What number should be subtracted from each of the numbers 54,71,75and 99 so that the remainders may be proportional

Options

A. 1
B. 2
C. 3
D. 6

Answer

To solve the problem of finding what number should be subtracted from each of the numbers 54, 71, 75, and 99 so that the remainders are proportional, we can follow these steps:

### Step 1: Understand Proportionality
We need to find a number \( x \) such that when subtracted from each of the numbers, the ratios of the remainders are equal. This means:
\[ \frac{54 - x}{71 - x} = \frac{75 - x}{99 - x} \]

### Step 2: Cross Multiply
To eliminate the fractions, we can cross-multiply:
\[ (54 - x)(99 - x) = (71 - x)(75 - x) \]

### Step 3: Expand Both Sides
Now, we will expand both sides of the equation:
- Left side:
\[ 54 \times 99 - 54x - 99x + x^2 = 5346 - 153x + x^2 \]
- Right side:
\[ 71 \times 75 - 71x - 75x + x^2 = 5325 - 146x + x^2 \]

### Step 4: Set the Equation
Now we can set the two expanded sides equal to each other:
\[ 5346 - 153x + x^2 = 5325 - 146x + x^2 \]

### Step 5: Simplify the Equation
We can subtract \( x^2 \) from both sides and simplify:
\[ 5346 - 153x = 5325 - 146x \]

### Step 6: Rearranging Terms
Rearranging the equation gives:
\[ 5346 - 5325 = 153x - 146x \]
\[ 21 = 7x \]

### Step 7: Solve for \( x \)
Dividing both sides by 7 gives:
\[ x = 3 \]

### Conclusion
The number that should be subtracted from each of the numbers 54, 71, 75, and 99 is **3**.

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