Question Simplify the expression $\frac{2}{3}\left(\frac{3}{4} x+1 \frac{1}{8}\right)+\frac{4}{5}\left(2 \frac{1}{2} x+15\right)$ .

Studdy Solution

STEP 1

Assumptions
1. We need to simplify the given expression.
2. The expression is $\frac{2}{3}\left(\frac{3}{4} x+1 \frac{1}{8}\right)+\frac{4}{5}\left(2 \frac{1}{2} x+15\right)$ .
3. We will follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

STEP 2

First, convert the mixed numbers to improper fractions to simplify the calculations.
$1 \frac{1}{8} = \frac{8}{8} + \frac{1}{8} = \frac{9}{8}$ $2 \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$

STEP 3

Replace the mixed numbers in the original expression with the improper fractions.
$\frac{2}{3}\left(\frac{3}{4} x + \frac{9}{8}\right) + \frac{4}{5}\left(\frac{5}{2} x + 15\right)$

STEP 4

Distribute the fractions outside the parentheses across the terms inside the parentheses.
$\frac{2}{3} \cdot \frac{3}{4} x + \frac{2}{3} \cdot \frac{9}{8} + \frac{4}{5} \cdot \frac{5}{2} x + \frac{4}{5} \cdot 15$

STEP 5

Simplify the coefficients by multiplying the fractions.
$\frac{2}{3} \cdot \frac{3}{4} = \frac{2 \cdot 3}{3 \cdot 4} = \frac{1 \cdot 3}{1 \cdot 4} = \frac{3}{4}$ $\frac{2}{3} \cdot \frac{9}{8} = \frac{2 \cdot 9}{3 \cdot 8} = \frac{1 \cdot 3}{1 \cdot 4} = \frac{3}{4}$ $\frac{4}{5} \cdot \frac{5}{2} = \frac{4 \cdot 5}{5 \cdot 2} = \frac{1 \cdot 4}{1 \cdot 2} = \frac{4}{2} = 2$ $\frac{4}{5} \cdot 15 = \frac{4 \cdot 15}{5} = \frac{1 \cdot 15}{1} = 15 \cdot 1 = 15$

STEP 6

Substitute the simplified coefficients back into the expression.
$\frac{3}{4} x + \frac{3}{4} + 2 x + 15$

STEP 7

Combine like terms by adding the coefficients of $x$ and the constant terms separately.
$\left(\frac{3}{4} + 2\right) x + \left(\frac{3}{4} + 15\right)$

STEP 8

Convert the whole number 2 to a fraction with the same denominator as $\frac{3}{4}$ to add the fractions.
$2 = \frac{2 \cdot 4}{4} = \frac{8}{4}$

STEP 9

Add the fractions with the common denominator.
$\frac{3}{4} + \frac{8}{4} = \frac{3 + 8}{4} = \frac{11}{4}$

STEP 10

Add the constant terms. Since 15 is a whole number, we can write it as $\frac{15 \cdot 4}{4}$ to have a common denominator with $\frac{3}{4}$ .
$\frac{3}{4} + 15 = \frac{3}{4} + \frac{60}{4} = \frac{3 + 60}{4} = \frac{63}{4}$

STEP 11

Substitute the results from STEP_9 and STEP_10 back into the expression.
$\frac{11}{4} x + \frac{63}{4}$

STEP 12

The expression is now simplified to:
$\frac{11}{4} x + \frac{63}{4}$
This is the simplified form of the original expression.

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