If two times the larger of the two numbers is divided by the smaller one, we get 3 as quotient and 5 as the remainder . Also, if ten times the smaller number is divided by the larger one, we get 4 as quotient and 2 as remainder. Find the numbers.

Options

A. 10 and 4
B. 8 and 5
C. 7 and 3
D. 10 and 3

Answer

To solve the problem, we need to set up equations based on the information given in the question. Let's denote the larger number as \( x \) and the smaller number as \( y \).

### Step 1: Set up the first equation
According to the problem, if two times the larger number is divided by the smaller number, we get a quotient of 3 and a remainder of 5. This can be expressed mathematically as:
\[ 2x = 3y + 5 \]
This is our first equation.

### Step 2: Set up the second equation
Next, the problem states that if ten times the smaller number is divided by the larger number, we get a quotient of 4 and a remainder of 2. This can be expressed as:
\[ 10y = 4x + 2 \]
This is our second equation.

### Step 3: Rearranging the equations
Now, we can rearrange both equations to express them in a standard form.

From the first equation:
\[ 2x - 3y = 5 \tag{1} \]

From the second equation:
\[ 10y - 4x = 2 \tag{2} \]

### Step 4: Simplifying the second equation
We can simplify equation (2) by dividing all terms by 2:
\[ 5y - 2x = 1 \tag{3} \]

### Step 5: Solve the system of equations
Now we have a system of equations:
1. \( 2x - 3y = 5 \)
2. \( 5y - 2x = 1 \)

We can add these two equations to eliminate \( x \):
\[ (2x - 3y) + (5y - 2x) = 5 + 1 \]
This simplifies to:
\[ 2y = 6 \]
Thus, we find:
\[ y = 3 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now that we have \( y \), we can substitute it back into equation (1) to find \( x \):
\[ 2x - 3(3) = 5 \]
This simplifies to:
\[ 2x - 9 = 5 \]
Adding 9 to both sides gives:
\[ 2x = 14 \]
Dividing by 2, we find:
\[ x = 7 \]

### Conclusion
The larger number \( x \) is 7, and the smaller number \( y \) is 3. Therefore, the two numbers are:
\[ \text{Larger number} = 7, \quad \text{Smaller number} = 3 \]