Practice · Smaller denominator means larger unit fraction · Sensim Math

Variant 1 easy

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{4} \times \frac{1}{9} < \frac{1}{\blacksquare} \times \frac{1}{6}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 4 times one over 9; on the right, one over the hidden number times one over 6. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{4} \times \frac{1}{9}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{6}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{4} \times \frac{1}{9} = \frac{1}{36}

The left side is one 36th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{6} = \frac{1}{\blacksquare \times 6}

The right side is one over 6 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 6 < 36

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 6 has to stay under 36, and 5 x 6 = 30 does while 6 x 6 = 36 does not.

\blacksquare = 5

The answer is 5.

Answer: 5

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{36}$ against $\frac{1}{30}$ -- and 30 is less than 36, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 36 divided by 6 is 6, and the greatest whole number below that is 5.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 2 easy

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{6} \times \frac{1}{11} < \frac{1}{\blacksquare} \times \frac{1}{9}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 6 times one over 11; on the right, one over the hidden number times one over 9. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{6} \times \frac{1}{11}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{9}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{6} \times \frac{1}{11} = \frac{1}{66}

The left side is one 66th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{9} = \frac{1}{\blacksquare \times 9}

The right side is one over 9 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 9 < 66

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 9 has to stay under 66, and 7 x 9 = 63 does while 8 x 9 = 72 does not.

\blacksquare = 7

The answer is 7.

Answer: 7

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{66}$ against $\frac{1}{63}$ -- and 63 is less than 66, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 66 divided by 9 is 7.33333, and the greatest whole number below that is 7.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 3 easy

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{7} \times \frac{1}{10} < \frac{1}{\blacksquare} \times \frac{1}{12}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 7 times one over 10; on the right, one over the hidden number times one over 12. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{7} \times \frac{1}{10}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{12}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{7} \times \frac{1}{10} = \frac{1}{70}

The left side is one 70th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{12} = \frac{1}{\blacksquare \times 12}

The right side is one over 12 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 12 < 70

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 12 has to stay under 70, and 5 x 12 = 60 does while 6 x 12 = 72 does not.

\blacksquare = 5

The answer is 5.

Answer: 5

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{70}$ against $\frac{1}{60}$ -- and 60 is less than 70, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 70 divided by 12 is 5.83333, and the greatest whole number below that is 5.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 4 easy

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{5} \times \frac{1}{12} < \frac{1}{\blacksquare} \times \frac{1}{8}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 5 times one over 12; on the right, one over the hidden number times one over 8. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{5} \times \frac{1}{12}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{8}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{5} \times \frac{1}{12} = \frac{1}{60}

The left side is one 60th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{8} = \frac{1}{\blacksquare \times 8}

The right side is one over 8 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 8 < 60

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 8 has to stay under 60, and 7 x 8 = 56 does while 8 x 8 = 64 does not.

\blacksquare = 7

The answer is 7.

Answer: 7

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{60}$ against $\frac{1}{56}$ -- and 56 is less than 60, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 60 divided by 8 is 7.5, and the greatest whole number below that is 7.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 5 medium

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{7} \times \frac{1}{12} < \frac{1}{\blacksquare} \times \frac{1}{13}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 7 times one over 12; on the right, one over the hidden number times one over 13. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{7} \times \frac{1}{12}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{13}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{7} \times \frac{1}{12} = \frac{1}{84}

The left side is one 84th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{13} = \frac{1}{\blacksquare \times 13}

The right side is one over 13 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 13 < 84

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 13 has to stay under 84, and 6 x 13 = 78 does while 7 x 13 = 91 does not.

\blacksquare = 6

The answer is 6.

Answer: 6

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{84}$ against $\frac{1}{78}$ -- and 78 is less than 84, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 84 divided by 13 is 6.46154, and the greatest whole number below that is 6.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 6 medium

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{3} \times \frac{1}{14} < \frac{1}{\blacksquare} \times \frac{1}{5}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 3 times one over 14; on the right, one over the hidden number times one over 5. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{3} \times \frac{1}{14}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{5}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{3} \times \frac{1}{14} = \frac{1}{42}

The left side is one 42th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{5} = \frac{1}{\blacksquare \times 5}

The right side is one over 5 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 5 < 42

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 5 has to stay under 42, and 8 x 5 = 40 does while 9 x 5 = 45 does not.

\blacksquare = 8

The answer is 8.

Answer: 8

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{42}$ against $\frac{1}{40}$ -- and 40 is less than 42, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 42 divided by 5 is 8.4, and the greatest whole number below that is 8.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 7 medium

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{9} \times \frac{1}{10} < \frac{1}{\blacksquare} \times \frac{1}{15}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 9 times one over 10; on the right, one over the hidden number times one over 15. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{9} \times \frac{1}{10}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{15}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{9} \times \frac{1}{10} = \frac{1}{90}

The left side is one 90th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{15} = \frac{1}{\blacksquare \times 15}

The right side is one over 15 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 15 < 90

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 15 has to stay under 90, and 5 x 15 = 75 does while 6 x 15 = 90 does not.

\blacksquare = 5

The answer is 5.

Answer: 5

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{90}$ against $\frac{1}{75}$ -- and 75 is less than 90, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 90 divided by 15 is 6, and the greatest whole number below that is 5.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 8 medium

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{5} \times \frac{1}{18} < \frac{1}{\blacksquare} \times \frac{1}{7}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 5 times one over 18; on the right, one over the hidden number times one over 7. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{5} \times \frac{1}{18}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{7}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{5} \times \frac{1}{18} = \frac{1}{90}

The left side is one 90th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{7} = \frac{1}{\blacksquare \times 7}

The right side is one over 7 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 7 < 90

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 7 has to stay under 90, and 12 x 7 = 84 does while 13 x 7 = 91 does not.

\blacksquare = 12

The answer is 12.

Answer: 12

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{90}$ against $\frac{1}{84}$ -- and 84 is less than 90, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 90 divided by 7 is 12.8571, and the greatest whole number below that is 12.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 9 hard

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{8} \times \frac{1}{15} < \frac{1}{\blacksquare} \times \frac{1}{20}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 8 times one over 15; on the right, one over the hidden number times one over 20. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{8} \times \frac{1}{15}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{20}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{8} \times \frac{1}{15} = \frac{1}{120}

The left side is one 120th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{20} = \frac{1}{\blacksquare \times 20}

The right side is one over 20 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 20 < 120

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 20 has to stay under 120, and 5 x 20 = 100 does while 6 x 20 = 120 does not.

\blacksquare = 5

The answer is 5.

Answer: 5

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{120}$ against $\frac{1}{100}$ -- and 100 is less than 120, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 120 divided by 20 is 6, and the greatest whole number below that is 5.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 10 hard

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{4} \times \frac{1}{21} < \frac{1}{\blacksquare} \times \frac{1}{11}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 4 times one over 21; on the right, one over the hidden number times one over 11. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{4} \times \frac{1}{21}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{11}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{4} \times \frac{1}{21} = \frac{1}{84}

The left side is one 84th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{11} = \frac{1}{\blacksquare \times 11}

The right side is one over 11 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 11 < 84

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 11 has to stay under 84, and 7 x 11 = 77 does while 8 x 11 = 88 does not.

\blacksquare = 7

The answer is 7.

Answer: 7

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{84}$ against $\frac{1}{77}$ -- and 77 is less than 84, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 84 divided by 11 is 7.63636, and the greatest whole number below that is 7.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 11 hard

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{10} \times \frac{1}{13} < \frac{1}{\blacksquare} \times \frac{1}{24}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 10 times one over 13; on the right, one over the hidden number times one over 24. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{10} \times \frac{1}{13}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{24}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{10} \times \frac{1}{13} = \frac{1}{130}

The left side is one 130th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{24} = \frac{1}{\blacksquare \times 24}

The right side is one over 24 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 24 < 130

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 24 has to stay under 130, and 5 x 24 = 120 does while 6 x 24 = 144 does not.

\blacksquare = 5

The answer is 5.

Answer: 5

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{130}$ against $\frac{1}{120}$ -- and 120 is less than 130, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 130 divided by 24 is 5.41667, and the greatest whole number below that is 5.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Variant 12 hard

Find the greatest natural number that can go in $\blacksquare$ .

$\frac{1}{6} \times \frac{1}{25} < \frac{1}{\blacksquare} \times \frac{1}{16}$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products of unit fractions are compared. On the left, one over 6 times one over 25; on the right, one over the hidden number times one over 16. The left must be the smaller. We want the biggest whole number that keeps it so.

Givens

The left side is $\frac{1}{6} \times \frac{1}{25}$ .
The right side is $\frac{1}{\blacksquare} \times \frac{1}{16}$ .
The left side must be strictly less than the right.

Unknowns

The greatest natural number the box can hold.

Constraints

The box is a natural number, so at least 1.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #6 Guess and Check#5 Look for a Pattern

Multiply each side out first. Both become single unit fractions, and unit fractions compare by their denominators alone -- the larger denominator being the smaller fraction.

3 · Execute4 carry out the plan

1Multiply the left side

#9 Solve an Easier Related Problem 5.NF.B.4

Multiplying unit fractions multiplies the denominators.

\frac{1}{6} \times \frac{1}{25} = \frac{1}{150}

The left side is one 150th.

2Multiply the right side

#9 Solve an Easier Related Problem 5.NF.B.4

The same on the right, leaving the box inside the denominator.

\frac{1}{\blacksquare} \times \frac{1}{16} = \frac{1}{\blacksquare \times 16}

The right side is one over 16 times the box.

3Compare two unit fractions

#9 Solve an Easier Related Problem 4.NF.A.2

With 1 on top of both, the smaller fraction is the one with the bigger bottom. The left is smaller, so its denominator must be the bigger one.

\blacksquare \times 16 < 150

The comparison flipped into one about whole numbers.

4Find the biggest whole number

#6 Guess and Check 3.NF.A.3

The box times 16 has to stay under 150, and 9 x 16 = 144 does while 10 x 16 = 160 does not.

\blacksquare = 9

The answer is 9.

Answer: 9

4 · Reviewdoes it hold up?

Put it back: $\frac{1}{150}$ against $\frac{1}{144}$ -- and 144 is less than 150, so the right side is indeed the larger fraction.

Another way: Dividing gives the bound straight away: 150 divided by 16 is 9.375, and the greatest whole number below that is 9.

Standardsmin grade 5

5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Multiplying each pair of unit fractions into one.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Turning the fraction comparison into one about denominators.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Checking the chosen number and the next one up.

💡Takeaway. Unit fractions compare backwards: the bigger the bottom, the smaller the piece.

Smaller denominator means larger unit fraction · 12 practice problems

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