Two numbers are in the ratio of 4 : 7. If 5 is subtracted from each, the ratio becomes 1 : 2. Find the greater number.

Options

A. 15
B. 40
C. 20
D. 35

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Variables
Let the two numbers be represented as \( x \) and \( y \). According to the problem, the ratio of the two numbers is given as:
\[ \frac{x}{y} = \frac{4}{7} \]

### Step 2: Express \( y \) in terms of \( x \)
From the ratio, we can express \( y \) in terms of \( x \):
\[ 7x = 4y \implies y = \frac{7}{4}x \]

### Step 3: Set up the equation after subtracting 5 from each number
When 5 is subtracted from each number, the new ratio becomes:
\[ \frac{x - 5}{y - 5} = \frac{1}{2} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 2(x - 5) = 1(y - 5) \]
Expanding this results in:
\[ 2x - 10 = y - 5 \]
Rearranging gives:
\[ 2x - y = 5 \quad \text{(Equation 1)} \]

### Step 5: Substitute \( y \) from Step 2 into Equation 1
Substituting \( y = \frac{7}{4}x \) into Equation 1:
\[ 2x - \frac{7}{4}x = 5 \]

### Step 6: Solve for \( x \)
To solve for \( x \), first find a common denominator:
\[ \frac{8}{4}x - \frac{7}{4}x = 5 \]
This simplifies to:
\[ \frac{1}{4}x = 5 \]
Multiplying both sides by 4 gives:
\[ x = 20 \]

### Step 7: Find \( y \)
Now that we have \( x \), we can find \( y \) using the expression for \( y \):
\[ y = \frac{7}{4}(20) = 35 \]

### Step 8: Identify the greater number
Comparing \( x \) and \( y \):
- \( x = 20 \)
- \( y = 35 \)

Thus, the greater number is:
\[ \text{Greater number} = 35 \]

### Final Answer
The greater number is **35**.

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