By what rational number should the sum of 18/5 and -7/15 be divided to get 47/6?2SEE ANSWERS

1. Break Down the Problem

To find the rational number $x$ such that: $\frac{\frac{18}{5} + \left(-\frac{7}{15}\right)}{x} = \frac{47}{6}$

2. Relevant Concepts

First, we need to simplify the sum $\frac{18}{5} + \left(-\frac{7}{15}\right)$. We will use the formula for adding fractions: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

3. Analysis and Detail

Step 1: Find a common denominator for $\frac{18}{5}$ and $-\frac{7}{15}$.

The least common multiple of 5 and 15 is 15.

Convert $\frac{18}{5}$ to fifteenths: $\frac{18}{5} = \frac{18 \times 3}{5 \times 3} = \frac{54}{15}$

Now add $\frac{54}{15}$ and $-\frac{7}{15}$: $\frac{54}{15} + \left(-\frac{7}{15}\right) = \frac{54 - 7}{15} = \frac{47}{15}$

Step 2: Set up the equation to solve for $x$: $\frac{\frac{47}{15}}{x} = \frac{47}{6}$

Step 3: Cross-multiply to eliminate the fractions: $47 \cdot 6 = 47 \cdot 15 \cdot x$ $282 = 47 \cdot 15 \cdot x$

4. Verify and Summarize

Step 4: Calculate $47 \cdot 15$: $47 \cdot 15 = 705$ Now substitute back into the equation: $282 = 705x$

Step 5: Solve for $x$: $x = \frac{282}{705}$ Now simplify $\frac{282}{705}$: The greatest common divisor (GCD) of 282 and 705 is 141. $x = \frac{282 \div 141}{705 \div 141} = \frac{2}{5}$

Final Answer

The rational number is: $\frac{2}{5}$