What should be added to `((3)/(4) + (2)/(5)) ` to get `(-8)/(15)` ?

Options

A. `(-103)/(60)`
B. `(103)/(60)`
C. `(-101)/(60)`
D. `(101)/(60)`

Answer

To find out what should be added to \(\frac{3}{4} + \frac{2}{5}\) to get \(-\frac{8}{15}\), we can follow these steps:

### Step 1: Calculate \(\frac{3}{4} + \frac{2}{5}\)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 5 is 20.

Convert each fraction:
- \(\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}\)
- \(\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}\)

Now, add the two fractions:
\[ \frac{15}{20} + \frac{8}{20} = \frac{15 + 8}{20} = \frac{23}{20} \]

### Step 2: Set up the equation

Let \(x\) be the number we need to add. We set up the equation:
\[ \frac{23}{20} + x = -\frac{8}{15} \]

### Step 3: Solve for \(x\)

Rearranging the equation gives:
\[ x = -\frac{8}{15} - \frac{23}{20} \]

### Step 4: Find a common denominator for the fractions

The LCM of 15 and 20 is 60. Convert both fractions:
- \(-\frac{8}{15} = -\frac{8 \times 4}{15 \times 4} = -\frac{32}{60}\)
- \(-\frac{23}{20} = -\frac{23 \times 3}{20 \times 3} = -\frac{69}{60}\)

### Step 5: Add the fractions

Now, we can add:
\[ x = -\frac{32}{60} - \frac{69}{60} = -\frac{32 + 69}{60} = -\frac{101}{60} \]

### Final Answer

Thus, the number that should be added is:
\[ x = -\frac{101}{60} \]

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