Sure! Let's solve the given mathematical expression step by step:
We start with the expression:
[tex]\[
\left(\frac{24}{4} \times \frac{2}{5}\right) - \left(\frac{1}{5} \times \frac{-10}{3}\right)
\][/tex]
Step 1: Simplify each fraction inside the parentheses separately.
First part:
[tex]\[
\frac{24}{4} \times \frac{2}{5}
\][/tex]
Calculate [tex]\(\frac{24}{4}\)[/tex]:
[tex]\[
\frac{24}{4} = 6
\][/tex]
Now multiply this result by [tex]\(\frac{2}{5}\)[/tex]:
[tex]\[
6 \times \frac{2}{5} = \frac{12}{5} = 2.4
\][/tex]
So the first part simplifies to [tex]\(2.4\)[/tex].
Second part:
[tex]\[
\frac{1}{5} \times \frac{-10}{3}
\][/tex]
Multiply the numerators and the denominators:
[tex]\[
\frac{1 \times -10}{5 \times 3} = \frac{-10}{15} = -\frac{2}{3}
\][/tex]
So the second part simplifies to [tex]\(-\frac{2}{3}\)[/tex].
Step 2: Now, we subtract the second part from the first part.
We have:
[tex]\[
2.4 - \left(-\frac{2}{3}\right)
\][/tex]
Since subtracting a negative is the same as adding a positive, we get:
[tex]\[
2.4 + \frac{2}{3}
\][/tex]
Step 3: To add these fractions, we need a common denominator for the fractions [tex]\(2.4\)[/tex] and [tex]\(\frac{2}{3}\)[/tex].
Convert [tex]\(2.4\)[/tex] to a fraction:
[tex]\[
2.4 = \frac{24}{10} = \frac{12}{5}
\][/tex]
To find a common denominator, we convert [tex]\(\frac{12}{5}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex] to equivalent fractions with a common denominator. The least common multiple (LCM) of 5 and 3 is 15.
Convert:
[tex]\[
\frac{12}{5} = \frac{12 \times 3}{5 \times 3} = \frac{36}{15}
\][/tex]
And:
[tex]\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\][/tex]
Now add the fractions:
[tex]\[
\frac{36}{15} + \frac{10}{15} = \frac{36 + 10}{15} = \frac{46}{15} \approx 3.066666666666667
\][/tex]
So the final simplified result of the expression is:
[tex]\[
3.066666666666667
\][/tex]
Therefore, the simplified result of [tex]\(\left(\frac{24}{4} \times \frac{2}{5}\right) - \left(\frac{1}{5} \times \frac{-10}{3}\right)\)[/tex] is approximately [tex]\(3.066666666666667\)[/tex].
