← Smaller difference means fractions are closer · Compare Fractions and Decimals by Structure

Smaller difference means fractions are closer · 12 practice problems

4.NF.A.15.NF.A.14.NF.A.2

Generated variants — 12

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

Variant 1 easy answer: 34\frac{3}{4}

Of 34\dfrac{3}{4} and 25\dfrac{2}{5}, find the fraction that is closer to 35\dfrac{3}{5}.

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 34\frac{3}{4} and 25\frac{2}{5}.
  • The target is 35\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.
34=34,25=25,35=35\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 34\frac{3}{4}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣34−35∣=320\left|\frac{3}{4} - \frac{3}{5}\right| = \frac{3}{20}
That gap is 320\frac{3}{20}.

3Measure the gap to 25\frac{2}{5}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣25−35∣=15\left|\frac{2}{5} - \frac{3}{5}\right| = \frac{1}{5}
That gap is 15\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.
320<15\frac{3}{20} < \frac{1}{5}
34\frac{3}{4} is closer.
Answer: 34\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 answer: 35\frac{3}{5}

Of 13\dfrac{1}{3} and 35\dfrac{3}{5}, find the fraction that is closer to 12\dfrac{1}{2}.

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 13\frac{1}{3} and 35\frac{3}{5}.
  • The target is 12\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.
13=13,35=35,12=12\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 13\frac{1}{3}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣13−12∣=16\left|\frac{1}{3} - \frac{1}{2}\right| = \frac{1}{6}
That gap is 16\frac{1}{6}.

3Measure the gap to 35\frac{3}{5}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣35−12∣=110\left|\frac{3}{5} - \frac{1}{2}\right| = \frac{1}{10}
That gap is 110\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.
110<16\frac{1}{10} < \frac{1}{6}
35\frac{3}{5} is closer.
Answer: 35\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 answer: 12\frac{1}{2}

Of 12\dfrac{1}{2} and 56\dfrac{5}{6}, find the fraction that is closer to 35\dfrac{3}{5}.

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 12\frac{1}{2} and 56\frac{5}{6}.
  • The target is 35\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.
12=12,56=56,35=35\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 12\frac{1}{2}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣12−35∣=110\left|\frac{1}{2} - \frac{3}{5}\right| = \frac{1}{10}
That gap is 110\frac{1}{10}.

3Measure the gap to 56\frac{5}{6}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣56−35∣=730\left|\frac{5}{6} - \frac{3}{5}\right| = \frac{7}{30}
That gap is 730\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.
110<730\frac{1}{10} < \frac{7}{30}
12\frac{1}{2} is closer.
Answer: 12\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 answer: 47\frac{4}{7}

Of 38\dfrac{3}{8} and 47\dfrac{4}{7}, find the fraction that is closer to 12\dfrac{1}{2}.

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 38\frac{3}{8} and 47\frac{4}{7}.
  • The target is 12\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.
38=38,47=47,12=12\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 38\frac{3}{8}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣38−12∣=18\left|\frac{3}{8} - \frac{1}{2}\right| = \frac{1}{8}
That gap is 18\frac{1}{8}.

3Measure the gap to 47\frac{4}{7}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣47−12∣=114\left|\frac{4}{7} - \frac{1}{2}\right| = \frac{1}{14}
That gap is 114\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.
114<18\frac{1}{14} < \frac{1}{8}
47\frac{4}{7} is closer.
Answer: 47\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 answer: 78\frac{7}{8}

Of 23\dfrac{2}{3} and 78\dfrac{7}{8}, find the fraction that is closer to 45\dfrac{4}{5}.

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 23\frac{2}{3} and 78\frac{7}{8}.
  • The target is 45\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.
23=23,78=78,45=45\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 23\frac{2}{3}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣23−45∣=215\left|\frac{2}{3} - \frac{4}{5}\right| = \frac{2}{15}
That gap is 215\frac{2}{15}.

3Measure the gap to 78\frac{7}{8}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣78−45∣=340\left|\frac{7}{8} - \frac{4}{5}\right| = \frac{3}{40}
That gap is 340\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.
340<215\frac{3}{40} < \frac{2}{15}
78\frac{7}{8} is closer.
Answer: 78\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 answer: 57\frac{5}{7}

Of 49\dfrac{4}{9} and 57\dfrac{5}{7}, find the fraction that is closer to 35\dfrac{3}{5}.

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 49\frac{4}{9} and 57\frac{5}{7}.
  • The target is 35\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.
49=49,57=57,35=35\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 49\frac{4}{9}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣49−35∣=745\left|\frac{4}{9} - \frac{3}{5}\right| = \frac{7}{45}
That gap is 745\frac{7}{45}.

3Measure the gap to 57\frac{5}{7}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣57−35∣=435\left|\frac{5}{7} - \frac{3}{5}\right| = \frac{4}{35}
That gap is 435\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.
435<745\frac{4}{35} < \frac{7}{45}
57\frac{5}{7} is closer.
Answer: 57\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 answer: 89\frac{8}{9}

Of 310\dfrac{3}{10} and 89\dfrac{8}{9}, find the fraction that is closer to 56\dfrac{5}{6}.

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 310\frac{3}{10} and 89\frac{8}{9}.
  • The target is 56\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.
310=310,89=89,56=56\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 310\frac{3}{10}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣310−56∣=815\left|\frac{3}{10} - \frac{5}{6}\right| = \frac{8}{15}
That gap is 815\frac{8}{15}.

3Measure the gap to 89\frac{8}{9}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣89−56∣=118\left|\frac{8}{9} - \frac{5}{6}\right| = \frac{1}{18}
That gap is 118\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.
118<815\frac{1}{18} < \frac{8}{15}
89\frac{8}{9} is closer.
Answer: 89\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 answer: 910\frac{9}{10}

Of 58\dfrac{5}{8} and 910\dfrac{9}{10}, find the fraction that is closer to 45\dfrac{4}{5}.

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 58\frac{5}{8} and 910\frac{9}{10}.
  • The target is 45\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.
58=58,910=910,45=45\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 58\frac{5}{8}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣58−45∣=740\left|\frac{5}{8} - \frac{4}{5}\right| = \frac{7}{40}
That gap is 740\frac{7}{40}.

3Measure the gap to 910\frac{9}{10}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣910−45∣=110\left|\frac{9}{10} - \frac{4}{5}\right| = \frac{1}{10}
That gap is 110\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.
110<740\frac{1}{10} < \frac{7}{40}
910\frac{9}{10} is closer.
Answer: 910\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 answer: 710\frac{7}{10}

Of 59\dfrac{5}{9} and 710\dfrac{7}{10}, find the fraction that is closer to 23\dfrac{2}{3}.

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 59\frac{5}{9} and 710\frac{7}{10}.
  • The target is 23\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.
59=59,710=710,23=23\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 59\frac{5}{9}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣59−23∣=19\left|\frac{5}{9} - \frac{2}{3}\right| = \frac{1}{9}
That gap is 19\frac{1}{9}.

3Measure the gap to 710\frac{7}{10}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣710−23∣=130\left|\frac{7}{10} - \frac{2}{3}\right| = \frac{1}{30}
That gap is 130\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.
130<19\frac{1}{30} < \frac{1}{9}
710\frac{7}{10} is closer.
Answer: 710\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 answer: 911\frac{9}{11}

Of 25\dfrac{2}{5} and 911\dfrac{9}{11}, find the fraction that is closer to 34\dfrac{3}{4}.

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 25\frac{2}{5} and 911\frac{9}{11}.
  • The target is 34\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.
25=25,911=911,34=34\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 25\frac{2}{5}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣25−34∣=720\left|\frac{2}{5} - \frac{3}{4}\right| = \frac{7}{20}
That gap is 720\frac{7}{20}.

3Measure the gap to 911\frac{9}{11}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣911−34∣=344\left|\frac{9}{11} - \frac{3}{4}\right| = \frac{3}{44}
That gap is 344\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.
344<720\frac{3}{44} < \frac{7}{20}
911\frac{9}{11} is closer.
Answer: 911\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 answer: 56\frac{5}{6}

Of 712\dfrac{7}{12} and 56\dfrac{5}{6}, find the fraction that is closer to 34\dfrac{3}{4}.

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 712\frac{7}{12} and 56\frac{5}{6}.
  • The target is 34\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.
712=712,56=56,34=34\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 712\frac{7}{12}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣712−34∣=16\left|\frac{7}{12} - \frac{3}{4}\right| = \frac{1}{6}
That gap is 16\frac{1}{6}.

3Measure the gap to 56\frac{5}{6}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣56−34∣=112\left|\frac{5}{6} - \frac{3}{4}\right| = \frac{1}{12}
That gap is 112\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.
112<16\frac{1}{12} < \frac{1}{6}
56\frac{5}{6} is closer.
Answer: 56\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 answer: 715\frac{7}{15}

Of 715\dfrac{7}{15} and 58\dfrac{5}{8}, find the fraction that is closer to 12\dfrac{1}{2}.

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 715\frac{7}{15} and 58\frac{5}{8}.
  • The target is 12\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.
715=715,58=58,12=12\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 715\frac{7}{15}

#7 Identify Subproblems 5.NF.A.1
Subtract the smaller from the larger to get the distance.
∣715−12∣=130\left|\frac{7}{15} - \frac{1}{2}\right| = \frac{1}{30}
That gap is 130\frac{1}{30}.

3Measure the gap to 58\frac{5}{8}

#7 Identify Subproblems 5.NF.A.1
Same again for the second candidate.
∣58−12∣=18\left|\frac{5}{8} - \frac{1}{2}\right| = \frac{1}{8}
That gap is 18\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.
130<18\frac{1}{30} < \frac{1}{8}
715\frac{7}{15} is closer.
Answer: 715\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.