One-half of a certain number is equal to 65% of the 2nd number. Find the ratio of 1st to 2nd number.

Options

A. `10 : 13`
B. `8 : 13`
C. `13 : 8`
D. `13 : 10`

Answer

To solve the problem step by step, we need to find the ratio of the first number (let's denote it as \( x \)) to the second number (denote it as \( y \)).

### Step 1: Set up the equation based on the problem statement.
According to the problem, one-half of the first number is equal to 65% of the second number. We can express this mathematically as:
\[ \frac{1}{2} x = 0.65 y \]

### Step 2: Convert 65% to a fraction.
Since 65% can be expressed as a fraction, we have:
\[ 0.65 = \frac{65}{100} = \frac{13}{20} \]
Thus, we can rewrite the equation as:
\[ \frac{1}{2} x = \frac{13}{20} y \]

### Step 3: Eliminate the fractions by multiplying both sides by a common denominator.
To eliminate the fractions, we can multiply both sides by 20 (the common denominator):
\[ 20 \cdot \frac{1}{2} x = 20 \cdot \frac{13}{20} y \]
This simplifies to:
\[ 10x = 13y \]

### Step 4: Rearrange the equation to find the ratio \( \frac{x}{y} \).
Now, we can rearrange this equation to express \( x \) in terms of \( y \):
\[ \frac{x}{y} = \frac{13}{10} \]

### Step 5: Write the ratio of the first number to the second number.
The ratio of the first number \( x \) to the second number \( y \) is:
\[ x : y = 13 : 10 \]

### Final Answer:
The ratio of the first number to the second number is \( 13 : 10 \).

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