What number should be added to or subtracted from each term of the ratio `17 : 24` so that it becomes equal to `1 : 2` ?

Options

A. 5 is subtracted
B. 10 is added
C. 7 is added
D. 10 is subtracted

Answer

To solve the problem of what number should be added to or subtracted from each term of the ratio \(17 : 24\) so that it becomes equal to \(1 : 2\), we can follow these steps:

### Step 1: Set up the equation
Let \(x\) be the number that we will add or subtract from both terms of the ratio. We can express the modified ratio as:
\[ \frac{17 + x}{24 + x} = \frac{1}{2} \]
or
\[ \frac{17 - x}{24 - x} = \frac{1}{2} \]

### Step 2: Solve the first equation
Let's first solve the equation where we add \(x\):
\[ \frac{17 + x}{24 + x} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2(17 + x) = 1(24 + x) \]
Expanding both sides:
\[ 34 + 2x = 24 + x \]
Now, isolate \(x\):
\[ 2x - x = 24 - 34 \]
\[ x = -10 \]

### Step 3: Verify the solution
Now, let's check if subtracting \(10\) from both terms gives us the desired ratio:
\[ \frac{17 - 10}{24 - 10} = \frac{7}{14} = \frac{1}{2} \]
This confirms that subtracting \(10\) works.

### Step 4: Solve the second equation (optional)
Now let's check the case where we add \(x\):
\[ \frac{17 - x}{24 - x} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2(17 - x) = 1(24 - x) \]
Expanding both sides:
\[ 34 - 2x = 24 - x \]
Isolating \(x\):
\[ -x = 24 - 34 + 2x \]
\[ -x - 2x = -10 \]
\[ -3x = -10 \]
\[ x = \frac{10}{3} \]
This value does not yield an integer, so it is not a valid solution in this context.

### Final Answer
The number that should be subtracted from each term of the ratio \(17 : 24\) to make it equal to \(1 : 2\) is \(10\).

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