Two numbers `x and y` are the ratio of `3 : 4`. If 10 is added to each number, the ratio becomes `5 : 6`. Then, the numbers `x and y` are

Options

A. 12 and 16
B. 15 and 20
C. 3 and 4
D. 30 and 40

Answer

To solve the problem step by step, we will follow the information given in the question.

### Step 1: Set up the initial ratio
We know that the two numbers \( x \) and \( y \) are in the ratio of \( 3:4 \). This can be expressed mathematically as:
\[ \frac{x}{y} = \frac{3}{4} \]
From this, we can derive the equation:
\[ 4x = 3y \quad \text{(Equation 1)} \]

**Hint:** Use the property of ratios to express one variable in terms of the other.

### Step 2: Set up the second ratio after adding 10
According to the problem, if we add 10 to each number, the new ratio becomes \( 5:6 \). This can be expressed as:
\[ \frac{x + 10}{y + 10} = \frac{5}{6} \]
Cross-multiplying gives us:
\[ 6(x + 10) = 5(y + 10) \]
Expanding this, we get:
\[ 6x + 60 = 5y + 50 \]
Rearranging this equation leads to:
\[ 6x - 5y = -10 \quad \text{(Equation 2)} \]

**Hint:** Cross-multiplication helps to eliminate the fractions and leads to a linear equation.

### Step 3: Solve the system of equations
Now we have two equations:
1. \( 4x - 3y = 0 \) (Equation 1)
2. \( 6x - 5y = -10 \) (Equation 2)

We can solve these equations simultaneously. Let's first express \( y \) in terms of \( x \) from Equation 1:
\[ 3y = 4x \implies y = \frac{4}{3}x \]

**Hint:** Isolate one variable to substitute into the other equation.

### Step 4: Substitute \( y \) into Equation 2
Now, substitute \( y \) in Equation 2:
\[ 6x - 5\left(\frac{4}{3}x\right) = -10 \]
This simplifies to:
\[ 6x - \frac{20}{3}x = -10 \]
To eliminate the fraction, multiply through by 3:
\[ 18x - 20x = -30 \]
This simplifies to:
\[ -2x = -30 \]
Thus,
\[ x = 15 \]

**Hint:** When you encounter fractions, multiplying through by a common denominator can simplify calculations.

### Step 5: Find \( y \)
Now that we have \( x \), we can find \( y \) using the equation \( y = \frac{4}{3}x \):
\[ y = \frac{4}{3} \times 15 = 20 \]

**Hint:** Substitute the value of \( x \) back into the equation to find \( y \).

### Conclusion
The two numbers are:
\[ x = 15 \quad \text{and} \quad y = 20 \]

**Final Answer:** The numbers \( x \) and \( y \) are \( 15 \) and \( 20 \) respectively.