It the same number be added to the numbers 5, 11, 15 and 31, the resulting numbers are in proportion. Find the number
a, b, c and d are in proportion `rArr =(a)/(b)=(c )/(d)`.

Answer

To solve the problem step by step, we will follow the process of setting up the equation based on the information given and solving for the unknown number.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find a number \( X \) that can be added to the numbers 5, 11, 15, and 31 such that the resulting numbers are in proportion.

2. **Setting Up the Proportion**: When the same number \( X \) is added to each of the numbers, the new numbers will be:
- \( 5 + X \)
- \( 11 + X \)
- \( 15 + X \)
- \( 31 + X \)

According to the property of proportions, we can set up the equation:
\[ \frac{5 + X}{11 + X} = \frac{15 + X}{31 + X} \]

3. **Cross Multiplying**: To eliminate the fractions, we will cross multiply:
\[ (5 + X)(31 + X) = (15 + X)(11 + X) \]

4. **Expanding Both Sides**:
- Left side:
\[ 5 \cdot 31 + 5X + 31X + X^2 = 155 + 36X + X^2 \]
- Right side:
\[ 15 \cdot 11 + 15X + 11X + X^2 = 165 + 26X + X^2 \]

5. **Setting the Equation**: Now we have:
\[ 155 + 36X + X^2 = 165 + 26X + X^2 \]

6. **Canceling \( X^2 \)**: We can subtract \( X^2 \) from both sides:
\[ 155 + 36X = 165 + 26X \]

7. **Rearranging the Equation**: Now, let's move all terms involving \( X \) to one side and constant terms to the other side:
\[ 36X - 26X = 165 - 155 \]
This simplifies to:
\[ 10X = 10 \]

8. **Solving for \( X \)**: Now, divide both sides by 10:
\[ X = 1 \]

### Final Answer:
The number that should be added to 5, 11, 15, and 31 is \( X = 1 \).