Question

what is 5/8+4/9 using benchmarks.

Answer 1

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:
- For \( \frac{5}{8} \):
\[
\frac{5 \times 9}{8 \times 9} = \frac{45}{72}
\]

- 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:
\[
\frac{77}{72} \text{ or } 1 \frac{5}{72}.
\]