What number should be added to each of 7, 17, 22 and 47 so that the resulting numbers will be in proportion in the given order?

Options

A. 1
B. 3
C. 2
D. 4

Answer

To solve the problem of what number should be added to each of the numbers 7, 17, 22, and 47 so that they are in proportion, we will follow these steps:

### Step 1: Set up the proportion
We need to find a number \( x \) such that when we add \( x \) to each of the numbers, they will be in proportion. This means we want:
\[ \frac{7 + x}{17 + x} = \frac{22 + x}{47 + x} \]

### Step 2: Cross-multiply
To eliminate the fractions, we will cross-multiply:
\[ (7 + x)(47 + x) = (17 + x)(22 + x) \]

### Step 3: Expand both sides
Now, we will expand both sides of the equation:
- Left side:
\[ 7 \cdot 47 + 7x + 47x + x^2 = 329 + 54x + x^2 \]
- Right side:
\[ 17 \cdot 22 + 17x + 22x + x^2 = 374 + 39x + x^2 \]

### Step 4: Set the equation
Now we can set the expanded forms equal to each other:
\[ 329 + 54x + x^2 = 374 + 39x + x^2 \]

### Step 5: Simplify the equation
We can subtract \( x^2 \) from both sides and simplify:
\[ 329 + 54x = 374 + 39x \]

### Step 6: Rearrange the equation
Now, we will move all terms involving \( x \) to one side and constant terms to the other:
\[ 54x - 39x = 374 - 329 \]
\[ 15x = 45 \]

### Step 7: Solve for \( x \)
Now, divide both sides by 15:
\[ x = \frac{45}{15} = 3 \]

### Conclusion
The number that should be added to each of the numbers 7, 17, 22, and 47 is **3**.

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