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Decompose fractions into unit sums

Learning goals

  • Writes a fraction as a sum of unit fractions with the same denominator.
  • Creates alternative decompositions of a fraction.
  • Represents decompositions with equations or models.
  • Checks that the parts recombine to the original fraction.

Click problems to add them to your worksheet.

Decompose fractions into unit sums 1
A strip is divided into \(12\) equal parts. Five of those parts make the fraction \(\frac{5}{12}\). a) Write \(\frac{5}{12}\) as a sum of unit fractions. b) Explain how the numerator \(5\) tells you how many unit fractions appear in the sum.

Hints

- What fraction represents one of the \(12\) equal parts? - Think of the numerator as a count of equal unit-fraction pieces. - Check that the number of addends matches the numerator.

Solution

1. One of \(12\) equal parts is the unit fraction \(\frac{1}{12}\). 2. Five equal parts are five copies of \(\frac{1}{12}\): \(\frac{5}{12}=\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\). 3. The numerator \(5\) counts the number of \(\frac{1}{12}\) unit fractions in \(\frac{5}{12}\).

Answer

a) \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\) b) The numerator \(5\) means there are five copies of \(\frac{1}{12}\).
Decompose fractions into unit sums 2
What fraction in twelfths is represented by the shaded bar? Write the amount first as a sum of unit fractions and then as one fraction.
Figure for problem 540599

Hints

- Determine the unit fraction represented by one part of the bar. - Count the shaded parts rather than using a count supplied in the text. - The number of unit-fraction copies becomes the numerator.

Solution

1. The bar is divided into \(12\) equal parts, so each shaded part represents \(\frac{1}{12}\). 2. There are \(9\) shaded parts, giving \(9\) copies of \(\frac{1}{12}\). 3. Thus the sum is \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}=\frac{9}{12}\).

Answer

\(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}=\frac{9}{12}\)
Decompose fractions into unit sums 3
Which unit-fraction sum is greater? Show how you know. \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\) or \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\)

Hints

- Identify the unit fraction used in both sums. - Count how many copies appear in each sum. - When the unit fractions are the same size, compare the numbers of copies.

Solution

1. The first sum has \(4\) copies of \(\frac{1}{6}\), so it equals \(\frac{4}{6}\). 2. The second sum has \(5\) copies of \(\frac{1}{6}\), so it equals \(\frac{5}{6}\). 3. The unit fractions are the same size, and \(5\) copies are more than \(4\) copies, so the second sum is greater.

Answer

The second sum is greater: \(\frac{5}{6}>\frac{4}{6}\).
Decompose fractions into unit sums 4
Use the bar to write the shaded amount as a sum of unit fractions. Then write the sum as one fraction.
Figure for problem 554417

Hints

- Count the equal parts in the whole to identify one unit fraction. - Count how many of those equal parts are shaded. - Write one copy of the unit fraction for each shaded part.

Solution

1. The bar is divided into \(5\) equal parts, so one part is \(\frac{1}{5}\). 2. Three parts are shaded, so the unit-fraction sum is \(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}\). 3. Three copies of \(\frac{1}{5}\) make \(\frac{3}{5}\).

Answer

\(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=\frac{3}{5}\)
Decompose fractions into unit sums 5
Write \(\frac{4}{5}\) as a sum of unit fractions with denominator \(5\).

Hints

- The denominator tells the size of the unit fraction. - The numerator tells how many copies are needed.

Solution

1. The denominator \(5\) tells us the unit fraction is \(\frac{1}{5}\). 2. The numerator \(4\) tells us to use four copies of that unit fraction. 3. Therefore, \(\frac{4}{5}=\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}\).

Answer

\(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}\)
Decompose fractions into unit sums 6
The model shows more than one whole shaded in sixth-sized parts. a) How many copies of \(\frac{1}{6}\) are shaded? b) Write the shaded amount as an improper fraction and as a mixed number.
Figure for problem 554419

Hints

- Identify the size of one equal part in the model. - Count every shaded part across the wholes. - Regroup enough unit fractions to make one whole.

Solution

1. The model shows \(7\) shaded sixth-sized parts. 2. Seven copies of \(\frac{1}{6}\) make \(\frac{7}{6}\). 3. Six sixths make one whole, leaving one sixth, so \(\frac{7}{6}=1\frac{1}{6}\).

Answer

a) \(7\) copies b) \(\frac{7}{6}=1\frac{1}{6}\)
Decompose fractions into unit sums 7
Write the shaded part of the bar as a sum of unit fractions. All the unit fractions must have the same denominator.
Figure for problem 558791

Hints

- One equal section is one unit fraction. - Count how many of those sections are shaded. - Write one term for every shaded section.

Solution

1. The bar has five equal parts, and three are shaded. 2. Each shaded part is \(\frac{1}{5}\), so the sum is \(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}\).

Answer

\(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}\)
Decompose fractions into unit sums 8
What single fraction is equal to \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\)? Do not simplify.

Hints

- All the pieces have the same size. - Count the unit-fraction terms without changing their denominator.

Solution

1. Every term represents one sixth of the same whole. 2. There are four sixths, so the sum is \(\frac{4}{6}\).

Answer

\(\frac{4}{6}\)
Decompose fractions into unit sums 9
Each bar represents one whole. Write the shaded amount as a sum of unit fourths. Then write the total as one fraction.
Figure for problem 558793

Hints

- Think of each shaded piece as its own fraction. - Use the same denominator for every piece, even across two bars. - How can you write the total without repeated addition?

Solution

1. The first bar has four fourths shaded, and the second has one fourth shaded. 2. The unit-fraction sum is \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\). 3. Five fourths make \(\frac{5}{4}\).

Answer

\(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{5}{4}\)
Decompose fractions into unit sums 10
Is \(\frac{2}{7}\) a unit fraction? If not, write it as a sum of unit fractions with denominator \(7\).

Hints

- Check what the numerator of a unit fraction must be. - Think of \(\frac{2}{7}\) as two equal seventh-sized pieces.

Solution

1. A unit fraction has numerator \(1\), so \(\frac{2}{7}\) is not a unit fraction. 2. Two sevenths are \(\frac{1}{7}+\frac{1}{7}\).

Answer

No. \(\frac{2}{7}=\frac{1}{7}+\frac{1}{7}\).
Decompose fractions into unit sums 11
Consider the fraction \(\frac{3}{10}\). a) Write it as a sum of unit fractions with the same denominator. b) How many unit-fraction addends are in your sum, and what does that number have to do with the numerator of \(\frac{3}{10}\)?

Hints

- A unit fraction has numerator \(1\). - Ask what one of the ten equal parts is called as a fraction. - Relate the number of equal parts being counted to the numerator.

Solution

1. One tenth is the unit fraction \(\frac{1}{10}\). 2. Three tenths are three copies of that unit fraction: \(\frac{3}{10}=\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\). 3. The sum has \(3\) addends. That count matches the numerator \(3\), which tells how many tenths are present.

Answer

a) \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\) b) There are \(3\) addends, matching the numerator \(3\).
Decompose fractions into unit sums 12
Write \(\frac{2}{3}\) using equal unit fractions in two ways. 1) As a sum of two equal unit fractions. 2) As a sum of four equal unit fractions. Explain how the parts changed.

Hints

- Start with the meaning of the numerator \(2\). - To get four equal pieces instead of two, divide each original piece in half. - Check that the four smaller pieces still have the original total value.

Solution

1. Two thirds are \(\frac{2}{3}=\frac{1}{3}+\frac{1}{3}\). 2. Split each third in half to get two sixths. This gives \(\frac{2}{3}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\). 3. Each new part is half the size, but there are twice as many parts. The total is unchanged.

Answer

1) \(\frac{1}{3}+\frac{1}{3}\) 2) \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\). Each third is split into two sixths.
Decompose fractions into unit sums 13
The shaded amount in the bar equals the two written unit twelfths plus a missing unit-fraction sum: \(\frac{1}{12}+\frac{1}{12}+\square\). What sum of unit twelfths belongs in the box?
Figure for problem 540596

Hints

- Read the total shaded amount from the bar. - Compare the number of shaded twelfths with the two unit twelfths already written. - Represent each remaining shaded part with one unit-fraction addend.

Solution

1. Read the bar: \(5\) of its \(12\) equal parts are shaded, so the target amount is \(\frac{5}{12}\). 2. The two written terms account for \(\frac{2}{12}\), leaving \(\frac{3}{12}\). 3. Write that missing amount as \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\).

Answer

\(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\)
Decompose fractions into unit sums 14
Leo looks at the shaded bar and writes \(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}\) for the shaded amount. Explain why Leo's sum is incorrect and write a correct unit-fraction sum for the bar.
Figure for problem 540597

Hints

- First determine what Leo's five identical addends total. - Use the bar to identify the size of one equal shaded part. - The number of shaded parts tells how many copies of that unit fraction are needed.

Solution

1. Five copies of \(\frac{1}{5}\) make \(\frac{5}{5}=1\), so Leo's sum represents a whole. 2. The displayed bar is divided into \(8\) equal parts, with \(5\) shaded. Each shaded part is therefore \(\frac{1}{8}\). 3. The correct sum is \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{5}{8}\).

Answer

Leo's sum equals \(1\), not the shaded amount. A correct sum is \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\).
Decompose fractions into unit sums 15
Use the model shown. a) Write the shaded amount as a sum made only of unit fractions. b) How many terms are in the sum?
Figure for problem 540598

Hints

- Identify the unit fraction represented by one equal part of a bar. - Count the shaded equal parts across both wholes. - Use one unit-fraction addend for each shaded part.

Solution

1. Each whole is divided into thirds, so each shaded part represents \(\frac{1}{3}\). 2. The model shows \(5\) shaded thirds in all, representing \(1\frac{2}{3}=\frac{5}{3}\). 3. Therefore, the unit-fraction sum is \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\), which has \(5\) terms.

Answer

a) \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\) b) \(5\) terms
Decompose fractions into unit sums 16
Two shaded parts are removed from the bar shown. a) Write the remaining amount as a unit-fraction sum. b) Write the remaining amount as one fraction.
Figure for problem 540600

Hints

- Read the starting number of shaded tenths from the bar. - Remove the stated number of shaded parts. - Use one \(\frac{1}{10}\) addend for each part that remains.

Solution

1. The bar is divided into tenths and begins with \(9\) shaded parts. Removing \(2\) leaves \(7\) shaded tenths. 2. The remaining unit-fraction sum is \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\). 3. Seven copies of \(\frac{1}{10}\) equal \(\frac{7}{10}\).

Answer

a) \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\) b) \(\frac{7}{10}\)
Decompose fractions into unit sums 17
The fraction \(\frac{5}{6}\) is five copies of \(\frac{1}{6}\). Two copies are removed. a) Write the remaining amount as a unit-fraction sum and as one fraction. b) Write the removed amount as one fraction. c) Write an equation showing that the remaining amount plus the removed amount equals \(\frac{5}{6}\).

Hints

- Think of the numerator \(5\) as a count of five unit sixths. - Keep track separately of the copies that remain and the copies that were removed. - For the final equation, use sixths so the two groups visibly rebuild the original five copies.

Solution

1. Removing two of the five unit sixths leaves three copies: \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{3}{6}=\frac{1}{2}\). 2. The two removed copies total \(\frac{2}{6}=\frac{1}{3}\). 3. Using denominator \(6\), the parts recombine as \(\frac{3}{6}+\frac{2}{6}=\frac{5}{6}\).

Answer

a) \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{3}{6}=\frac{1}{2}\) b) \(\frac{2}{6}=\frac{1}{3}\) c) \(\frac{3}{6}+\frac{2}{6}=\frac{5}{6}\)
Decompose fractions into unit sums 18
Nora writes \(\frac{1}{2}=\frac{1}{6}+\frac{2}{6}\) and says the right side is a unit-fraction sum. a) Explain why her description is incorrect. b) Rewrite only \(\frac{2}{6}\) as unit fractions so the equation becomes a valid unit-fraction sum.

Hints

- Check the numerator of each addend against the definition of a unit fraction. - Ask how many copies of \(\frac{1}{6}\) make \(\frac{2}{6}\). - Verify that the rewritten right side still equals one half.

Solution

1. A unit fraction has numerator \(1\), so \(\frac{2}{6}\) is not a unit fraction. 2. The fraction \(\frac{2}{6}\) is two copies of \(\frac{1}{6}\). 3. Therefore, \(\frac{1}{2}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\), which is a valid unit-fraction sum because \(\frac{3}{6}=\frac{1}{2}\).

Answer

a) \(\frac{2}{6}\) is not a unit fraction because its numerator is not \(1\). b) \(\frac{1}{2}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\)
Decompose fractions into unit sums 19
The two models show the same amount. a) Write the amount as a sum of unit halves. b) Write the amount as a sum of unit fourths. c) Which representation uses more terms, and how many more?
Figure for problem 540604

Hints

- Count the shaded equal parts in model a) and identify the unit fraction for one part. - Do the same for model b). - Compare the two numbers of unit-fraction addends.

Solution

1. Model a) shows \(3\) shaded halves, so the amount is \(\frac{3}{2}=1\frac{1}{2}\) and the unit-half sum is \(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\). 2. Model b) shows the same amount as \(6\) shaded fourths, so the unit-fourth sum is \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\). 3. The fourths representation uses \(6-3=3\) more terms.

Answer

a) \(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\) b) \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\) c) The fourths representation uses \(3\) more terms.
Decompose fractions into unit sums 20
Riley says the shaded bar is represented by eight copies of \(\frac{1}{12}\). Is Riley correct? If not, add or remove the fewest possible copies of \(\frac{1}{12}\) to make the unit-fraction sum match the bar.
Figure for problem 540605

Hints

- Read the shaded fraction from the bar first. - Rename that fraction in twelfths to find how many unit twelfths it contains. - Compare that count with Riley's eight copies.

Solution

1. The bar shows \(\frac{3}{4}\) shaded. 2. In twelfths, \(\frac{3}{4}=\frac{9}{12}\), so the bar represents nine copies of \(\frac{1}{12}\). 3. Riley used eight copies, so one more \(\frac{1}{12}\) is needed.

Answer

Riley is not correct. Add one copy of \(\frac{1}{12}\); the correct sum has nine copies of \(\frac{1}{12}\).
Decompose fractions into unit sums 21
Maya looks at the model and says the shaded amount is \(1\frac{1}{3}\). Is Maya correct? Show how the unit fractions can be regrouped into a whole and a remaining fraction.
Figure for problem 540607

Hints

- Use the model to identify the unit fraction and count its shaded copies. - Regroup enough copies to make one whole. - Simplify the fraction represented by the copies left over.

Solution

1. Each shaded part is \(\frac{1}{6}\), and the model contains \(8\) shaded parts, so the total is \(\frac{8}{6}\). 2. Six sixths make one whole, leaving \(\frac{2}{6}\). 3. Since \(\frac{2}{6}=\frac{1}{3}\), the total is \(1\frac{1}{3}\). Maya is correct.

Answer

Yes. \(\frac{8}{6}=1\frac{2}{6}=1\frac{1}{3}\).
Decompose fractions into unit sums 22
Twelve copies of \(\frac{1}{12}\) are divided into three equal groups. a) Write the unit-fraction sum in each group. b) What fraction does each group equal?

Hints

- Divide the number of identical terms equally among the three groups. - For part a), write one unit fraction for each term in a group. - For part b), combine and simplify the four twelfths.

Solution

1. Divide the twelve terms equally among three groups: \(12\div3=4\) terms per group. 2. Each group is \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}=\frac{4}{12}\). 3. Since \(\frac{4}{12}=\frac{1}{3}\), each group equals \(\frac{1}{3}\).

Answer

a) \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\) b) \(\frac{1}{3}\)
Decompose fractions into unit sums 23
The bar represents one whole. a) Write the orange part as a sum of unit fractions. Do the same for the green part. b) Combine your two sums. Is the result equal to the blue part? Explain using unit fractions.
Figure for problem 540723

Hints

- What unit fraction does one small part of the bar represent? - Use one term for each orange piece, then for each green piece. - Compare the number of unit-fraction terms with the blue pieces.

Solution

1. Orange covers two eighths, so its unit-fraction sum is \(\frac{1}{8}+\frac{1}{8}\). Green covers one eighth, or \(\frac{1}{8}\). 2. Combining those sums gives \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{8}\). 3. Blue also covers three eighths. Both amounts contain three units of size \(\frac{1}{8}\), so they are equal.

Answer

a) Orange: \(\frac{1}{8}+\frac{1}{8}\); green: \(\frac{1}{8}\). b) Yes. \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{8}\), the same as blue.
Decompose fractions into unit sums 24
Each small section in the bar represents \(\frac{1}{12}\) of the whole. The colors separate the shaded unit twelfths into three groups. a) How many copies of \(\frac{1}{12}\) are blue, orange, and green? b) Write the entire shaded amount as a unit-fraction sum and as one fraction. c) Write the unshaded amount as a unit-fraction sum and as one fraction.
Figure for problem 540724

Hints

- Count the small sections in each color group; every small section has the same value. - For the shaded total, combine the counts before writing the numerator. - For the unshaded amount, count the white sections and use the same unit fraction.

Solution

1. The blue group has \(3\) unit twelfths, the orange group has \(2\), and the green group has \(4\). 2. Altogether there are \(3+2+4=9\) shaded unit twelfths. The expanded sum is \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}=\frac{9}{12}=\frac{3}{4}\). 3. Three sections are unshaded, so the unshaded amount is \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}=\frac{3}{12}=\frac{1}{4}\).

Answer

a) Blue: \(3\); orange: \(2\); green: \(4\) b) \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}=\frac{9}{12}=\frac{3}{4}\) c) \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}=\frac{3}{12}=\frac{1}{4}\)
Decompose fractions into unit sums 25
Jordan writes \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{12}\) because he adds the denominators as well as the numerators. a) Explain why the denominator should not change when equal-sized fourths are joined. b) Write the correct fraction for the unit-fraction sum.

Hints

- Ask what the denominator tells you about the size of each equal part. - Joining several fourth-sized parts does not cut those parts into a new size. - Count the number of unit fourths to determine the numerator of the sum.

Solution

1. Each addend is one fourth, so all three pieces have the same size. Joining them changes only how many fourths there are, not the size of each part. 2. There are three copies of \(\frac{1}{4}\), so the sum is \(\frac{3}{4}\), not \(\frac{3}{12}\).

Answer

a) The denominator stays \(4\) because every addend is still a fourth-sized part. b) \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\)
Decompose fractions into unit sums 26
For each expression, choose one classification: - repeated copies of the same unit fraction; - all terms are unit fractions, but they are not all the same unit fraction; - at least one term is not a unit fraction. For expressions in the first category, also write the total as one fraction. <table><tr><th>Label</th><th>Expression</th></tr><tr><td>A</td><td>\(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\)</td></tr><tr><td>B</td><td>\(\frac{1}{4}+\frac{1}{8}+\frac{1}{8}\)</td></tr><tr><td>C</td><td>\(\frac{2}{5}+\frac{1}{5}\)</td></tr><tr><td>D</td><td>\(\frac{1}{3}+\frac{1}{3}\)</td></tr><tr><td>E</td><td>\(\frac{1}{2}+\frac{1}{4}+\frac{1}{4}\)</td></tr></table>

Hints

- First check whether every numerator is \(1\). - If every term is a unit fraction, check whether all the denominators are the same. - Only when the same unit fraction repeats do you need to combine the count into one numerator.

Solution

1. A uses three copies of the same unit fraction \(\frac{1}{8}\), so it is in the first category and totals \(\frac{3}{8}\). 2. B contains only unit fractions, but it uses both \(\frac{1}{4}\) and \(\frac{1}{8}\), so it is in the second category. 3. C contains \(\frac{2}{5}\), whose numerator is not \(1\), so it is in the third category. 4. D uses two copies of the same unit fraction \(\frac{1}{3}\), so it is in the first category and totals \(\frac{2}{3}\). 5. E contains only unit fractions, but it uses \(\frac{1}{2}\) and \(\frac{1}{4}\), so it is in the second category.

Answer

A: repeated copies of the same unit fraction; \(\frac{3}{8}\) B: all terms are unit fractions, but they are not all the same unit fraction C: at least one term is not a unit fraction D: repeated copies of the same unit fraction; \(\frac{2}{3}\) E: all terms are unit fractions, but they are not all the same unit fraction
Decompose fractions into unit sums 27
A unit-fraction decomposition is summarized in the table. <table><tr><th>Group</th><th>Number of \(\frac{1}{12}\) terms</th></tr><tr><td>A</td><td>3</td></tr><tr><td>B</td><td>1</td></tr><tr><td>C</td><td>5</td></tr></table> a) Write the complete expanded unit-fraction sum. b) Write the total as one fraction in simplest form. c) Which group contributes the greatest fraction of the whole, and what fraction does it contribute?

Hints

- Treat each table count as a number of copies of the same unit fraction. - Add the counts to determine the numerator of the total in twelfths. - For part c), compare the group counts because every term has the same size.

Solution

1. Group A contributes three \(\frac{1}{12}\) terms, group B contributes one, and group C contributes five. The complete expanded sum is \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\). 2. The total is \(\frac{9}{12}=\frac{3}{4}\). 3. Group C has \(5\) copies, so it contributes \(\frac{5}{12}\), which is greater than group A's \(\frac{3}{12}\) and group B's \(\frac{1}{12}\).

Answer

a) \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\) b) \(\frac{9}{12}=\frac{3}{4}\) c) Group C, \(\frac{5}{12}\)
Decompose fractions into unit sums 28
Write \(\frac{3}{4}\) as exactly six identical unit fractions. What is the unit fraction?

Hints

- Six identical unit fractions will create an equivalent fraction with numerator \(6\). - Think about how the numerator \(3\) can become \(6\) without changing the fraction's value. - Apply the same scale factor to the denominator.

Solution

1. Six identical unit fractions combine to a fraction whose numerator is \(6\). 2. Find an equivalent fraction for \(\frac{3}{4}\) with numerator \(6\): multiply the numerator and denominator by \(2\). 3. \(\frac{3}{4}=\frac{6}{8}\), so the six identical terms are each \(\frac{1}{8}\).

Answer

\(\frac{1}{8}\)
Decompose fractions into unit sums 29
The two colors in the bar form one shaded part. Write the shaded amount as two fractions with denominator \(8\), one for each color. Then express it as a sum of unit eighths.
Figure for problem 558795

Hints

- Count eighths in each colored group separately. - Keep the eighth-sized unit for both groups. - Expand each group into copies of one eighth.

Solution

1. The blue section is \(\frac{4}{8}\) and the orange section is \(\frac{2}{8}\). 2. Together they are \(\frac{4}{8}+\frac{2}{8}=\frac{6}{8}\). 3. As unit eighths, this is \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\).

Answer

\(\frac{4}{8}+\frac{2}{8}=\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\)
Decompose fractions into unit sums 30
Both sums equal \(\frac{5}{9}\). Ada writes \(\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}\). Ben writes \(\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{2}{9}\). Which sum uses only unit fractions? Rewrite the other sum using only unit fractions.

Hints

- What must the numerator of a unit fraction be? - Check each term, not just the total of a sum. - If a term is not a unit fraction, how can you replace it without changing the total?

Solution

1. A unit fraction has numerator \(1\). All five terms in Ada's sum are unit fractions. 2. Ben's final term, \(\frac{2}{9}\), is not a unit fraction. It equals \(\frac{1}{9}+\frac{1}{9}\). 3. Replacing that term gives Ben the same five-term sum as Ada, equal to \(\frac{5}{9}\).

Answer

Ada's sum. Ben's corrected sum is \(\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}\).
Decompose fractions into unit sums 31
Nina writes \(\frac{4}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\). What is missing from her sum? Correct it.

Hints

- Count the terms on the right. - Compare that count with the numerator on the left. - Keep every term the same unit size.

Solution

1. The right side contains only three sixths, or \(\frac{3}{6}\). 2. Four sixths need four copies of one sixth. 3. Add one more \(\frac{1}{6}\).

Answer

One more \(\frac{1}{6}\) is missing. Correct: \(\frac{4}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\).
Decompose fractions into unit sums 32
Riya colors part of a paper strip red and then colors more of it blue. She says the blue part is \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\). Is she right? Write a unit-fraction sum for each color. How many tenths are still uncolored?
Figure for problem 558798

Hints

- Check the blue section on its own before counting all the shading. - How can each small shaded section be written as a unit fraction? - Compare the total number of colored sections with all the sections in the strip.

Solution

1. The red part is \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\). 2. The blue part is \(\frac{1}{10}+\frac{1}{10}\), so Riya's sum for blue has one term too many. 3. The colors cover five tenths altogether. Five of the ten parts are still uncolored.

Answer

No. Red: \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\). Blue: \(\frac{1}{10}+\frac{1}{10}\). Riya counted one extra blue tenth. \(5\) tenths are uncolored.
Decompose fractions into unit sums 33
Break \(\frac{5}{12}\) into parts in two ways. a) Write it as a sum of five unit fractions with denominator \(12\). b) Write it as two fractions with denominator \(12\), neither equal to \(0\). Then show how each part in your answer is made from unit twelfths.

Hints

- A unit twelfth is a single one of twelve equal parts. - In part b), split five copies of that part into two nonempty groups. - Expand each group to show the unit fractions inside it.

Solution

1. Five copies of \(\frac{1}{12}\) make \(\frac{5}{12}\). 2. One choice for b) is \(\frac{2}{12}+\frac{3}{12}\). 3. The first part is two unit twelfths and the second is three unit twelfths. Together they make five unit twelfths.

Answer

a) \(\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\) b) For example, \(\frac{2}{12}+\frac{3}{12}\), where \(\frac{2}{12}=\frac{1}{12}+\frac{1}{12}\) and \(\frac{3}{12}=\frac{1}{12}+\frac{1}{12}+\frac{1}{12}\).
Decompose fractions into unit sums 34
Nine identical unit fractions add to the shaded amount shown in the bar. What is the unit fraction?
Figure for problem 540601

Hints

- Read the shaded fraction from the bar. - Look for an equivalent fraction whose numerator matches the number of identical addends. - The denominator of that equivalent fraction tells the size of one unit fraction.

Solution

1. The bar shows \(3\) of \(4\) equal parts shaded, so the shaded amount is \(\frac{3}{4}\). 2. To express that amount with \(9\) equal unit-fraction terms, rename \(\frac{3}{4}\) with numerator \(9\): \(\frac{3}{4}=\frac{9}{12}\). 3. Therefore, the nine identical addends are copies of \(\frac{1}{12}\).

Answer

\(\frac{1}{12}\)
Decompose fractions into unit sums 35
What is the smallest number of identical unit fractions that can add to the shaded amount shown? Name the unit fraction and write the sum.
Figure for problem 540609

Hints

- Read the shaded fraction from the bar and consider its simplest form. - In a repeated unit-fraction sum, the numerator counts the number of identical terms. - Ask what happens to that numerator when you make an equivalent fraction with more equal parts.

Solution

1. The bar shows \(\frac{3}{4}\). 2. In simplest form, \(\frac{3}{4}\) means \(3\) copies of the unit fraction \(\frac{1}{4}\). 3. Any equivalent fraction with a different denominator would multiply the numerator \(3\) by a whole number and therefore use more than \(3\) identical unit-fraction copies. 4. The smallest possible number of terms is \(3\): \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\).

Answer

\(3\) terms: \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\)
Decompose fractions into unit sums 36
Camden writes \(\frac{3}{4}=\frac{2}{4}+\frac{1}{4}\) and calls both addends unit fractions because both have denominator \(4\). a) Explain Camden's mistake. b) Rewrite \(\frac{3}{4}\) in two valid ways using only equal-sized unit fractions within each sum.

Hints

- Check the numerator of each proposed unit fraction. - Write the original fraction as repeated fourths. - For a different sum, divide each fourth into equal smaller parts first.

Solution

1. A unit fraction must have numerator \(1\). The term \(\frac{2}{4}\) is not a unit fraction. 2. One valid sum is \(\frac{3}{4}=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\). 3. Another is \(\frac{3}{4}=\frac{6}{8}=\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\).

Answer

a) \(\frac{2}{4}\) is not a unit fraction because its numerator is not \(1\). b) \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\) and \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\).
Decompose fractions into unit sums 37
Use the bar to explain how the shaded fraction can be written as a sum of copies of one unit fraction. Then explain in words why the same idea works for any proper fraction: what does its numerator count, and what does its denominator tell you?
Figure for problem 540738

Hints

- Count all equal parts in the bar to identify the unit fraction. - Count the shaded parts to identify how many copies of that unit fraction are used. - Generalize from what the two numbers in a fraction tell you about equal parts.

Solution

1. The bar has \(8\) equal parts, so one part is \(\frac{1}{8}\). Five parts are shaded, so the shaded fraction is \(\frac{5}{8}\). 2. Therefore, \(\frac{5}{8}=\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\). 3. In any fraction, the denominator tells the size of one equal part, so it determines the unit fraction. The numerator tells how many of those equal parts are counted.

Answer

The model shows \(\frac{5}{8}=\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\). In general, the denominator names the unit fraction and the numerator tells how many copies of that unit fraction are present.
Decompose fractions into unit sums 38
Can \(\frac{2}{3}\) be written as exactly five identical unit fractions? Explain why or why not using equivalent fractions.

Hints

- If five identical unit fractions are combined, what number would count the copies in the numerator? - List a few numerators of fractions equivalent to \(\frac{2}{3}\). - Decide whether a numerator of \(5\) can occur in that pattern.

Solution

1. Five identical unit fractions would combine into one fraction whose numerator is \(5\), because there would be five equal unit-fraction copies. 2. Equivalent fractions for \(\frac{2}{3}\) are made by multiplying the numerator and denominator by the same whole number. Their numerators are \(2,4,6,8,\ldots\). 3. Since \(5\) is not one of those possible numerators, no equivalent fraction for \(\frac{2}{3}\) can represent exactly five identical unit-fraction copies.

Answer

No. \(\frac{2}{3}\) cannot be written as exactly five identical unit fractions.

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