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Decompose fractions into unit sums

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5114224
Write \(\frac{3}{10}\) as a sum of unit fractions in two ways. a) Use the same unit fraction more than once. b) Use two different unit fractions.

Hints

- A unit fraction has a numerator of \(1\). - For part a), think about how many tenths make \(\frac{3}{10}\). - For part b), split \(\frac{3}{10}\) into two fractions and simplify one of them.

Solution

1. For a), the numerator tells how many copies of \(\frac{1}{10}\) are needed: \(\frac{3}{10}=\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\). 2. For b), decompose \(\frac{3}{10}\) as \(\frac{2}{10}+\frac{1}{10}\). Since \(\frac{2}{10}=\frac{1}{5}\), \(\frac{3}{10}=\frac{1}{5}+\frac{1}{10}\).

Answer

a) \(\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\) b) \(\frac{1}{5}+\frac{1}{10}\)
5106224
Write \(\frac{5}{12}\) in two different ways as a sum of two different unit fractions. A unit fraction has numerator \(1\), such as \(\frac{1}{2}\) or \(\frac{1}{10}\).

Hints

- Can you split the numerator \(5\) into two numbers that are factors of \(12\)? - What happens when you simplify a fraction such as \(\frac{4}{12}\)? - Look for two different decompositions of \(5\) that lead to unit fractions.

Solution

1. Look for ways to split \(\frac{5}{12}\) into two fractions that can each simplify to a unit fraction. 2. Since \(5=4+1\), \(\frac{5}{12}=\frac{4}{12}+\frac{1}{12}=\frac{1}{3}+\frac{1}{12}\). 3. Since \(5=3+2\), \(\frac{5}{12}=\frac{3}{12}+\frac{2}{12}=\frac{1}{4}+\frac{1}{6}\).

Answer

Two possible answers are \(\frac{1}{3}+\frac{1}{12}\) and \(\frac{1}{4}+\frac{1}{6}\).
5107054
In music, a \(\frac{4}{4}\) measure represents one whole measure. A half note has a value of \(\frac{1}{2}\), a quarter note has a value of \(\frac{1}{4}\), and an eighth note has a value of \(\frac{1}{8}\). A measure already contains one half note and one eighth note. a) What fraction of the measure is still unfilled? b) Give two different ways to fill the remaining part using only quarter notes and eighth notes.

Hints

- Think of one whole measure as eight equal eighth-note units. - Rewrite the half note in eighths. - How many eighths are missing from one whole measure? - Which combinations of quarter notes and eighth notes make that missing fraction?

Solution

1. The notes already use \(\frac{1}{2}+\frac{1}{8}=\frac{4}{8}+\frac{1}{8}=\frac{5}{8}\) of the measure. 2. For a), the unfilled part is \(1-\frac{5}{8}=\frac{3}{8}\). 3. For b), one quarter note and one eighth note fill \(\frac{2}{8}+\frac{1}{8}=\frac{3}{8}\). 4. Another way is three eighth notes: \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{8}\).

Answer

a) \(\frac{3}{8}\) b) One quarter note and one eighth note; or three eighth notes.

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