The difference of two number is `45%` of their sum. The ratio of the greater number to the smaller number will be

Options

A. `27:11`
B. `29:11`
C. `29:13`
D. `143:55`

Answer

To solve the problem, we need to find the ratio of the greater number to the smaller number given that the difference of the two numbers is 45% of their sum.

Let’s denote the two numbers as \( x \) (the greater number) and \( y \) (the smaller number).

### Step 1: Set Up the Equation
According to the problem, the difference of the two numbers can be expressed as:
\[ x - y \]
And the sum of the two numbers can be expressed as:
\[ x + y \]
The problem states that the difference is 45% of the sum:
\[ x - y = 0.45(x + y) \]

### Step 2: Expand the Equation
Now, we can expand the equation:
\[ x - y = 0.45x + 0.45y \]

### Step 3: Rearrange the Equation
Next, we will rearrange the equation to isolate terms involving \( x \) and \( y \):
\[ x - 0.45x = y + 0.45y \]
This simplifies to:
\[ 0.55x = 1.45y \]

### Step 4: Solve for the Ratio
Now, we can express the ratio of \( x \) to \( y \):
\[ \frac{x}{y} = \frac{1.45}{0.55} \]
To simplify this, we can multiply both the numerator and the denominator by 100 to eliminate the decimals:
\[ \frac{x}{y} = \frac{145}{55} \]
Now, we can simplify \( \frac{145}{55} \):
\[ \frac{145 \div 5}{55 \div 5} = \frac{29}{11} \]

### Step 5: Conclusion
Thus, the ratio of the greater number to the smaller number is:
\[ \frac{x}{y} = \frac{29}{11} \]

### Final Answer
The ratio of the greater number to the smaller number is \( 29:11 \).

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