Let's solve the given expression step-by-step: [tex]\(\left(\frac{4}{5} - \frac{3}{7}\right) \times 1 \frac{2}{5}\)[/tex].
1. Convert the mixed number to an improper fraction:
- The mixed number is [tex]\(1 \frac{2}{5}\)[/tex].
- The whole number is 1.
- The fraction part is [tex]\(\frac{2}{5}\)[/tex].
- To convert a mixed number to an improper fraction:
[tex]\[
1 \frac{2}{5} = \frac{1 \times 5 + 2}{5} = \frac{5 + 2}{5} = \frac{7}{5}
\][/tex]
- So, [tex]\(1 \frac{2}{5} = \frac{7}{5}\)[/tex].
2. Perform the fraction subtraction:
- The fractions to subtract are [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{3}{7}\)[/tex].
- To subtract fractions, they must have a common denominator.
- The denominators are 5 and 7. The common denominator is their product, [tex]\(5 \times 7 = 35\)[/tex].
- Convert each fraction to have the common denominator:
[tex]\[
\frac{4}{5} = \frac{4 \times 7}{5 \times 7} = \frac{28}{35}
\][/tex]
[tex]\[
\frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35}
\][/tex]
- Subtract the numerators:
[tex]\[
\frac{28}{35} - \frac{15}{35} = \frac{28 - 15}{35} = \frac{13}{35}
\][/tex]
3. Multiply the result by the improper fraction:
- The result from the subtraction is [tex]\(\frac{13}{35}\)[/tex].
- We need to multiply this by [tex]\(\frac{7}{5}\)[/tex]:
[tex]\[
\frac{13}{35} \times \frac{7}{5} = \frac{13 \times 7}{35 \times 5} = \frac{91}{175}
\][/tex]
4. Simplify the resulting fraction:
- To simplify [tex]\(\frac{91}{175}\)[/tex], we need to find the greatest common divisor (GCD) of 91 and 175.
- The GCD of 91 and 175 is 7.
- Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{91 \div 7}{175 \div 7} = \frac{13}{25}
\][/tex]
Therefore, the simplified result of the expression [tex]\(\left(\frac{4}{5} - \frac{3}{7} \right) \times 1 \frac{2}{5}\)[/tex] is [tex]\(\frac{13}{25}\)[/tex].
