70 as possible.

💡Takeaway. Closest to 70 means smallest gap - so just check the numbers right below and right above 70 and subtract!