To add the fractions \( \frac{5}{8} \) and \( \frac{4}{9} \) using benchmarks, we'll start by estimating each fraction and then proceed with the actual addition.
Step 1: Identify benchmarks.
- \( \frac{5}{8} \) is close to \( \frac{1}{2} \) (which is \( \frac{4}{8} \)) and just above \( \frac{3}{4} \) (which is \( \frac{6}{8} \)). So, we can approximate \( \frac{5}{8} \) as a little more than \( 0.5 \) and less than \( 0.75 \).
- \( \frac{4}{9} \) is close to \( \frac{1}{2} \) (which is \( \frac{4.5}{9} \)). So, we can also approximate \( \frac{4}{9} \) as a little less than \( 0.5 \).
Step 2: Estimate the sum using benchmarks.
- Since \( \frac{5}{8} \) is a little more than \( 0.5 \) (around \( 0.625 \)) and \( \frac{4}{9} \) is a little less than \( 0.5 \) (around \( 0.444 \)), the combined estimate would be slightly below \( 1.0 \).
Step 3: Find a common denominator and add. The least common denominator (LCD) of \( 8 \) and \( 9 \) is \( 72 \).
Convert both fractions to have the same denominator:
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For \( \frac{5}{8} \): \[ \frac{5 \times 9}{8 \times 9} = \frac{45}{72} \]
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For \( \frac{4}{9} \): \[ \frac{4 \times 8}{9 \times 8} = \frac{32}{72} \]
Step 4: Add the fractions. Now we can add the two fractions with the same denominator: \[ \frac{45}{72} + \frac{32}{72} = \frac{45 + 32}{72} = \frac{77}{72} \]
Step 5: Simplify if necessary. The fraction \( \frac{77}{72} \) is an improper fraction and can also be expressed as: \[ 1 \frac{5}{72} \]
Therefore, the sum of \( \frac{5}{8} + \frac{4}{9} \) is: \[ \frac{77}{72} \text{ or } 1 \frac{5}{72}. \]
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