Practice · Decompose into sums of unit fractions · Sensim Math

Variant 1 easy

Write $\dfrac{7}{12}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 7 over 12 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{7}{12}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{7}{12}

is

\frac{1}{2}

: one more 1th would overshoot.

\frac{7}{12} - \frac{1}{2} = \frac{1}{12}

\frac{1}{12}

is still to be shared out.

2Split the leftover $\frac{1}{12}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{1}{12}

is

\frac{1}{21}

, and what remains is already a unit fraction.

\frac{1}{12} - \frac{1}{21} = \frac{1}{28}

The last piece is

\frac{1}{28}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 28 to confirm they rebuild the target.

\frac{1}{2} + \frac{1}{21} + \frac{1}{28} = \frac{7}{12}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{2} + \frac{1}{21} + \frac{1}{28}

4 · Reviewdoes it hold up?

All three denominators differ (2, 21, 28) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 2 easy

Write $\dfrac{5}{12}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 5 over 12 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{5}{12}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{5}{12}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{5}{12} - \frac{1}{3} = \frac{1}{12}

\frac{1}{12}

is still to be shared out.

2Split the leftover $\frac{1}{12}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{1}{12}

is

\frac{1}{21}

, and what remains is already a unit fraction.

\frac{1}{12} - \frac{1}{21} = \frac{1}{28}

The last piece is

\frac{1}{28}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 28 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{21} + \frac{1}{28} = \frac{5}{12}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{21} + \frac{1}{28}

4 · Reviewdoes it hold up?

All three denominators differ (3, 21, 28) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 3 easy

Write $\dfrac{5}{14}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 5 over 14 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{5}{14}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{5}{14}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{5}{14} - \frac{1}{3} = \frac{1}{42}

\frac{1}{42}

is still to be shared out.

2Split the leftover $\frac{1}{42}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{1}{42}

is

\frac{1}{78}

, and what remains is already a unit fraction.

\frac{1}{42} - \frac{1}{78} = \frac{1}{91}

The last piece is

\frac{1}{91}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 91 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{78} + \frac{1}{91} = \frac{5}{14}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{78} + \frac{1}{91}

4 · Reviewdoes it hold up?

All three denominators differ (3, 78, 91) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 4 easy

Write $\dfrac{7}{15}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 7 over 15 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{7}{15}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{7}{15}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{7}{15} - \frac{1}{3} = \frac{2}{15}

\frac{2}{15}

is still to be shared out.

2Split the leftover $\frac{2}{15}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{2}{15}

is

\frac{1}{12}

, and what remains is already a unit fraction.

\frac{2}{15} - \frac{1}{12} = \frac{1}{20}

The last piece is

\frac{1}{20}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 20 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{12} + \frac{1}{20} = \frac{7}{15}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{12} + \frac{1}{20}

4 · Reviewdoes it hold up?

All three denominators differ (3, 12, 20) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 5 medium

Write $\dfrac{5}{16}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 5 over 16 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{5}{16}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{5}{16}

is

\frac{1}{4}

: one more 3th would overshoot.

\frac{5}{16} - \frac{1}{4} = \frac{1}{16}

\frac{1}{16}

is still to be shared out.

2Split the leftover $\frac{1}{16}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{1}{16}

is

\frac{1}{24}

, and what remains is already a unit fraction.

\frac{1}{16} - \frac{1}{24} = \frac{1}{48}

The last piece is

\frac{1}{48}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 48 to confirm they rebuild the target.

\frac{1}{4} + \frac{1}{24} + \frac{1}{48} = \frac{5}{16}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{4} + \frac{1}{24} + \frac{1}{48}

4 · Reviewdoes it hold up?

All three denominators differ (4, 24, 48) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 6 medium

Write $\dfrac{7}{18}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 7 over 18 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{7}{18}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{7}{18}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{7}{18} - \frac{1}{3} = \frac{1}{18}

\frac{1}{18}

is still to be shared out.

2Split the leftover $\frac{1}{18}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{1}{18}

is

\frac{1}{30}

, and what remains is already a unit fraction.

\frac{1}{18} - \frac{1}{30} = \frac{1}{45}

The last piece is

\frac{1}{45}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 45 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{30} + \frac{1}{45} = \frac{7}{18}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{30} + \frac{1}{45}

4 · Reviewdoes it hold up?

All three denominators differ (3, 30, 45) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 7 medium

Write $\dfrac{9}{20}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 9 over 20 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{9}{20}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{9}{20}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{9}{20} - \frac{1}{3} = \frac{7}{60}

\frac{7}{60}

is still to be shared out.

2Split the leftover $\frac{7}{60}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{7}{60}

is

\frac{1}{15}

, and what remains is already a unit fraction.

\frac{7}{60} - \frac{1}{15} = \frac{1}{20}

The last piece is

\frac{1}{20}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 20 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{15} + \frac{1}{20} = \frac{9}{20}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{15} + \frac{1}{20}

4 · Reviewdoes it hold up?

All three denominators differ (3, 15, 20) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 8 medium

Write $\dfrac{9}{22}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 9 over 22 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{9}{22}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{9}{22}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{9}{22} - \frac{1}{3} = \frac{5}{66}

\frac{5}{66}

is still to be shared out.

2Split the leftover $\frac{5}{66}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{5}{66}

is

\frac{1}{22}

, and what remains is already a unit fraction.

\frac{5}{66} - \frac{1}{22} = \frac{1}{33}

The last piece is

\frac{1}{33}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 33 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{22} + \frac{1}{33} = \frac{9}{22}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{22} + \frac{1}{33}

4 · Reviewdoes it hold up?

All three denominators differ (3, 22, 33) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 9 hard

Write $\dfrac{11}{24}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 11 over 24 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{11}{24}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{11}{24}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{11}{24} - \frac{1}{3} = \frac{1}{8}

\frac{1}{8}

is still to be shared out.

2Split the leftover $\frac{1}{8}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{1}{8}

is

\frac{1}{12}

, and what remains is already a unit fraction.

\frac{1}{8} - \frac{1}{12} = \frac{1}{24}

The last piece is

\frac{1}{24}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 24 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{12} + \frac{1}{24} = \frac{11}{24}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{12} + \frac{1}{24}

4 · Reviewdoes it hold up?

All three denominators differ (3, 12, 24) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 10 hard

Write $\dfrac{13}{28}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 13 over 28 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{13}{28}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{13}{28}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{13}{28} - \frac{1}{3} = \frac{11}{84}

\frac{11}{84}

is still to be shared out.

2Split the leftover $\frac{11}{84}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{11}{84}

is

\frac{1}{12}

, and what remains is already a unit fraction.

\frac{11}{84} - \frac{1}{12} = \frac{1}{21}

The last piece is

\frac{1}{21}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 21 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{12} + \frac{1}{21} = \frac{13}{28}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{12} + \frac{1}{21}

4 · Reviewdoes it hold up?

All three denominators differ (3, 12, 21) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 11 hard

Write $\dfrac{11}{30}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 11 over 30 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{11}{30}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{11}{30}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{11}{30} - \frac{1}{3} = \frac{1}{30}

\frac{1}{30}

is still to be shared out.

2Split the leftover $\frac{1}{30}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{1}{30}

is

\frac{1}{55}

, and what remains is already a unit fraction.

\frac{1}{30} - \frac{1}{55} = \frac{1}{66}

The last piece is

\frac{1}{66}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 66 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{55} + \frac{1}{66} = \frac{11}{30}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{55} + \frac{1}{66}

4 · Reviewdoes it hold up?

All three denominators differ (3, 55, 66) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Variant 12 hard

Write $\dfrac{13}{36}$ as a sum of three different unit fractions $\dfrac{1}{\bigstar}+\dfrac{1}{\heartsuit}+\dfrac{1}{\blacktriangle}$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

We must write 13 over 36 as three unit fractions -- fractions with 1 on top -- added together, all with different bottoms.

Givens

The target is $\frac{13}{36}$ .
Exactly three unit fractions, all different.

Unknowns

The three denominators.

Constraints

Each piece has numerator 1, and no two may be the same.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Peel off the biggest unit fraction that still fits, then face the same question with a smaller leftover. Being greedy keeps the pieces distinct and shrinking, so it cannot loop.

3 · Execute3 carry out the plan

1Peel off the first unit fraction

#7 Identify Subproblems 4.NF.B.3

The largest unit fraction not bigger than

\frac{13}{36}

is

\frac{1}{3}

: one more 2th would overshoot.

\frac{13}{36} - \frac{1}{3} = \frac{1}{36}

\frac{1}{36}

is still to be shared out.

2Split the leftover $\frac{1}{36}$ into two unit fractions

#6 Guess and Check 5.NF.A.1

Repeat the move: the biggest unit fraction fitting inside

\frac{1}{36}

is

\frac{1}{63}

, and what remains is already a unit fraction.

\frac{1}{36} - \frac{1}{63} = \frac{1}{84}

The last piece is

\frac{1}{84}

.

3Put the three pieces together and check

#6 Guess and Check 5.NF.A.1

Add all three over 84 to confirm they rebuild the target.

\frac{1}{3} + \frac{1}{63} + \frac{1}{84} = \frac{13}{36}

Three different unit fractions, and they add back exactly.

Answer:

\frac{1}{3} + \frac{1}{63} + \frac{1}{84}

4 · Reviewdoes it hold up?

All three denominators differ (3, 63, 84) and each piece is smaller than the one before, which is what taking the biggest first guarantees.

Another way: Other splits exist -- this is not the only answer -- but the greedy route always finds one, and finds it without searching.

Standardsmin grade 5

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Reading the target as a sum of pieces with 1 on top.
5.NF.A.1 Add and subtract fractions with unlike denominators — Each subtraction, and the final addition that checks the split.

💡Takeaway. Take the biggest piece that fits, then ask the same question about what is left. The leftovers shrink until nothing remains.

Decompose into sums of unit fractions · 12 practice problems

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