In a bag of jellybeans, `(1)/(3)` are cherry and `(1)/(4)` are licorice . If the remainning 20 jellybeans are orange, how many jellybeans are in the bag?

Options

A. `16`
B. `32`
C. `36`
D. `48`

Answer

To find the total number of jellybeans in the bag, we can set up an equation based on the information given in the problem. Let's go through the steps to solve it.

### Step-by-Step Solution:

1. **Define the total number of jellybeans**:
Let \( X \) be the total number of jellybeans in the bag.

2. **Determine the number of cherry and licorice jellybeans**:
- The number of cherry jellybeans is \( \frac{1}{3}X \).
- The number of licorice jellybeans is \( \frac{1}{4}X \).

3. **Calculate the total number of jellybeans that are either cherry or licorice**:
To find the total number of jellybeans that are cherry and licorice, we add the two fractions:
\[ \text{Total cherry and licorice} = \frac{1}{3}X + \frac{1}{4}X \]
To add these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12. Thus, we convert the fractions:
\[ \frac{1}{3}X = \frac{4}{12}X \quad \text{and} \quad \frac{1}{4}X = \frac{3}{12}X \]
Now we can add them:
\[ \frac{4}{12}X + \frac{3}{12}X = \frac{7}{12}X \]

4. **Set up the equation**:
The problem states that the remaining jellybeans are orange, and there are 20 of them. Therefore, we can write the equation:
\[ X - \left(\frac{7}{12}X\right) = 20 \]

5. **Simplify the equation**:
Simplifying the left side gives:
\[ \frac{5}{12}X = 20 \]

6. **Solve for \( X \)**:
To find \( X \), multiply both sides by \( \frac{12}{5} \):
\[ X = 20 \times \frac{12}{5} \]
\[ X = 20 \times 2.4 = 48 \]

7. **Conclusion**:
The total number of jellybeans in the bag is \( 48 \).

### Final Answer:
There are **48 jellybeans** in the bag.