Practice · Smaller difference means fractions are closer · Sensim Math

Variant 1 easy

Of $\dfrac{3}{4}$ and $\dfrac{2}{5}$ , find the fraction that is closer to $\dfrac{3}{5}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 3 over 4 and 2 over 5, are each some distance from 3 over 5. We must say which one is nearer.

Givens

The two candidates are $\frac{3}{4}$ and $\frac{2}{5}$ .
The target is $\frac{3}{5}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 20ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 20, the smallest number all three denominators divide.

\frac{3}{4} = \frac{3}{4},\quad \frac{2}{5} = \frac{2}{5},\quad \frac{3}{5} = \frac{3}{5}

Every mark on the ruler is one 20th.

2Measure the gap to $\frac{3}{4}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{3}{4} - \frac{3}{5}\right| = \frac{3}{20}

That gap is

\frac{3}{20}

.

3Measure the gap to $\frac{2}{5}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{2}{5} - \frac{3}{5}\right| = \frac{1}{5}

That gap is

\frac{1}{5}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{3}{20} < \frac{1}{5}

\frac{3}{4}

is closer.

Answer:

\frac{3}{4}

4 · Reviewdoes it hold up?

On the 20ths ruler the target sits at 3, and the two candidates at 3 and 2 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 20.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 2 easy

Of $\dfrac{1}{3}$ and $\dfrac{3}{5}$ , find the fraction that is closer to $\dfrac{1}{2}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 1 over 3 and 3 over 5, are each some distance from 1 over 2. We must say which one is nearer.

Givens

The two candidates are $\frac{1}{3}$ and $\frac{3}{5}$ .
The target is $\frac{1}{2}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 30ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 30, the smallest number all three denominators divide.

\frac{1}{3} = \frac{1}{3},\quad \frac{3}{5} = \frac{3}{5},\quad \frac{1}{2} = \frac{1}{2}

Every mark on the ruler is one 30th.

2Measure the gap to $\frac{1}{3}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{1}{3} - \frac{1}{2}\right| = \frac{1}{6}

That gap is

\frac{1}{6}

.

3Measure the gap to $\frac{3}{5}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{3}{5} - \frac{1}{2}\right| = \frac{1}{10}

That gap is

\frac{1}{10}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{1}{10} < \frac{1}{6}

\frac{3}{5}

is closer.

Answer:

\frac{3}{5}

4 · Reviewdoes it hold up?

On the 30ths ruler the target sits at 1, and the two candidates at 1 and 3 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 30.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 3 easy

Of $\dfrac{1}{2}$ and $\dfrac{5}{6}$ , find the fraction that is closer to $\dfrac{3}{5}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 1 over 2 and 5 over 6, are each some distance from 3 over 5. We must say which one is nearer.

Givens

The two candidates are $\frac{1}{2}$ and $\frac{5}{6}$ .
The target is $\frac{3}{5}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 30ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 30, the smallest number all three denominators divide.

\frac{1}{2} = \frac{1}{2},\quad \frac{5}{6} = \frac{5}{6},\quad \frac{3}{5} = \frac{3}{5}

Every mark on the ruler is one 30th.

2Measure the gap to $\frac{1}{2}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{1}{2} - \frac{3}{5}\right| = \frac{1}{10}

That gap is

\frac{1}{10}

.

3Measure the gap to $\frac{5}{6}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{5}{6} - \frac{3}{5}\right| = \frac{7}{30}

That gap is

\frac{7}{30}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{1}{10} < \frac{7}{30}

\frac{1}{2}

is closer.

Answer:

\frac{1}{2}

4 · Reviewdoes it hold up?

On the 30ths ruler the target sits at 3, and the two candidates at 1 and 5 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 30.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 4 medium

Of $\dfrac{3}{8}$ and $\dfrac{4}{7}$ , find the fraction that is closer to $\dfrac{1}{2}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 3 over 8 and 4 over 7, are each some distance from 1 over 2. We must say which one is nearer.

Givens

The two candidates are $\frac{3}{8}$ and $\frac{4}{7}$ .
The target is $\frac{1}{2}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 56ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 56, the smallest number all three denominators divide.

\frac{3}{8} = \frac{3}{8},\quad \frac{4}{7} = \frac{4}{7},\quad \frac{1}{2} = \frac{1}{2}

Every mark on the ruler is one 56th.

2Measure the gap to $\frac{3}{8}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{3}{8} - \frac{1}{2}\right| = \frac{1}{8}

That gap is

\frac{1}{8}

.

3Measure the gap to $\frac{4}{7}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{4}{7} - \frac{1}{2}\right| = \frac{1}{14}

That gap is

\frac{1}{14}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{1}{14} < \frac{1}{8}

\frac{4}{7}

is closer.

Answer:

\frac{4}{7}

4 · Reviewdoes it hold up?

On the 56ths ruler the target sits at 1, and the two candidates at 3 and 4 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 56.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 5 easy

Of $\dfrac{2}{3}$ and $\dfrac{7}{8}$ , find the fraction that is closer to $\dfrac{4}{5}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 2 over 3 and 7 over 8, are each some distance from 4 over 5. We must say which one is nearer.

Givens

The two candidates are $\frac{2}{3}$ and $\frac{7}{8}$ .
The target is $\frac{4}{5}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 120ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 120, the smallest number all three denominators divide.

\frac{2}{3} = \frac{2}{3},\quad \frac{7}{8} = \frac{7}{8},\quad \frac{4}{5} = \frac{4}{5}

Every mark on the ruler is one 120th.

2Measure the gap to $\frac{2}{3}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{2}{3} - \frac{4}{5}\right| = \frac{2}{15}

That gap is

\frac{2}{15}

.

3Measure the gap to $\frac{7}{8}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{7}{8} - \frac{4}{5}\right| = \frac{3}{40}

That gap is

\frac{3}{40}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{3}{40} < \frac{2}{15}

\frac{7}{8}

is closer.

Answer:

\frac{7}{8}

4 · Reviewdoes it hold up?

On the 120ths ruler the target sits at 4, and the two candidates at 2 and 7 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 120.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 6 medium

Of $\dfrac{4}{9}$ and $\dfrac{5}{7}$ , find the fraction that is closer to $\dfrac{3}{5}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 4 over 9 and 5 over 7, are each some distance from 3 over 5. We must say which one is nearer.

Givens

The two candidates are $\frac{4}{9}$ and $\frac{5}{7}$ .
The target is $\frac{3}{5}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 315ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 315, the smallest number all three denominators divide.

\frac{4}{9} = \frac{4}{9},\quad \frac{5}{7} = \frac{5}{7},\quad \frac{3}{5} = \frac{3}{5}

Every mark on the ruler is one 315th.

2Measure the gap to $\frac{4}{9}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{4}{9} - \frac{3}{5}\right| = \frac{7}{45}

That gap is

\frac{7}{45}

.

3Measure the gap to $\frac{5}{7}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{5}{7} - \frac{3}{5}\right| = \frac{4}{35}

That gap is

\frac{4}{35}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{4}{35} < \frac{7}{45}

\frac{5}{7}

is closer.

Answer:

\frac{5}{7}

4 · Reviewdoes it hold up?

On the 315ths ruler the target sits at 3, and the two candidates at 4 and 5 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 315.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 7 hard

Of $\dfrac{3}{10}$ and $\dfrac{8}{9}$ , find the fraction that is closer to $\dfrac{5}{6}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 3 over 10 and 8 over 9, are each some distance from 5 over 6. We must say which one is nearer.

Givens

The two candidates are $\frac{3}{10}$ and $\frac{8}{9}$ .
The target is $\frac{5}{6}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 90ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 90, the smallest number all three denominators divide.

\frac{3}{10} = \frac{3}{10},\quad \frac{8}{9} = \frac{8}{9},\quad \frac{5}{6} = \frac{5}{6}

Every mark on the ruler is one 90th.

2Measure the gap to $\frac{3}{10}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{3}{10} - \frac{5}{6}\right| = \frac{8}{15}

That gap is

\frac{8}{15}

.

3Measure the gap to $\frac{8}{9}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{8}{9} - \frac{5}{6}\right| = \frac{1}{18}

That gap is

\frac{1}{18}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{1}{18} < \frac{8}{15}

\frac{8}{9}

is closer.

Answer:

\frac{8}{9}

4 · Reviewdoes it hold up?

On the 90ths ruler the target sits at 5, and the two candidates at 3 and 8 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 90.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 8 medium

Of $\dfrac{5}{8}$ and $\dfrac{9}{10}$ , find the fraction that is closer to $\dfrac{4}{5}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 5 over 8 and 9 over 10, are each some distance from 4 over 5. We must say which one is nearer.

Givens

The two candidates are $\frac{5}{8}$ and $\frac{9}{10}$ .
The target is $\frac{4}{5}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 40ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 40, the smallest number all three denominators divide.

\frac{5}{8} = \frac{5}{8},\quad \frac{9}{10} = \frac{9}{10},\quad \frac{4}{5} = \frac{4}{5}

Every mark on the ruler is one 40th.

2Measure the gap to $\frac{5}{8}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{5}{8} - \frac{4}{5}\right| = \frac{7}{40}

That gap is

\frac{7}{40}

.

3Measure the gap to $\frac{9}{10}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{9}{10} - \frac{4}{5}\right| = \frac{1}{10}

That gap is

\frac{1}{10}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{1}{10} < \frac{7}{40}

\frac{9}{10}

is closer.

Answer:

\frac{9}{10}

4 · Reviewdoes it hold up?

On the 40ths ruler the target sits at 4, and the two candidates at 5 and 9 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 40.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 9 medium

Of $\dfrac{5}{9}$ and $\dfrac{7}{10}$ , find the fraction that is closer to $\dfrac{2}{3}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 5 over 9 and 7 over 10, are each some distance from 2 over 3. We must say which one is nearer.

Givens

The two candidates are $\frac{5}{9}$ and $\frac{7}{10}$ .
The target is $\frac{2}{3}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 90ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 90, the smallest number all three denominators divide.

\frac{5}{9} = \frac{5}{9},\quad \frac{7}{10} = \frac{7}{10},\quad \frac{2}{3} = \frac{2}{3}

Every mark on the ruler is one 90th.

2Measure the gap to $\frac{5}{9}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{5}{9} - \frac{2}{3}\right| = \frac{1}{9}

That gap is

\frac{1}{9}

.

3Measure the gap to $\frac{7}{10}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{7}{10} - \frac{2}{3}\right| = \frac{1}{30}

That gap is

\frac{1}{30}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{1}{30} < \frac{1}{9}

\frac{7}{10}

is closer.

Answer:

\frac{7}{10}

4 · Reviewdoes it hold up?

On the 90ths ruler the target sits at 2, and the two candidates at 5 and 7 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 90.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 10 hard

Of $\dfrac{2}{5}$ and $\dfrac{9}{11}$ , find the fraction that is closer to $\dfrac{3}{4}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 2 over 5 and 9 over 11, are each some distance from 3 over 4. We must say which one is nearer.

Givens

The two candidates are $\frac{2}{5}$ and $\frac{9}{11}$ .
The target is $\frac{3}{4}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 220ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 220, the smallest number all three denominators divide.

\frac{2}{5} = \frac{2}{5},\quad \frac{9}{11} = \frac{9}{11},\quad \frac{3}{4} = \frac{3}{4}

Every mark on the ruler is one 220th.

2Measure the gap to $\frac{2}{5}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{2}{5} - \frac{3}{4}\right| = \frac{7}{20}

That gap is

\frac{7}{20}

.

3Measure the gap to $\frac{9}{11}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{9}{11} - \frac{3}{4}\right| = \frac{3}{44}

That gap is

\frac{3}{44}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{3}{44} < \frac{7}{20}

\frac{9}{11}

is closer.

Answer:

\frac{9}{11}

4 · Reviewdoes it hold up?

On the 220ths ruler the target sits at 3, and the two candidates at 2 and 9 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 220.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 11 hard

Of $\dfrac{7}{12}$ and $\dfrac{5}{6}$ , find the fraction that is closer to $\dfrac{3}{4}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 7 over 12 and 5 over 6, are each some distance from 3 over 4. We must say which one is nearer.

Givens

The two candidates are $\frac{7}{12}$ and $\frac{5}{6}$ .
The target is $\frac{3}{4}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 12ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 12, the smallest number all three denominators divide.

\frac{7}{12} = \frac{7}{12},\quad \frac{5}{6} = \frac{5}{6},\quad \frac{3}{4} = \frac{3}{4}

Every mark on the ruler is one 12th.

2Measure the gap to $\frac{7}{12}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{7}{12} - \frac{3}{4}\right| = \frac{1}{6}

That gap is

\frac{1}{6}

.

3Measure the gap to $\frac{5}{6}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{5}{6} - \frac{3}{4}\right| = \frac{1}{12}

That gap is

\frac{1}{12}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{1}{12} < \frac{1}{6}

\frac{5}{6}

is closer.

Answer:

\frac{5}{6}

4 · Reviewdoes it hold up?

On the 12ths ruler the target sits at 3, and the two candidates at 7 and 5 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 12.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Variant 12 hard

Of $\dfrac{7}{15}$ and $\dfrac{5}{8}$ , find the fraction that is closer to $\dfrac{1}{2}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Two fractions, 7 over 15 and 5 over 8, are each some distance from 1 over 2. We must say which one is nearer.

Givens

The two candidates are $\frac{7}{15}$ and $\frac{5}{8}$ .
The target is $\frac{1}{2}$ .

Unknowns

Which candidate is closer to the target.

Constraints

Closer means a smaller gap, whichever side of the target it is on.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Put all three on one ruler marked in 120ths. With a shared scale the gaps become plain counts of the same-sized pieces, so comparing them needs no further work.

3 · Execute4 carry out the plan

1Put all three fractions on one ruler

#1 Draw a Diagram 4.NF.A.1

Rewrite each with denominator 120, the smallest number all three denominators divide.

\frac{7}{15} = \frac{7}{15},\quad \frac{5}{8} = \frac{5}{8},\quad \frac{1}{2} = \frac{1}{2}

Every mark on the ruler is one 120th.

2Measure the gap to $\frac{7}{15}$

#7 Identify Subproblems 5.NF.A.1

Subtract the smaller from the larger to get the distance.

\left|\frac{7}{15} - \frac{1}{2}\right| = \frac{1}{30}

That gap is

\frac{1}{30}

.

3Measure the gap to $\frac{5}{8}$

#7 Identify Subproblems 5.NF.A.1

Same again for the second candidate.

\left|\frac{5}{8} - \frac{1}{2}\right| = \frac{1}{8}

That gap is

\frac{1}{8}

.

4Compare the two gaps

#1 Draw a Diagram 4.NF.A.2

Over the same denominator the smaller gap is simply the one with the smaller top.

\frac{1}{30} < \frac{1}{8}

\frac{7}{15}

is closer.

Answer:

\frac{7}{15}

4 · Reviewdoes it hold up?

On the 120ths ruler the target sits at 1, and the two candidates at 7 and 5 -- the winner is the fewer marks away.

Another way: Converting all three to decimals gives the same verdict, but the common denominator keeps the numbers exact.

Standardsmin grade 5

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Rewriting all three fractions over 120.
5.NF.A.1 Add and subtract fractions with unlike denominators — Subtracting to measure each gap.
4.NF.A.2 Compare two fractions with different numerators and different denominators — Comparing the two gaps to pick the nearer fraction.

💡Takeaway. Closeness is a gap, and gaps only compare fairly when both are measured in the same-sized pieces.

Smaller difference means fractions are closer · 12 practice problems

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