← Largest difference uses max and min bounds · Pin Down a Number from Digit and Range Conditions

Largest difference uses max and min bounds · 12 practice problems

3.NBT.A.16.EE.B.84.NBT.B.4

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Generated variants — 12

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Variant 1 easy

When the amount Minchae saved is rounded to the nearest thousand won, it is $23000$ won. When the amount Junho saved is rounded to the nearest hundred won, it is $15000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 23000 won at the nearest thousand; Junho's round to 15000 won at the nearest hundred. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 23000 won (nearest thousand).
Junho's amount rounds to 15000 won (nearest hundred).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 22500 up rounds to 23000. The top of that range is 500 short of the next thousand -- one won more and it would round up instead.

23000 + 500 - 1 = 23499

At most 23499 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 50 below 15000: at that amount it rounds up to 15000, and anything less rounds down.

15000 - 50 = 14950

At least 14950 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

22500 \le M \le 23499,\quad 14950 \le J \le 15049

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

23499 - 14950 = 8549

The greatest possible difference is 8549 won.

Answer: 8549 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 23499 rounds to 23000 at the nearest thousand, and 14950 rounds to 15000 at the nearest hundred. One won either way would break it.

Another way: The gap between the rounded figures is 8000 won; the extra 549 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 2 easy

When the amount Minchae saved is rounded to the nearest hundred won, it is $35000$ won. When the amount Junho saved is rounded to the nearest thousand won, it is $28000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 35000 won at the nearest hundred; Junho's round to 28000 won at the nearest thousand. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 35000 won (nearest hundred).
Junho's amount rounds to 28000 won (nearest thousand).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 34950 up rounds to 35000. The top of that range is 50 short of the next hundred -- one won more and it would round up instead.

35000 + 50 - 1 = 35049

At most 35049 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 500 below 28000: at that amount it rounds up to 28000, and anything less rounds down.

28000 - 500 = 27500

At least 27500 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

34950 \le M \le 35049,\quad 27500 \le J \le 28499

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

35049 - 27500 = 7549

The greatest possible difference is 7549 won.

Answer: 7549 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 35049 rounds to 35000 at the nearest hundred, and 27500 rounds to 28000 at the nearest thousand. One won either way would break it.

Another way: The gap between the rounded figures is 7000 won; the extra 549 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 3 easy

When the amount Minchae saved is rounded to the nearest thousand won, it is $39000$ won. When the amount Junho saved is rounded to the nearest hundred won, it is $17000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 39000 won at the nearest thousand; Junho's round to 17000 won at the nearest hundred. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 39000 won (nearest thousand).
Junho's amount rounds to 17000 won (nearest hundred).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 38500 up rounds to 39000. The top of that range is 500 short of the next thousand -- one won more and it would round up instead.

39000 + 500 - 1 = 39499

At most 39499 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 50 below 17000: at that amount it rounds up to 17000, and anything less rounds down.

17000 - 50 = 16950

At least 16950 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

38500 \le M \le 39499,\quad 16950 \le J \le 17049

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

39499 - 16950 = 22549

The greatest possible difference is 22549 won.

Answer: 22549 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 39499 rounds to 39000 at the nearest thousand, and 16950 rounds to 17000 at the nearest hundred. One won either way would break it.

Another way: The gap between the rounded figures is 22000 won; the extra 549 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 4 easy

When the amount Minchae saved is rounded to the nearest hundred won, it is $42000$ won. When the amount Junho saved is rounded to the nearest thousand won, it is $31000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 42000 won at the nearest hundred; Junho's round to 31000 won at the nearest thousand. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 42000 won (nearest hundred).
Junho's amount rounds to 31000 won (nearest thousand).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 41950 up rounds to 42000. The top of that range is 50 short of the next hundred -- one won more and it would round up instead.

42000 + 50 - 1 = 42049

At most 42049 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 500 below 31000: at that amount it rounds up to 31000, and anything less rounds down.

31000 - 500 = 30500

At least 30500 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

41950 \le M \le 42049,\quad 30500 \le J \le 31499

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

42049 - 30500 = 11549

The greatest possible difference is 11549 won.

Answer: 11549 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 42049 rounds to 42000 at the nearest hundred, and 30500 rounds to 31000 at the nearest thousand. One won either way would break it.

Another way: The gap between the rounded figures is 11000 won; the extra 549 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 5 medium

When the amount Minchae saved is rounded to the nearest thousand won, it is $46000$ won. When the amount Junho saved is rounded to the nearest hundred won, it is $21000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 46000 won at the nearest thousand; Junho's round to 21000 won at the nearest hundred. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 46000 won (nearest thousand).
Junho's amount rounds to 21000 won (nearest hundred).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 45500 up rounds to 46000. The top of that range is 500 short of the next thousand -- one won more and it would round up instead.

46000 + 500 - 1 = 46499

At most 46499 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 50 below 21000: at that amount it rounds up to 21000, and anything less rounds down.

21000 - 50 = 20950

At least 20950 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

45500 \le M \le 46499,\quad 20950 \le J \le 21049

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

46499 - 20950 = 25549

The greatest possible difference is 25549 won.

Answer: 25549 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 46499 rounds to 46000 at the nearest thousand, and 20950 rounds to 21000 at the nearest hundred. One won either way would break it.

Another way: The gap between the rounded figures is 25000 won; the extra 549 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 6 medium

When the amount Minchae saved is rounded to the nearest ten thousand won, it is $50000$ won. When the amount Junho saved is rounded to the nearest thousand won, it is $28000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 50000 won at the nearest ten thousand; Junho's round to 28000 won at the nearest thousand. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 50000 won (nearest ten thousand).
Junho's amount rounds to 28000 won (nearest thousand).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 45000 up rounds to 50000. The top of that range is 5000 short of the next ten thousand -- one won more and it would round up instead.

50000 + 5000 - 1 = 54999

At most 54999 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 500 below 28000: at that amount it rounds up to 28000, and anything less rounds down.

28000 - 500 = 27500

At least 27500 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

45000 \le M \le 54999,\quad 27500 \le J \le 28499

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

54999 - 27500 = 27499

The greatest possible difference is 27499 won.

Answer: 27499 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 54999 rounds to 50000 at the nearest ten thousand, and 27500 rounds to 28000 at the nearest thousand. One won either way would break it.

Another way: The gap between the rounded figures is 22000 won; the extra 5499 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 7 medium

When the amount Minchae saved is rounded to the nearest thousand won, it is $58000$ won. When the amount Junho saved is rounded to the nearest hundred won, it is $26000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 58000 won at the nearest thousand; Junho's round to 26000 won at the nearest hundred. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 58000 won (nearest thousand).
Junho's amount rounds to 26000 won (nearest hundred).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 57500 up rounds to 58000. The top of that range is 500 short of the next thousand -- one won more and it would round up instead.

58000 + 500 - 1 = 58499

At most 58499 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 50 below 26000: at that amount it rounds up to 26000, and anything less rounds down.

26000 - 50 = 25950

At least 25950 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

57500 \le M \le 58499,\quad 25950 \le J \le 26049

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

58499 - 25950 = 32549

The greatest possible difference is 32549 won.

Answer: 32549 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 58499 rounds to 58000 at the nearest thousand, and 25950 rounds to 26000 at the nearest hundred. One won either way would break it.

Another way: The gap between the rounded figures is 32000 won; the extra 549 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 8 medium

When the amount Minchae saved is rounded to the nearest hundred won, it is $61000$ won. When the amount Junho saved is rounded to the nearest thousand won, it is $44000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 61000 won at the nearest hundred; Junho's round to 44000 won at the nearest thousand. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 61000 won (nearest hundred).
Junho's amount rounds to 44000 won (nearest thousand).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 60950 up rounds to 61000. The top of that range is 50 short of the next hundred -- one won more and it would round up instead.

61000 + 50 - 1 = 61049

At most 61049 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 500 below 44000: at that amount it rounds up to 44000, and anything less rounds down.

44000 - 500 = 43500

At least 43500 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

60950 \le M \le 61049,\quad 43500 \le J \le 44499

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

61049 - 43500 = 17549

The greatest possible difference is 17549 won.

Answer: 17549 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 61049 rounds to 61000 at the nearest hundred, and 43500 rounds to 44000 at the nearest thousand. One won either way would break it.

Another way: The gap between the rounded figures is 17000 won; the extra 549 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 9 hard

When the amount Minchae saved is rounded to the nearest hundred won, it is $67000$ won. When the amount Junho saved is rounded to the nearest ten thousand won, it is $40000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 67000 won at the nearest hundred; Junho's round to 40000 won at the nearest ten thousand. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 67000 won (nearest hundred).
Junho's amount rounds to 40000 won (nearest ten thousand).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 66950 up rounds to 67000. The top of that range is 50 short of the next hundred -- one won more and it would round up instead.

67000 + 50 - 1 = 67049

At most 67049 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 5000 below 40000: at that amount it rounds up to 40000, and anything less rounds down.

40000 - 5000 = 35000

At least 35000 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

66950 \le M \le 67049,\quad 35000 \le J \le 44999

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

67049 - 35000 = 32049

The greatest possible difference is 32049 won.

Answer: 32049 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 67049 rounds to 67000 at the nearest hundred, and 35000 rounds to 40000 at the nearest ten thousand. One won either way would break it.

Another way: The gap between the rounded figures is 27000 won; the extra 5049 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 10 hard

When the amount Minchae saved is rounded to the nearest hundred won, it is $74000$ won. When the amount Junho saved is rounded to the nearest thousand won, it is $33000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 74000 won at the nearest hundred; Junho's round to 33000 won at the nearest thousand. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 74000 won (nearest hundred).
Junho's amount rounds to 33000 won (nearest thousand).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 73950 up rounds to 74000. The top of that range is 50 short of the next hundred -- one won more and it would round up instead.

74000 + 50 - 1 = 74049

At most 74049 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 500 below 33000: at that amount it rounds up to 33000, and anything less rounds down.

33000 - 500 = 32500

At least 32500 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

73950 \le M \le 74049,\quad 32500 \le J \le 33499

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

74049 - 32500 = 41549

The greatest possible difference is 41549 won.

Answer: 41549 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 74049 rounds to 74000 at the nearest hundred, and 32500 rounds to 33000 at the nearest thousand. One won either way would break it.

Another way: The gap between the rounded figures is 41000 won; the extra 549 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 11 hard

When the amount Minchae saved is rounded to the nearest hundred won, it is $85000$ won. When the amount Junho saved is rounded to the nearest ten thousand won, it is $60000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 85000 won at the nearest hundred; Junho's round to 60000 won at the nearest ten thousand. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 85000 won (nearest hundred).
Junho's amount rounds to 60000 won (nearest ten thousand).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 84950 up rounds to 85000. The top of that range is 50 short of the next hundred -- one won more and it would round up instead.

85000 + 50 - 1 = 85049

At most 85049 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 5000 below 60000: at that amount it rounds up to 60000, and anything less rounds down.

60000 - 5000 = 55000

At least 55000 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

84950 \le M \le 85049,\quad 55000 \le J \le 64999

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

85049 - 55000 = 30049

The greatest possible difference is 30049 won.

Answer: 30049 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 85049 rounds to 85000 at the nearest hundred, and 55000 rounds to 60000 at the nearest ten thousand. One won either way would break it.

Another way: The gap between the rounded figures is 25000 won; the extra 5049 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Variant 12 hard

When the amount Minchae saved is rounded to the nearest ten thousand won, it is $90000$ won. When the amount Junho saved is rounded to the nearest thousand won, it is $52000$ won. Find the greatest possible difference between the two amounts they saved.

Your answer

Show solution

1 · Understandwhat's really being asked

Minchae's savings round to 90000 won at the nearest ten thousand; Junho's round to 52000 won at the nearest thousand. Neither exact amount is known, only its range. We want the biggest gap the two could have.

Givens

Minchae's amount rounds to 90000 won (nearest ten thousand).
Junho's amount rounds to 52000 won (nearest thousand).

Unknowns

The greatest possible difference between the two amounts.

Constraints

Rounding half up, so an amount exactly half a place above rounds up and is outside the range.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems#5 Look for a Pattern

A difference is largest when the first amount is as big as possible and the second as small as possible, so find each extreme separately and then subtract.

3 · Execute4 carry out the plan

1Largest amount Minchae could have

#6 Guess and Check 3.NBT.A.1

Anything from 85000 up rounds to 90000. The top of that range is 5000 short of the next ten thousand -- one won more and it would round up instead.

90000 + 5000 - 1 = 94999

At most 94999 won.

2Smallest amount Junho could have

#6 Guess and Check 3.NBT.A.1

The bottom of Junho's range is exactly 500 below 52000: at that amount it rounds up to 52000, and anything less rounds down.

52000 - 500 = 51500

At least 51500 won.

3Combine the extremes for the biggest gap

#7 Identify Subproblems 6.EE.B.8

Pair the biggest possible first amount with the smallest possible second one.

85000 \le M \le 94999,\quad 51500 \le J \le 52499

Extremes on opposite ends give the widest spread.

4Subtract to find the greatest difference

#5 Look for a Pattern 4.NBT.B.4

Take the smallest Junho from the largest Minchae.

94999 - 51500 = 43499

The greatest possible difference is 43499 won.

Answer: 43499 won

4 · Reviewdoes it hold up?

Check both bounds round as claimed: 94999 rounds to 90000 at the nearest ten thousand, and 51500 rounds to 52000 at the nearest thousand. One won either way would break it.

Another way: The gap between the rounded figures is 38000 won; the extra 5499 comes from stretching each amount to its own end of its range.

Standardsmin grade 6

3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reading each rounded figure back into a range of possible amounts.
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Writing each range as an inequality and picking its extreme.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Subtracting the two extremes.

💡Takeaway. A rounded number is a range, not a value. The biggest gap takes the top of one range and the bottom of the other.

Where next?

← Back to the problem: Largest difference uses max and min bounds