The ratio between two numbers is 4 :5. If each number is increased by 4, the ratio becomes 6 : 7. Find the difference between the numbers.

Options

A. 2
B. 9
C. 3
D. 7

Answer

To solve the problem step by step, we will follow the given information and derive the required values.

### Step 1: Define the Variables
Let the two numbers be represented as:
- First number = \( x \)
- Second number = \( y \)

### Step 2: Set Up the Initial Ratio
According to the problem, the ratio between the two numbers is given as \( 4:5 \). This can be expressed mathematically as:
\[ \frac{x}{y} = \frac{4}{5} \]
Cross-multiplying gives us:
\[ 5x = 4y \quad \text{(Equation 1)} \]

### Step 3: Set Up the New Ratio After Increasing Each Number
When each number is increased by 4, the new ratio becomes \( 6:7 \). This can be expressed as:
\[ \frac{x + 4}{y + 4} = \frac{6}{7} \]
Cross-multiplying gives us:
\[ 7(x + 4) = 6(y + 4) \]
Expanding this, we get:
\[ 7x + 28 = 6y + 24 \]
Rearranging gives us:
\[ 7x - 6y = -4 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( 5x - 4y = 0 \)
2. \( 7x - 6y = -4 \)

We can solve these equations using the elimination method. First, we will multiply Equation 1 by 7:
\[ 7(5x - 4y) = 0 \implies 35x - 28y = 0 \quad \text{(Equation 3)} \]

Now, we will multiply Equation 2 by 5:
\[ 5(7x - 6y) = -20 \implies 35x - 30y = -20 \quad \text{(Equation 4)} \]

### Step 5: Subtract the Equations
Now we subtract Equation 3 from Equation 4:
\[ (35x - 30y) - (35x - 28y) = -20 - 0 \]
This simplifies to:
\[ -30y + 28y = -20 \implies -2y = -20 \]
Dividing both sides by -2 gives:
\[ y = 10 \]

### Step 6: Substitute to Find \( x \)
Now we can substitute \( y = 10 \) back into Equation 1 to find \( x \):
\[ 5x - 4(10) = 0 \implies 5x - 40 = 0 \implies 5x = 40 \implies x = 8 \]

### Step 7: Find the Difference Between the Numbers
Now we have:
- \( x = 8 \)
- \( y = 10 \)

The difference between the two numbers is:
\[ |x - y| = |8 - 10| = 2 \]

### Final Answer
The difference between the two numbers is **2**.

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