Question

Estimate the product. Then find the exact product.
{2 5/8+1/4}X{5 1/2 -1/8}

Answer 1

To solve the expression \((2 \frac{5}{8} + \frac{1}{4}) \times (5 \frac{1}{2} - \frac{1}{8})\), we can first estimate the product by rounding the mixed numbers to the nearest whole numbers.

### Estimation

1. Round \(2 \frac{5}{8}\) to \(3\) (since \(2 \frac{5}{8} \approx 3\)).
2. Round \( \frac{1}{4} \) to \(0\) (minimal effect on the sum).
3. Round \(5 \frac{1}{2}\) to \(6\) (since \(5 \frac{1}{2} \approx 6\)).
4. Round \(-\frac{1}{8}\) to \(0\) (minimal effect on the subtraction).

Now the estimation becomes:
\[
3 \times 6 = 18
\]

### Exact Calculation

Now let's compute the exact values step by step.

1. **Calculate \(2 \frac{5}{8} + \frac{1}{4}\)**:
- Convert \(2 \frac{5}{8}\) to an improper fraction:
\[
2 \frac{5}{8} = \frac{16}{8} + \frac{5}{8} = \frac{21}{8}
\]
- Convert \(\frac{1}{4}\) to a fraction with a denominator of \(8\):
\[
\frac{1}{4} = \frac{2}{8}
\]
- Now add them together:
\[
\frac{21}{8} + \frac{2}{8} = \frac{23}{8}
\]

2. **Calculate \(5 \frac{1}{2} - \frac{1}{8}\)**:
- Convert \(5 \frac{1}{2}\) to an improper fraction:
\[
5 \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2}
\]
- Convert \(\frac{11}{2}\) to a fraction with a denominator of \(8\):
\[
\frac{11}{2} = \frac{44}{8}
\]
- Now subtract:
\[
\frac{44}{8} - \frac{1}{8} = \frac{43}{8}
\]

3. **Multiply the results**:
\[
\left(\frac{23}{8}\right) \times \left(\frac{43}{8}\right) = \frac{23 \times 43}{8 \times 8}
\]
- Calculate \(23 \times 43\):
\[
23 \times 43 = 989
\]
- Calculate \(8 \times 8\):
\[
8 \times 8 = 64
\]
- Therefore:
\[
\frac{989}{64}
\]

Thus, the final answer is:
\[
\boxed{\frac{989}{64}}
\]