Solving the Fraction Expression

This solution explains how to find the value of the given mathematical expression involving fractions:

$ \frac{5}{2} + \frac{1}{(1 + \frac{3}{8})} - \frac{3}{5} $

Step-by-Step Calculation

Step 1: Simplify the term in the parenthesis

First, we need to simplify the denominator of the second term, which is $1 + \frac{3}{8}$. To add these, we find a common denominator, which is 8.

$ 1 + \frac{3}{8} = \frac{8}{8} + \frac{3}{8} = \frac{8 + 3}{8} = \frac{11}{8} $

Step 2: Simplify the second term

Now, substitute the simplified denominator back into the second term:

$ \frac{1}{(1 + \frac{3}{8})} = \frac{1}{\frac{11}{8}} $

Dividing 1 by a fraction is the same as taking the reciprocal of the fraction:

$ \frac{1}{\frac{11}{8}} = \frac{8}{11} $

Step 3: Rewrite the expression

The original expression now simplifies to:

$ \frac{5}{2} + \frac{8}{11} - \frac{3}{5} $

Step 4: Find a common denominator

To add and subtract these fractions, we need a common denominator. The least common multiple (LCM) of the denominators 2, 11, and 5 is $2 \times 11 \times 5 = 110$.

Step 5: Convert fractions to the common denominator

Convert each fraction so it has the denominator 110:

• For $\frac{5}{2}$: Multiply the numerator and denominator by 55.

  $ \frac{5}{2} = \frac{5 \times 55}{2 \times 55} = \frac{275}{110} $

• For $\frac{8}{11}$: Multiply the numerator and denominator by 10.

  $ \frac{8}{11} = \frac{8 \times 10}{11 \times 10} = \frac{80}{110} $

• For $\frac{3}{5}$: Multiply the numerator and denominator by 22.

  $ \frac{3}{5} = \frac{3 \times 22}{5 \times 22} = \frac{66}{110} $

Step 6: Perform the addition and subtraction

Finally, perform the operations using the fractions with the common denominator:

$ \frac{275}{110} + \frac{80}{110} - \frac{66}{110} = \frac{275 + 80 - 66}{110} $

Combine the numerators:

$ \frac{(275 + 80) - 66}{110} = \frac{355 - 66}{110} $

Calculate the final numerator:

$ \frac{289}{110} $

Thus, the value of the expression is $\frac{289}{110}$.