What should be subtracted from the numbers 16, 29, 21 and 39, so that they are in proportion?

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem of what should be subtracted from the numbers 16, 29, 21, and 39 so that they are in proportion, we will follow these steps:

### Step 1: Understand the concept of proportion
Proportions mean that the ratios of the numbers are equal. For four numbers \( a, b, c, d \) to be in proportion, the following condition must hold:
\[ \frac{a}{b} = \frac{c}{d} \]
This can be rewritten as:
\[ a \cdot d = b \cdot c \]

### Step 2: Let \( x \) be the number to be subtracted
Let \( x \) be the number that we need to subtract from each of the four numbers. Therefore, the new numbers will be:
- \( 16 - x \)
- \( 29 - x \)
- \( 21 - x \)
- \( 39 - x \)

### Step 3: Set up the proportion
According to the proportion condition, we can set up the equation:
\[ (16 - x)(39 - x) = (29 - x)(21 - x) \]

### Step 4: Expand both sides
Now we will expand both sides of the equation:
- Left side:
\[ (16 - x)(39 - x) = 624 - 16x - 39x + x^2 = x^2 - 55x + 624 \]
- Right side:
\[ (29 - x)(21 - x) = 609 - 29x - 21x + x^2 = x^2 - 50x + 609 \]

### Step 5: Set the equation
Now we can set both sides equal to each other:
\[ x^2 - 55x + 624 = x^2 - 50x + 609 \]

### Step 6: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ -55x + 624 = -50x + 609 \]
Now, add \( 55x \) to both sides:
\[ 624 = 5x + 609 \]
Subtract 609 from both sides:
\[ 15 = 5x \]

### Step 7: Solve for \( x \)
Now divide both sides by 5:
\[ x = 3 \]

### Conclusion
The number that should be subtracted from 16, 29, 21, and 39 to make them proportional is **3**.