What should be added to `-(3)/(8)` to get `(5)/(6)` ?

Options

A. `(29)/(24)`
B. `(24)/(29)`
C. `(27)/(33)`
D. `(39)/(41)`

Answer

To solve the problem of what should be added to \(-\frac{3}{8}\) to get \(\frac{5}{6}\), we can follow these steps:

### Step 1: Set up the equation
We want to find a number \(x\) such that:
\[ -\frac{3}{8} + x = \frac{5}{6} \]

### Step 2: Isolate \(x\)
To find \(x\), we can rearrange the equation:
\[ x = \frac{5}{6} + \frac{3}{8} \]

### Step 3: Find the LCM of the denominators
The denominators are 6 and 8. We need to find the least common multiple (LCM) of these two numbers.

- The prime factorization of 6 is \(2 \times 3\).
- The prime factorization of 8 is \(2^3\).

The LCM will take the highest power of each prime:
- For 2, the highest power is \(2^3\).
- For 3, the highest power is \(3^1\).

Thus, the LCM of 6 and 8 is:
\[ LCM = 2^3 \times 3^1 = 8 \times 3 = 24 \]

### Step 4: Convert each fraction to have the common denominator
Now, we convert \(\frac{5}{6}\) and \(\frac{3}{8}\) to have a denominator of 24.

- For \(\frac{5}{6}\):
\[ \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24} \]

- For \(\frac{3}{8}\):
\[ \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \]

### Step 5: Add the fractions
Now we can add the two fractions:
\[ x = \frac{20}{24} + \frac{9}{24} = \frac{20 + 9}{24} = \frac{29}{24} \]

### Step 6: Conclusion
Thus, the number that should be added to \(-\frac{3}{8}\) to get \(\frac{5}{6}\) is:
\[ \boxed{\frac{29}{24}} \]

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