Question

Estimate the sum by rounding each mixed number to the nearest whole number then find the exact sum

11 3/8+9 7/10+21 1/4

Answer 1

To estimate the sum by rounding each mixed number to the nearest whole number, we can follow these steps:

1. **Round each mixed number:**
- \( 11 \frac{3}{8} \): The fractional part \( \frac{3}{8} \) is less than \( \frac{1}{2} \), so we round down to \( 11 \).
- \( 9 \frac{7}{10} \): The fractional part \( \frac{7}{10} \) is greater than \( \frac{1}{2} \), so we round up to \( 10 \).
- \( 21 \frac{1}{4} \): The fractional part \( \frac{1}{4} \) is less than \( \frac{1}{2} \), so we round down to \( 21 \).

Now we add the rounded values:
\[
11 + 10 + 21 = 42
\]

### Exact Sum Calculation:
Now let's calculate the exact sum of the mixed numbers:
\[
11 \frac{3}{8} + 9 \frac{7}{10} + 21 \frac{1}{4}
\]

First, convert each mixed number into an improper fraction:

1. \( 11 \frac{3}{8} = \frac{11 \times 8 + 3}{8} = \frac{88 + 3}{8} = \frac{91}{8} \)
2. \( 9 \frac{7}{10} = \frac{9 \times 10 + 7}{10} = \frac{90 + 7}{10} = \frac{97}{10} \)
3. \( 21 \frac{1}{4} = \frac{21 \times 4 + 1}{4} = \frac{84 + 1}{4} = \frac{85}{4} \)

Now, we need a common denominator to add these fractions. The least common multiple of \( 8, 10, \) and \( 4 \) is \( 40 \).

Convert each fraction to have a denominator of \( 40 \):

1. \( \frac{91}{8} = \frac{91 \times 5}{8 \times 5} = \frac{455}{40} \)
2. \( \frac{97}{10} = \frac{97 \times 4}{10 \times 4} = \frac{388}{40} \)
3. \( \frac{85}{4} = \frac{85 \times 10}{4 \times 10} = \frac{850}{40} \)

Now, add these fractions:
\[
\frac{455}{40} + \frac{388}{40} + \frac{850}{40} = \frac{455 + 388 + 850}{40} = \frac{1693}{40}
\]

To convert this back to a mixed number:
- Divide \( 1693 \) by \( 40 \):
- \( 1693 \div 40 = 42 \) with a remainder \((1693 - 40 \times 42)\).
- \( 1693 - 1680 = 13 \)

So, \( 1693 \div 40 = 42 \frac{13}{40} \).

Thus, the **exact sum** is:
\[
\frac{1693}{40} \text{ or } 42 \frac{13}{40}
\]

### Summary:
- Estimated sum: **42**
- Exact sum: **42 \frac{13}{40}** or **\(\frac{1693}{40}\)**.