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Add and subtract fractions with like denominators

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5106104
Calculate each expression. Write each answer in simplest form. a) \(\frac{7}{24} + \frac{5}{24} + \frac{11}{24}\) b) \(\frac{31}{45} - \frac{11}{45} + \frac{4}{45}\) c) \(\frac{19}{60} + \frac{23}{60} - \frac{7}{60}\)

Hints

- What stays the same when you add or subtract fractions with the same denominator? - Can you combine the numerators first and keep the denominator? - Check whether the numerator and denominator of each final answer have a common factor.

Solution

1. For a), the denominators are the same, so combine the numerators: \(\frac{7+5+11}{24} = \frac{23}{24}\). 2. For b), combine the numerators: \(\frac{31-11+4}{45} = \frac{24}{45} = \frac{8}{15}\). 3. For c), combine the numerators: \(\frac{19+23-7}{60} = \frac{35}{60} = \frac{7}{12}\).

Answer

a) \(\frac{23}{24}\) b) \(\frac{8}{15}\) c) \(\frac{7}{12}\)
5107064
A dotted quarter note has the time value of three eighth notes, or \(\frac{3}{8}\) of a whole note. A musician wants to fill a \(\frac{2}{4}\) measure and has already written one dotted quarter note. How many sixteenth notes, each worth \(\frac{1}{16}\), are needed to complete the measure exactly?

Hints

- Rewrite the measure and the dotted quarter note with the same denominator. - Find the fraction of the measure that is still missing. - How many unit fractions of \(\frac{1}{16}\) make that missing amount?

Solution

1. Rewrite the full measure in sixteenths: \(\frac{2}{4}=\frac{8}{16}\). 2. Rewrite the dotted quarter note: \(\frac{3}{8}=\frac{6}{16}\). 3. Find the missing part: \(\frac{8}{16}-\frac{6}{16}=\frac{2}{16}\). 4. Since each sixteenth note represents \(\frac{1}{16}\), exactly \(2\) sixteenth notes are needed.

Answer

\(2\) sixteenth notes are needed.
5317424
Two ribbons, \(D\) and \(E\), are shown on equally scaled number lines. Each interval between whole numbers is divided into eighths. a) Find the length of ribbon \(D\). b) Find the length of ribbon \(E\). c) Which ribbon is longer, and by how much? d) Find the total length of both ribbons.
Figure for problem 531742

Hints

- Each small step represents \(\frac{1}{8}\). - Subtract the starting value from the ending value for each ribbon. - Use eighths to compare, subtract, and add the lengths.

Solution

1. Ribbon \(D\) runs from \(\frac{2}{8}\) to \(\frac{11}{8}\), so its length is \(\frac{11}{8}-\frac{2}{8}=\frac{9}{8}=1\frac{1}{8}\). 2. Ribbon \(E\) runs from \(\frac{5}{8}\) to \(\frac{15}{8}\), so its length is \(\frac{15}{8}-\frac{5}{8}=\frac{10}{8}=\frac{5}{4}=1\frac{1}{4}\). 3. Since \(1\frac{1}{4}=1\frac{2}{8}\), ribbon \(E\) is longer by \(\frac{1}{8}\). 4. The total length is \(1\frac{1}{8}+1\frac{1}{4}=1\frac{1}{8}+1\frac{2}{8}=2\frac{3}{8}\).

Answer

a) \(1\frac{1}{8}\) b) \(1\frac{1}{4}\) c) Ribbon \(E\) is longer by \(\frac{1}{8}\). d) \(2\frac{3}{8}\)
5317694
Points \(A\) and \(B\) are marked on the number line. a) What values are marked by \(A\) and \(B\)? b) Find the distance between the points. Write the result as a fraction in simplest form or as a mixed number.
Figure for problem 531769

Hints

- Count the equal intervals between consecutive whole numbers. - Read each point by counting steps from \(0\). - Subtract the smaller value from the larger value to find the distance.

Solution

1. Each interval between whole numbers is divided into thirds, so one step is \(\frac{1}{3}\). 2. Point \(A\) is at \(\frac{2}{3}\), and point \(B\) is at \(2\frac{1}{3}=\frac{7}{3}\). 3. The distance is \(\frac{7}{3}-\frac{2}{3}=\frac{5}{3}=1\frac{2}{3}\).

Answer

a) \(A=\frac{2}{3}\), \(B=2\frac{1}{3}\) b) \(1\frac{2}{3}\)

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