← When the largest digit is wanted, only the top of the window matters · Inequality Range Membership

When the largest digit is wanted, only the top of the window matters · 12 practice problems

6.NS.B.36.EE.B.85.NBT.A.4

Generated variants — 12

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

Variant 1 easy answer: 1

Rounded to the nearest whole number, the quotient below is $2$ . Find the largest digit that $\blacksquare$ can be.

$7.\blacksquare 7 \div 2.9$

Show solution

1 · Understandwhat's really being asked

The quotient 7 point box 7, divided by 2.9, rounds to 2. We want the largest digit the box can hold.

Givens

The dividend is 7 point box 7.
The divisor is 2.9.
The quotient rounds to 2.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 1.5 up to but not including 2.5 rounds to 2.

1.5 \le \text{quotient} < 2.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 2.5 \times 2.9 = 7.25

The dividend has to stay under 7.25.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 1 the dividend is 7.17, still under; with 2 it is 7.27, over.

7.17 < 7.25 < 7.27

1 is the last one that fits.

Answer: 1

4 · Reviewdoes it hold up?

Checking the quotient itself: 7.17 over 2.9 is about 2.472, which rounds to 2.

Another way: Trying all ten digits also finds 1, and shows that 0, 1 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 2 easy answer: 4

Rounded to the nearest whole number, the quotient below is $2$ . Find the largest digit that $\blacksquare$ can be.

$4.\blacksquare 8 \div 1.8$

Show solution

1 · Understandwhat's really being asked

The quotient 4 point box 8, divided by 1.8, rounds to 2. We want the largest digit the box can hold.

Givens

The dividend is 4 point box 8.
The divisor is 1.8.
The quotient rounds to 2.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 1.5 up to but not including 2.5 rounds to 2.

1.5 \le \text{quotient} < 2.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 2.5 \times 1.8 = 4.5

The dividend has to stay under 4.5.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 4 the dividend is 4.48, still under; with 5 it is 4.58, over.

4.48 < 4.5 < 4.58

4 is the last one that fits.

Answer: 4

4 · Reviewdoes it hold up?

Checking the quotient itself: 4.48 over 1.8 is about 2.489, which rounds to 2.

Another way: Trying all ten digits also finds 4, and shows that 0, 1, 2, 3, 4 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 3 easy answer: 5

Rounded to the nearest whole number, the quotient below is $4$ . Find the largest digit that $\blacksquare$ can be.

$7.\blacksquare 8 \div 1.7$

Show solution

1 · Understandwhat's really being asked

The quotient 7 point box 8, divided by 1.7, rounds to 4. We want the largest digit the box can hold.

Givens

The dividend is 7 point box 8.
The divisor is 1.7.
The quotient rounds to 4.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 3.5 up to but not including 4.5 rounds to 4.

3.5 \le \text{quotient} < 4.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 4.5 \times 1.7 = 7.65

The dividend has to stay under 7.65.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 5 the dividend is 7.58, still under; with 6 it is 7.68, over.

7.58 < 7.65 < 7.68

5 is the last one that fits.

Answer: 5

4 · Reviewdoes it hold up?

Checking the quotient itself: 7.58 over 1.7 is about 4.459, which rounds to 4.

Another way: Trying all ten digits also finds 5, and shows that 0, 1, 2, 3, 4, 5 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 4 easy answer: 4

Rounded to the nearest whole number, the quotient below is $4$ . Find the largest digit that $\blacksquare$ can be.

$8.\blacksquare 8 \div 1.9$

Show solution

1 · Understandwhat's really being asked

The quotient 8 point box 8, divided by 1.9, rounds to 4. We want the largest digit the box can hold.

Givens

The dividend is 8 point box 8.
The divisor is 1.9.
The quotient rounds to 4.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 3.5 up to but not including 4.5 rounds to 4.

3.5 \le \text{quotient} < 4.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 4.5 \times 1.9 = 8.55

The dividend has to stay under 8.55.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 4 the dividend is 8.48, still under; with 5 it is 8.58, over.

8.48 < 8.55 < 8.58

4 is the last one that fits.

Answer: 4

4 · Reviewdoes it hold up?

Checking the quotient itself: 8.48 over 1.9 is about 4.463, which rounds to 4.

Another way: Trying all ten digits also finds 4, and shows that 0, 1, 2, 3, 4 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 5 medium answer: 0

Rounded to the nearest whole number, the quotient below is $5$ . Find the largest digit that $\blacksquare$ can be.

$7.\blacksquare 9 \div 1.3$

Show solution

1 · Understandwhat's really being asked

The quotient 7 point box 9, divided by 1.3, rounds to 5. We want the largest digit the box can hold.

Givens

The dividend is 7 point box 9.
The divisor is 1.3.
The quotient rounds to 5.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 4.5 up to but not including 5.5 rounds to 5.

4.5 \le \text{quotient} < 5.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 5.5 \times 1.3 = 7.15

The dividend has to stay under 7.15.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 0 the dividend is 7.09, still under; with 1 it is 7.19, over.

7.09 < 7.15 < 7.19

0 is the last one that fits.

Answer: 0

4 · Reviewdoes it hold up?

Checking the quotient itself: 7.09 over 1.3 is about 5.454, which rounds to 5.

Another way: Trying all ten digits also finds 0, and shows that 0 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 6 medium answer: 5

Rounded to the nearest whole number, the quotient below is $4$ . Find the largest digit that $\blacksquare$ can be.

$7.\blacksquare 9 \div 1.7$

Show solution

1 · Understandwhat's really being asked

The quotient 7 point box 9, divided by 1.7, rounds to 4. We want the largest digit the box can hold.

Givens

The dividend is 7 point box 9.
The divisor is 1.7.
The quotient rounds to 4.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 3.5 up to but not including 4.5 rounds to 4.

3.5 \le \text{quotient} < 4.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 4.5 \times 1.7 = 7.65

The dividend has to stay under 7.65.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 5 the dividend is 7.59, still under; with 6 it is 7.69, over.

7.59 < 7.65 < 7.69

5 is the last one that fits.

Answer: 5

4 · Reviewdoes it hold up?

Checking the quotient itself: 7.59 over 1.7 is about 4.465, which rounds to 4.

Another way: Trying all ten digits also finds 5, and shows that 0, 1, 2, 3, 4, 5 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 7 medium answer: 1

Rounded to the nearest whole number, the quotient below is $3$ . Find the largest digit that $\blacksquare$ can be.

$11.\blacksquare 5 \div 3.2$

Show solution

1 · Understandwhat's really being asked

The quotient 11 point box 5, divided by 3.2, rounds to 3. We want the largest digit the box can hold.

Givens

The dividend is 11 point box 5.
The divisor is 3.2.
The quotient rounds to 3.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 2.5 up to but not including 3.5 rounds to 3.

2.5 \le \text{quotient} < 3.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 3.5 \times 3.2 = 11.2

The dividend has to stay under 11.2.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 1 the dividend is 11.15, still under; with 2 it is 11.25, over.

11.15 < 11.2 < 11.25

1 is the last one that fits.

Answer: 1

4 · Reviewdoes it hold up?

Checking the quotient itself: 11.15 over 3.2 is about 3.484, which rounds to 3.

Another way: Trying all ten digits also finds 1, and shows that 0, 1 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 8 medium answer: 6

Rounded to the nearest whole number, the quotient below is $7$ . Find the largest digit that $\blacksquare$ can be.

$12.\blacksquare 8 \div 1.7$

Show solution

1 · Understandwhat's really being asked

The quotient 12 point box 8, divided by 1.7, rounds to 7. We want the largest digit the box can hold.

Givens

The dividend is 12 point box 8.
The divisor is 1.7.
The quotient rounds to 7.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 6.5 up to but not including 7.5 rounds to 7.

6.5 \le \text{quotient} < 7.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 7.5 \times 1.7 = 12.75

The dividend has to stay under 12.75.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 6 the dividend is 12.68, still under; with 7 it is 12.78, over.

12.68 < 12.75 < 12.78

6 is the last one that fits.

Answer: 6

4 · Reviewdoes it hold up?

Checking the quotient itself: 12.68 over 1.7 is about 7.459, which rounds to 7.

Another way: Trying all ten digits also finds 6, and shows that 0, 1, 2, 3, 4, 5, 6 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 9 hard answer: 6

Rounded to the nearest whole number, the quotient below is $5$ . Find the largest digit that $\blacksquare$ can be.

$13.\blacksquare 9 \div 2.5$

Show solution

1 · Understandwhat's really being asked

The quotient 13 point box 9, divided by 2.5, rounds to 5. We want the largest digit the box can hold.

Givens

The dividend is 13 point box 9.
The divisor is 2.5.
The quotient rounds to 5.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 4.5 up to but not including 5.5 rounds to 5.

4.5 \le \text{quotient} < 5.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 5.5 \times 2.5 = 13.75

The dividend has to stay under 13.75.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 6 the dividend is 13.69, still under; with 7 it is 13.79, over.

13.69 < 13.75 < 13.79

6 is the last one that fits.

Answer: 6

4 · Reviewdoes it hold up?

Checking the quotient itself: 13.69 over 2.5 is about 5.476, which rounds to 5.

Another way: Trying all ten digits also finds 6, and shows that 0, 1, 2, 3, 4, 5, 6 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 10 hard answer: 4

Rounded to the nearest whole number, the quotient below is $7$ . Find the largest digit that $\blacksquare$ can be.

$13.\blacksquare 5 \div 1.8$

Show solution

1 · Understandwhat's really being asked

The quotient 13 point box 5, divided by 1.8, rounds to 7. We want the largest digit the box can hold.

Givens

The dividend is 13 point box 5.
The divisor is 1.8.
The quotient rounds to 7.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 6.5 up to but not including 7.5 rounds to 7.

6.5 \le \text{quotient} < 7.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 7.5 \times 1.8 = 13.5

The dividend has to stay under 13.5.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 4 the dividend is 13.45, still under; with 5 it is 13.55, over.

13.45 < 13.5 < 13.55

4 is the last one that fits.

Answer: 4

4 · Reviewdoes it hold up?

Checking the quotient itself: 13.45 over 1.8 is about 7.472, which rounds to 7.

Another way: Trying all ten digits also finds 4, and shows that 0, 1, 2, 3, 4 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 11 hard answer: 7

Rounded to the nearest whole number, the quotient below is $5$ . Find the largest digit that $\blacksquare$ can be.

$14.\blacksquare 8 \div 2.7$

Show solution

1 · Understandwhat's really being asked

The quotient 14 point box 8, divided by 2.7, rounds to 5. We want the largest digit the box can hold.

Givens

The dividend is 14 point box 8.
The divisor is 2.7.
The quotient rounds to 5.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 4.5 up to but not including 5.5 rounds to 5.

4.5 \le \text{quotient} < 5.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 5.5 \times 2.7 = 14.85

The dividend has to stay under 14.85.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 7 the dividend is 14.78, still under; with 8 it is 14.88, over.

14.78 < 14.85 < 14.88

7 is the last one that fits.

Answer: 7

4 · Reviewdoes it hold up?

Checking the quotient itself: 14.78 over 2.7 is about 5.474, which rounds to 5.

Another way: Trying all ten digits also finds 7, and shows that 0, 1, 2, 3, 4, 5, 6, 7 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.

Variant 12 hard answer: 3

Rounded to the nearest whole number, the quotient below is $4$ . Find the largest digit that $\blacksquare$ can be.

$14.\blacksquare 5 \div 3.2$

Show solution

1 · Understandwhat's really being asked

The quotient 14 point box 5, divided by 3.2, rounds to 4. We want the largest digit the box can hold.

Givens

The dividend is 14 point box 5.
The divisor is 3.2.
The quotient rounds to 4.

Unknowns

The largest digit that fits in the box.

Constraints

The box holds a single digit, 0 to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

A bigger digit makes a bigger dividend and a bigger quotient, so only the top of the rounding window can stop it. Cap the dividend there and read the digit off, instead of trying all ten.

3 · Execute3 carry out the plan

1Say what rounding to that value allows

#11 Work Backwards 5.NBT.A.4

Any quotient from 3.5 up to but not including 4.5 rounds to 4.

3.5 \le \text{quotient} < 4.5

Half a unit each side.

2Use only the top of that window

#3 Eliminate Possibilities 6.EE.B.8

The larger the digit, the larger the quotient, so the lower bound cannot be what stops it.

\text{dividend} < 4.5 \times 3.2 = 14.4

The dividend has to stay under 14.4.

3Try the digit that cap allows

#8 Analyze the Units 6.NS.B.3

With 3 the dividend is 14.35, still under; with 4 it is 14.45, over.

14.35 < 14.4 < 14.45

3 is the last one that fits.

Answer: 3

4 · Reviewdoes it hold up?

Checking the quotient itself: 14.35 over 3.2 is about 4.484, which rounds to 4.

Another way: Trying all ten digits also finds 3, and shows that 0, 1, 2, 3 all work -- the largest of them being the answer.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing the decimal dividend by the decimal divisor.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Turning the rounding condition into a bound on the dividend.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. If you want the biggest value that works, find what stops it from being bigger. That is the only end of the range you need.