← Dividing by a whole number multiplies by its reciprocal · Division as the Inverse of Multiplication

Dividing by a whole number multiplies by its reciprocal · 12 practice problems

5.NF.B.76.NS.A.1

Generated variants — 12

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

Variant 1 easy answer:

\frac{9}{10}

$2$ bottles each holding $2\frac{1}{4}\ \text{L}$ of juice are bought and shared equally among $5$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

2 bottles hold 2 and 1/4 litres each. All the juice is split evenly between 5 people, and we want one person's share.

Givens

There are 2 bottles.
Each bottle holds 2 and 1/4 litres.
5 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 5 equal shares. Splitting each bottle separately gives the same answer through 2 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

2 wholes are 8 4ths, plus 1 more.

2\frac{1}{4} = \frac{9}{4}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

2 bottles of the same size.

\frac{9}{4} \times 2 = 4\frac{1}{2}\ \text{L}

There are 4 and 1/2 litres in all.

3Divide the total among 5

#8 Analyze the Units 5.NF.B.7

Dividing by 5 is multiplying by 1 over 5: the same amount of juice cut into 5 equal parts.

4\frac{1}{2} \div 5 = 4\frac{1}{2} \times \frac{1}{5} = \frac{9}{10}

One person gets 9/10 of a litre.

Answer:

\frac{9}{10}

4 · Reviewdoes it hold up?

5 shares of 9/10 litres come back to 4 and 1/2 litres, which is the 2 bottles. And one share is smaller than one bottle, as it must be with 5 people and only 2 bottles.

Another way: Splitting each bottle first gives 9/20 litres per bottle per person; 2 of those add to the same 9/10.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 2 easy answer:

\frac{24}{35}

$3$ bottles each holding $1\frac{3}{5}\ \text{L}$ of juice are bought and shared equally among $7$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

3 bottles hold 1 and 3/5 litres each. All the juice is split evenly between 7 people, and we want one person's share.

Givens

There are 3 bottles.
Each bottle holds 1 and 3/5 litres.
7 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 7 equal shares. Splitting each bottle separately gives the same answer through 3 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

1 wholes are 5 5ths, plus 3 more.

1\frac{3}{5} = \frac{8}{5}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

3 bottles of the same size.

\frac{8}{5} \times 3 = 4\frac{4}{5}\ \text{L}

There are 4 and 4/5 litres in all.

3Divide the total among 7

#8 Analyze the Units 5.NF.B.7

Dividing by 7 is multiplying by 1 over 7: the same amount of juice cut into 7 equal parts.

4\frac{4}{5} \div 7 = 4\frac{4}{5} \times \frac{1}{7} = \frac{24}{35}

One person gets 24/35 of a litre.

Answer:

\frac{24}{35}

4 · Reviewdoes it hold up?

7 shares of 24/35 litres come back to 4 and 4/5 litres, which is the 3 bottles. And one share is smaller than one bottle, as it must be with 7 people and only 3 bottles.

Another way: Splitting each bottle first gives 8/35 litres per bottle per person; 3 of those add to the same 24/35.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 3 easy answer:

1\frac{1}{14}

$2$ bottles each holding $3\frac{3}{4}\ \text{L}$ of juice are bought and shared equally among $7$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

2 bottles hold 3 and 3/4 litres each. All the juice is split evenly between 7 people, and we want one person's share.

Givens

There are 2 bottles.
Each bottle holds 3 and 3/4 litres.
7 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 7 equal shares. Splitting each bottle separately gives the same answer through 2 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

3 wholes are 12 4ths, plus 3 more.

3\frac{3}{4} = \frac{15}{4}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

2 bottles of the same size.

\frac{15}{4} \times 2 = 7\frac{1}{2}\ \text{L}

There are 7 and 1/2 litres in all.

3Divide the total among 7

#8 Analyze the Units 5.NF.B.7

Dividing by 7 is multiplying by 1 over 7: the same amount of juice cut into 7 equal parts.

7\frac{1}{2} \div 7 = 7\frac{1}{2} \times \frac{1}{7} = 1\frac{1}{14}

One person gets 1 and 1/14 of a litre.

Answer:

1\frac{1}{14}

4 · Reviewdoes it hold up?

7 shares of 1 and 1/14 litres come back to 7 and 1/2 litres, which is the 2 bottles. And one share is smaller than one bottle, as it must be with 7 people and only 2 bottles.

Another way: Splitting each bottle first gives 15/28 litres per bottle per person; 2 of those add to the same 1 and 1/14.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 4 easy answer:

\frac{9}{10}

$3$ bottles each holding $2\frac{2}{5}\ \text{L}$ of juice are bought and shared equally among $8$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

3 bottles hold 2 and 2/5 litres each. All the juice is split evenly between 8 people, and we want one person's share.

Givens

There are 3 bottles.
Each bottle holds 2 and 2/5 litres.
8 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 8 equal shares. Splitting each bottle separately gives the same answer through 3 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

2 wholes are 10 5ths, plus 2 more.

2\frac{2}{5} = \frac{12}{5}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

3 bottles of the same size.

\frac{12}{5} \times 3 = 7\frac{1}{5}\ \text{L}

There are 7 and 1/5 litres in all.

3Divide the total among 8

#8 Analyze the Units 5.NF.B.7

Dividing by 8 is multiplying by 1 over 8: the same amount of juice cut into 8 equal parts.

7\frac{1}{5} \div 8 = 7\frac{1}{5} \times \frac{1}{8} = \frac{9}{10}

One person gets 9/10 of a litre.

Answer:

\frac{9}{10}

4 · Reviewdoes it hold up?

8 shares of 9/10 litres come back to 7 and 1/5 litres, which is the 3 bottles. And one share is smaller than one bottle, as it must be with 8 people and only 3 bottles.

Another way: Splitting each bottle first gives 3/10 litres per bottle per person; 3 of those add to the same 9/10.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 5 medium answer:

1\frac{7}{48}

$5$ bottles each holding $1\frac{5}{6}\ \text{L}$ of juice are bought and shared equally among $8$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

5 bottles hold 1 and 5/6 litres each. All the juice is split evenly between 8 people, and we want one person's share.

Givens

There are 5 bottles.
Each bottle holds 1 and 5/6 litres.
8 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 8 equal shares. Splitting each bottle separately gives the same answer through 5 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

1 wholes are 6 6ths, plus 5 more.

1\frac{5}{6} = \frac{11}{6}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

5 bottles of the same size.

\frac{11}{6} \times 5 = 9\frac{1}{6}\ \text{L}

There are 9 and 1/6 litres in all.

3Divide the total among 8

#8 Analyze the Units 5.NF.B.7

Dividing by 8 is multiplying by 1 over 8: the same amount of juice cut into 8 equal parts.

9\frac{1}{6} \div 8 = 9\frac{1}{6} \times \frac{1}{8} = 1\frac{7}{48}

One person gets 1 and 7/48 of a litre.

Answer:

1\frac{7}{48}

4 · Reviewdoes it hold up?

8 shares of 1 and 7/48 litres come back to 9 and 1/6 litres, which is the 5 bottles. And one share is smaller than one bottle, as it must be with 8 people and only 5 bottles.

Another way: Splitting each bottle first gives 11/48 litres per bottle per person; 5 of those add to the same 1 and 7/48.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 6 medium answer:

\frac{16}{27}

$4$ bottles each holding $1\frac{1}{3}\ \text{L}$ of juice are bought and shared equally among $9$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

4 bottles hold 1 and 1/3 litres each. All the juice is split evenly between 9 people, and we want one person's share.

Givens

There are 4 bottles.
Each bottle holds 1 and 1/3 litres.
9 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 9 equal shares. Splitting each bottle separately gives the same answer through 4 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

1 wholes are 3 3ths, plus 1 more.

1\frac{1}{3} = \frac{4}{3}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

4 bottles of the same size.

\frac{4}{3} \times 4 = 5\frac{1}{3}\ \text{L}

There are 5 and 1/3 litres in all.

3Divide the total among 9

#8 Analyze the Units 5.NF.B.7

Dividing by 9 is multiplying by 1 over 9: the same amount of juice cut into 9 equal parts.

5\frac{1}{3} \div 9 = 5\frac{1}{3} \times \frac{1}{9} = \frac{16}{27}

One person gets 16/27 of a litre.

Answer:

\frac{16}{27}

4 · Reviewdoes it hold up?

9 shares of 16/27 litres come back to 5 and 1/3 litres, which is the 4 bottles. And one share is smaller than one bottle, as it must be with 9 people and only 4 bottles.

Another way: Splitting each bottle first gives 4/27 litres per bottle per person; 4 of those add to the same 16/27.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 7 medium answer:

1\frac{1}{18}

$4$ bottles each holding $2\frac{3}{8}\ \text{L}$ of juice are bought and shared equally among $9$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

4 bottles hold 2 and 3/8 litres each. All the juice is split evenly between 9 people, and we want one person's share.

Givens

There are 4 bottles.
Each bottle holds 2 and 3/8 litres.
9 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 9 equal shares. Splitting each bottle separately gives the same answer through 4 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

2 wholes are 16 8ths, plus 3 more.

2\frac{3}{8} = \frac{19}{8}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

4 bottles of the same size.

\frac{19}{8} \times 4 = 9\frac{1}{2}\ \text{L}

There are 9 and 1/2 litres in all.

3Divide the total among 9

#8 Analyze the Units 5.NF.B.7

Dividing by 9 is multiplying by 1 over 9: the same amount of juice cut into 9 equal parts.

9\frac{1}{2} \div 9 = 9\frac{1}{2} \times \frac{1}{9} = 1\frac{1}{18}

One person gets 1 and 1/18 of a litre.

Answer:

1\frac{1}{18}

4 · Reviewdoes it hold up?

9 shares of 1 and 1/18 litres come back to 9 and 1/2 litres, which is the 4 bottles. And one share is smaller than one bottle, as it must be with 9 people and only 4 bottles.

Another way: Splitting each bottle first gives 19/72 litres per bottle per person; 4 of those add to the same 1 and 1/18.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 8 medium answer:

\frac{5}{6}

$5$ bottles each holding $1\frac{1}{2}\ \text{L}$ of juice are bought and shared equally among $9$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

5 bottles hold 1 and 1/2 litres each. All the juice is split evenly between 9 people, and we want one person's share.

Givens

There are 5 bottles.
Each bottle holds 1 and 1/2 litres.
9 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 9 equal shares. Splitting each bottle separately gives the same answer through 5 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

1 wholes are 2 2ths, plus 1 more.

1\frac{1}{2} = \frac{3}{2}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

5 bottles of the same size.

\frac{3}{2} \times 5 = 7\frac{1}{2}\ \text{L}

There are 7 and 1/2 litres in all.

3Divide the total among 9

#8 Analyze the Units 5.NF.B.7

Dividing by 9 is multiplying by 1 over 9: the same amount of juice cut into 9 equal parts.

7\frac{1}{2} \div 9 = 7\frac{1}{2} \times \frac{1}{9} = \frac{5}{6}

One person gets 5/6 of a litre.

Answer:

\frac{5}{6}

4 · Reviewdoes it hold up?

9 shares of 5/6 litres come back to 7 and 1/2 litres, which is the 5 bottles. And one share is smaller than one bottle, as it must be with 9 people and only 5 bottles.

Another way: Splitting each bottle first gives 1/6 litres per bottle per person; 5 of those add to the same 5/6.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 9 hard answer:

1\frac{1}{27}

$2$ bottles each holding $4\frac{2}{3}\ \text{L}$ of juice are bought and shared equally among $9$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

2 bottles hold 4 and 2/3 litres each. All the juice is split evenly between 9 people, and we want one person's share.

Givens

There are 2 bottles.
Each bottle holds 4 and 2/3 litres.
9 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 9 equal shares. Splitting each bottle separately gives the same answer through 2 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

4 wholes are 12 3ths, plus 2 more.

4\frac{2}{3} = \frac{14}{3}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

2 bottles of the same size.

\frac{14}{3} \times 2 = 9\frac{1}{3}\ \text{L}

There are 9 and 1/3 litres in all.

3Divide the total among 9

#8 Analyze the Units 5.NF.B.7

Dividing by 9 is multiplying by 1 over 9: the same amount of juice cut into 9 equal parts.

9\frac{1}{3} \div 9 = 9\frac{1}{3} \times \frac{1}{9} = 1\frac{1}{27}

One person gets 1 and 1/27 of a litre.

Answer:

1\frac{1}{27}

4 · Reviewdoes it hold up?

9 shares of 1 and 1/27 litres come back to 9 and 1/3 litres, which is the 2 bottles. And one share is smaller than one bottle, as it must be with 9 people and only 2 bottles.

Another way: Splitting each bottle first gives 14/27 litres per bottle per person; 2 of those add to the same 1 and 1/27.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 10 hard answer:

\frac{42}{55}

$6$ bottles each holding $1\frac{2}{5}\ \text{L}$ of juice are bought and shared equally among $11$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

6 bottles hold 1 and 2/5 litres each. All the juice is split evenly between 11 people, and we want one person's share.

Givens

There are 6 bottles.
Each bottle holds 1 and 2/5 litres.
11 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 11 equal shares. Splitting each bottle separately gives the same answer through 6 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

1 wholes are 5 5ths, plus 2 more.

1\frac{2}{5} = \frac{7}{5}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

6 bottles of the same size.

\frac{7}{5} \times 6 = 8\frac{2}{5}\ \text{L}

There are 8 and 2/5 litres in all.

3Divide the total among 11

#8 Analyze the Units 5.NF.B.7

Dividing by 11 is multiplying by 1 over 11: the same amount of juice cut into 11 equal parts.

8\frac{2}{5} \div 11 = 8\frac{2}{5} \times \frac{1}{11} = \frac{42}{55}

One person gets 42/55 of a litre.

Answer:

\frac{42}{55}

4 · Reviewdoes it hold up?

11 shares of 42/55 litres come back to 8 and 2/5 litres, which is the 6 bottles. And one share is smaller than one bottle, as it must be with 11 people and only 6 bottles.

Another way: Splitting each bottle first gives 7/55 litres per bottle per person; 6 of those add to the same 42/55.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 11 hard answer:

\frac{35}{48}

$7$ bottles each holding $1\frac{1}{4}\ \text{L}$ of juice are bought and shared equally among $12$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

7 bottles hold 1 and 1/4 litres each. All the juice is split evenly between 12 people, and we want one person's share.

Givens

There are 7 bottles.
Each bottle holds 1 and 1/4 litres.
12 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 12 equal shares. Splitting each bottle separately gives the same answer through 7 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

1 wholes are 4 4ths, plus 1 more.

1\frac{1}{4} = \frac{5}{4}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

7 bottles of the same size.

\frac{5}{4} \times 7 = 8\frac{3}{4}\ \text{L}

There are 8 and 3/4 litres in all.

3Divide the total among 12

#8 Analyze the Units 5.NF.B.7

Dividing by 12 is multiplying by 1 over 12: the same amount of juice cut into 12 equal parts.

8\frac{3}{4} \div 12 = 8\frac{3}{4} \times \frac{1}{12} = \frac{35}{48}

One person gets 35/48 of a litre.

Answer:

\frac{35}{48}

4 · Reviewdoes it hold up?

12 shares of 35/48 litres come back to 8 and 3/4 litres, which is the 7 bottles. And one share is smaller than one bottle, as it must be with 12 people and only 7 bottles.

Another way: Splitting each bottle first gives 5/48 litres per bottle per person; 7 of those add to the same 35/48.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.

Variant 12 hard answer:

\frac{19}{26}

$3$ bottles each holding $3\frac{1}{6}\ \text{L}$ of juice are bought and shared equally among $13$ people. How many liters does each person drink?

Show solution

1 · Understandwhat's really being asked

3 bottles hold 3 and 1/6 litres each. All the juice is split evenly between 13 people, and we want one person's share.

Givens

There are 3 bottles.
Each bottle holds 3 and 1/6 litres.
13 people share all of it equally.

Unknowns

How many litres one person drinks.

Constraints

Every share is the same size, and none of the juice is left over.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

Pour all the bottles together first, then cut that one amount into 13 equal shares. Splitting each bottle separately gives the same answer through 3 smaller divisions and an addition.

3 · Execute3 carry out the plan

1Write the bottle as one fraction

#7 Identify Subproblems 6.NS.A.1

3 wholes are 18 6ths, plus 1 more.

3\frac{1}{6} = \frac{19}{6}

Improper fractions multiply cleanly; mixed numbers do not.

2Total the juice

#7 Identify Subproblems 6.NS.A.1

3 bottles of the same size.

\frac{19}{6} \times 3 = 9\frac{1}{2}\ \text{L}

There are 9 and 1/2 litres in all.

3Divide the total among 13

#8 Analyze the Units 5.NF.B.7

Dividing by 13 is multiplying by 1 over 13: the same amount of juice cut into 13 equal parts.

9\frac{1}{2} \div 13 = 9\frac{1}{2} \times \frac{1}{13} = \frac{19}{26}

One person gets 19/26 of a litre.

Answer:

\frac{19}{26}

4 · Reviewdoes it hold up?

13 shares of 19/26 litres come back to 9 and 1/2 litres, which is the 3 bottles. And one share is smaller than one bottle, as it must be with 13 people and only 3 bottles.

Another way: Splitting each bottle first gives 19/78 litres per bottle per person; 3 of those add to the same 19/26.

Standardsmin grade 6

5.NF.B.7 Apply and extend understanding of division to divide unit fractions by whole numbers — Dividing a fraction by a whole number by multiplying by its reciprocal.
6.NS.A.1 Divide fractions by fractions — Turning the mixed number into a fraction and scaling it up.

💡Takeaway. Cutting something into equal parts is multiplying by one over the number of parts.