Problem 33 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$4\left(\frac{3}{5}-\frac{8}{15}\right)+9\left(\frac{1}{3}+\frac{1}{4}\right)$$

Step-by-Step Solution

Answer

Question: Calculate the value of the expression: $$4\left(\frac{3}{5}-\frac{8}{15}\right)+9\left(\frac{1}{3}+\frac{1}{4}\right)$$ Answer: The value of the expression is $$\frac{331}{60}$$.

Step 1: Simplify the expressions inside the parentheses

Calculate the values inside the parentheses: $$\frac{3}{5}-\frac{8}{15}\longrightarrow\text{Find a common denominator (15)}\longrightarrow\frac{3\cdot3}{5\cdot3}-\frac{8}{15}\longrightarrow\frac{9}{15}-\frac{8}{15}$$ $$\frac{1}{3}+\frac{1}{4}\longrightarrow\text{Find a common denominator (12)}\longrightarrow\frac{1\cdot4}{3\cdot4}+\frac{1\cdot3}{4\cdot3}\longrightarrow\frac{4}{12}+\frac{3}{12}$$

Step 2: Add/subtract the fractions inside the parentheses

Now, add or subtract the fractions inside the parentheses: $$\frac{9}{15}-\frac{8}{15}=\frac{1}{15}$$ $$\frac{4}{12}+\frac{3}{12}=\frac{7}{12}$$ So now we have the simplified expression: $$4\left(\frac{1}{15}\right)+9\left(\frac{7}{12}\right)$$

Step 3: Perform multiplication

Multiply the numbers by the respective fractions: $$4\cdot\frac{1}{15}=\frac{4}{15}$$ $$9\cdot\frac{7}{12}=\frac{63}{12}$$ Now, we have the simplified expression: $$\frac{4}{15}+\frac{63}{12}$$

Step 4: Add the two fractions

Find a common denominator (60) and add the two fractions: $$\frac{4}{15}+\frac{63}{12}\longrightarrow\frac{4\cdot4}{15\cdot4}+\frac{63\cdot5}{12\cdot5}\longrightarrow\frac{16}{60}+\frac{315}{60}$$ Now, add the two fractions: $$\frac{16}{60}+\frac{315}{60}=\frac{331}{60}$$

Step 5: Simplify the final result (if possible)

In this case, there is no common factor between 331 and 60, so the final answer cannot be simplified further: $$4\left(\frac{3}{5}-\frac{8}{15}\right)+9\left(\frac{1}{3}+\frac{1}{4}\right)=\frac{331}{60}$$