To solve the problem of adding [tex]\(10 \frac{3}{7}\)[/tex] and [tex]\(19 \frac{5}{9}\)[/tex], follow these detailed steps:
1. Convert the mixed numbers to improper fractions:
- For [tex]\(10 \frac{3}{7}\)[/tex], first convert the integer part to a fraction:
[tex]\[
10 \frac{3}{7} = 10 + \frac{3}{7}
\][/tex]
To add these, convert 10 into a fraction with a denominator of 7:
[tex]\[
10 = \frac{70}{7}
\][/tex]
Now add the fractions:
[tex]\[
10 \frac{3}{7} = \frac{70}{7} + \frac{3}{7} = \frac{70 + 3}{7} = \frac{73}{7}
\][/tex]
- For [tex]\(19 \frac{5}{9}\)[/tex], convert the integer part to a fraction:
[tex]\[
19 \frac{5}{9} = 19 + \frac{5}{9}
\][/tex]
Convert 19 into a fraction with a denominator of 9:
[tex]\[
19 = \frac{171}{9}
\][/tex]
Add the fractions:
[tex]\[
19 \frac{5}{9} = \frac{171}{9} + \frac{5}{9} = \frac{176}{9}
\][/tex]
2. Add the improper fractions:
[tex]\[
\frac{73}{7} + \frac{176}{9}
\][/tex]
To add these fractions, find a common denominator. The least common multiple (LCM) of 7 and 9 is 63.
- Convert each fraction to have this common denominator:
[tex]\[
\frac{73}{7} = \frac{73 \times 9}{7 \times 9} = \frac{657}{63}
\][/tex]
[tex]\[
\frac{176}{9} = \frac{176 \times 7}{9 \times 7} = \frac{1232}{63}
\][/tex]
Now add the numerators together:
[tex]\[
\frac{657}{63} + \frac{1232}{63} = \frac{657 + 1232}{63} = \frac{1889}{63}
\][/tex]
3. Convert the result back to a mixed number:
- Divide [tex]\(1889\)[/tex] by [tex]\(63\)[/tex] to find the whole number and the fraction:
[tex]\[
1889 \div 63 \approx 29.984126984126988
\][/tex]
This implies that the mixed number will have 29 as the integer part, and the fractional part is calculated as:
[tex]\[
1889 - (29 \times 63) = 1889 - 1827 = 62
\][/tex]
So the fraction part is:
[tex]\[
\frac{62}{63}
\][/tex]
4. Simplify if needed:
- The fraction [tex]\(\frac{62}{63}\)[/tex] is already in its simplest form since 62 and 63 are coprime (i.e., their greatest common divisor is 1).
5. Write the final answer:
The final result of the addition [tex]\(10 \frac{3}{7} + 19 \frac{5}{9}\)[/tex] is:
[tex]\[
29 \frac{62}{63}
\][/tex]
So, [tex]\( 10 \frac{3}{7} + 19 \frac{5}{9} = 29 \frac{62}{63} \)[/tex].
