To add the fractions [tex]\( 4 \frac{2}{5} \)[/tex] and [tex]\( \frac{4}{15} \)[/tex], follow these steps:
1. Convert the mixed number [tex]\( 4 \frac{2}{5} \)[/tex] to an improper fraction:
- A mixed number consists of an integer part and a fractional part. The mixed number [tex]\( 4 \frac{2}{5} \)[/tex] has an integer part of 4 and a fractional part of [tex]\( \frac{2}{5} \)[/tex].
- Convert the integer part to a fraction: [tex]\( 4 = \frac{4 \cdot 5}{5} = \frac{20}{5} \)[/tex].
- Add the fractional part: [tex]\( \frac{20}{5} + \frac{2}{5} = \frac{22}{5} \)[/tex].
So, [tex]\( 4 \frac{2}{5} = \frac{22}{5} \)[/tex].
2. Find a common denominator for the fractions [tex]\( \frac{22}{5} \)[/tex] and [tex]\( \frac{4}{15} \)[/tex]:
- The denominators are 5 and 15. The least common multiple (LCM) of 5 and 15 is 15.
- To reach a common denominator, you should convert both fractions so that they have the same denominator.
3. Convert the fractions to have the common denominator:
- For [tex]\( \frac{22}{5} \)[/tex]:
- Multiply both the numerator and the denominator by [tex]\( 15 \div 5 = 3 \)[/tex]:
[tex]\[
\frac{22}{5} = \frac{22 \times 3}{5 \times 3} = \frac{66}{15}
\][/tex]
- [tex]\( \frac{4}{15} \)[/tex] already has the common denominator of 15.
4. Add the fractions:
- Now add the numerators:
[tex]\[
\frac{66}{15} + \frac{4}{15} = \frac{66 + 4}{15} = \frac{70}{15}
\][/tex]
5. Simplify the fraction [tex]\( \frac{70}{15} \)[/tex]:
- Find the greatest common divisor (GCD) of 70 and 15, which is 5.
- Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{70}{15} = \frac{70 \div 5}{15 \div 5} = \frac{14}{3}
\][/tex]
The final result of adding the fractions [tex]\( 4 \frac{2}{5} \)[/tex] and [tex]\( \frac{4}{15} \)[/tex] is:
[tex]\[
\boxed{\frac{14}{3}}
\][/tex]
