Let's solve the problem step by step.
### Step 1: Estimate the answer
To estimate the sum of the mixed numbers \( 15 \frac{3}{4} \) and \( 10 \frac{2}{7} \), we can round them to the nearest whole number:
- \( 15 \frac{3}{4} \) is close to 16.
- \( 10 \frac{2}{7} \) is close to 10.
So, an estimate for the sum is:
[tex]\[ 16 + 10 = 26 \][/tex]
### Step 2: Convert mixed numbers to improper fractions
For \( 15 \frac{3}{4} \):
- The whole number part is 15.
- The fractional part is \( \frac{3}{4} \).
- Convert to an improper fraction: \( 15 \times 4 + 3 = 60 + 3 = 63 \).
- So, \( 15 \frac{3}{4} = \frac{63}{4} \).
For \( 10 \frac{2}{7} \):
- The whole number part is 10.
- The fractional part is \( \frac{2}{7} \).
- Convert to an improper fraction: \( 10 \times 7 + 2 = 70 + 2 = 72 \).
- So, \( 10 \frac{2}{7} = \frac{72}{7} \).
### Step 3: Find a common denominator
The denominators are 4 and 7. The least common multiple (LCM) of 4 and 7 is 28.
### Step 4: Adjust the numerators
Convert the fractions to have a common denominator of 28:
- With \(\frac{63}{4}\):
[tex]\[
\frac{63 \times 7}{4 \times 7} = \frac{441}{28}
\][/tex]
- With \(\frac{72}{7}\):
[tex]\[
\frac{72 \times 4}{7 \times 4} = \frac{288}{28}
\][/tex]
### Step 5: Add the numerators
Add the numerators of the fractions:
[tex]\[
441 + 288 = 729
\][/tex]
So, the result is:
[tex]\[
\frac{729}{28}
\][/tex]
### Step 6: Simplify the fraction and convert back to a mixed number
1. Simplify the fraction:
- The greatest common divisor (GCD) of 729 and 28 is 1 (since they don't share any common factors other than 1), so the fraction is already in simplest form.
2. Convert to a mixed number:
- Divide 729 by 28 to find the whole number part:
[tex]\[
729 \div 28 \approx 26.
\][/tex]
- The remainder is \( 729 - (26 \times 28) = 729 - 728 = 1 \).
So, the mixed number is:
[tex]\[
26 \frac{1}{28}
\][/tex]
### Final Answer:
The exact sum of [tex]\( 15 \frac{3}{4} \)[/tex] and [tex]\( 10 \frac{2}{7} \)[/tex] is [tex]\( 26 \frac{1}{28} \)[/tex].
