To find the total distance Richard rides his bike along both Union Street and Broad Street, we need to add the lengths of the two streets together.
- Union Street: \(5 \frac{5}{18}\) miles
- Broad Street: \(5 \frac{3}{4}\) miles
First, we can convert the mixed numbers into improper fractions:
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For Union Street: \[ 5 \frac{5}{18} = \frac{5 \times 18 + 5}{18} = \frac{90 + 5}{18} = \frac{95}{18} \]
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For Broad Street: \[ 5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4} \]
Next, we need a common denominator to add these two fractions. The denominators are 18 and 4. The least common multiple of 18 and 4 is 36.
Now, we convert both fractions to have a denominator of 36:
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For \( \frac{95}{18} \): \[ \frac{95}{18} = \frac{95 \times 2}{18 \times 2} = \frac{190}{36} \]
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For \( \frac{23}{4} \): \[ \frac{23}{4} = \frac{23 \times 9}{4 \times 9} = \frac{207}{36} \]
Now we can add the two fractions: \[ \frac{190}{36} + \frac{207}{36} = \frac{190 + 207}{36} = \frac{397}{36} \]
Now, we can convert this improper fraction back into a mixed number. To do this, divide 397 by 36: \[ 397 \div 36 = 11 , \text{remainder} , 1 \]
Thus, we can express this as: \[ \frac{397}{36} = 11 \frac{1}{36} \]
Therefore, the total distance Richard rides is \( \boxed{11 \frac{1}{36}} \) miles.
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