What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional ?

Answer

To find the number that must be added to each of the numbers 6, 15, 20, and 43 to make them proportional, we can follow these steps:

### Step 1: Define the variable
Let \( x \) be the number that we need to add to each of the numbers.

### Step 2: Write the new numbers
After adding \( x \), the numbers become:
- \( 6 + x \)
- \( 15 + x \)
- \( 20 + x \)
- \( 43 + x \)

### Step 3: Set up the proportion
For the numbers to be proportional, the following condition must hold:
\[ \frac{6 + x}{15 + x} = \frac{20 + x}{43 + x} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ (6 + x)(43 + x) = (20 + x)(15 + x) \]

### Step 5: Expand both sides
Expanding both sides:
- Left side:
\[ 6 \cdot 43 + 6x + 43x + x^2 = 258 + 49x + x^2 \]
- Right side:
\[ 20 \cdot 15 + 20x + 15x + x^2 = 300 + 35x + x^2 \]

### Step 6: Simplify the equation
Now we have:
\[ 258 + 49x + x^2 = 300 + 35x + x^2 \]
We can cancel \( x^2 \) from both sides:
\[ 258 + 49x = 300 + 35x \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 49x - 35x = 300 - 258 \]
\[ 14x = 42 \]

### Step 8: Solve for \( x \)
Dividing both sides by 14:
\[ x = \frac{42}{14} = 3 \]

### Conclusion
The number that must be added to each of the numbers 6, 15, 20, and 43 to make them proportional is \( \boxed{3} \).

### Verification
To verify, we can substitute \( x = 3 \) back into the new numbers:
- \( 6 + 3 = 9 \)
- \( 15 + 3 = 18 \)
- \( 20 + 3 = 23 \)
- \( 43 + 3 = 46 \)

Now checking the proportions:
\[ \frac{9}{18} = \frac{1}{2} \quad \text{and} \quad \frac{23}{46} = \frac{1}{2} \]
Both ratios are equal, confirming that our solution is correct.