A number is divided into two parts In such a way that 80% of 1st part is 3 more than 60% of 2nd part and 80% of 2nd part is 6 more than 90% of the 1st part. Then the number is

Options

A. 125
B. 130
C. 135
D. 145

Answer

To solve the problem, we will break it down step by step.

### Step 1: Define the Variables
Let the first part be \( x \) and the second part be \( y \).

### Step 2: Set Up the Equations
According to the problem:
1. \( 80\% \) of the first part is \( 3 \) more than \( 60\% \) of the second part.
\[ 0.8x = 0.6y + 3 \quad \text{(Equation 1)} \]

2. \( 80\% \) of the second part is \( 6 \) more than \( 90\% \) of the first part.
\[ 0.8y = 0.9x + 6 \quad \text{(Equation 2)} \]

### Step 3: Rearranging the Equations
Rearranging both equations to isolate terms:
1. From Equation 1:
\[ 0.8x - 0.6y = 3 \quad \text{(Equation 1)} \]

2. From Equation 2:
\[ 0.9x - 0.8y = -6 \quad \text{(Equation 2)} \]

### Step 4: Eliminate One Variable
To eliminate one variable, we can multiply Equation 1 by \( 9 \) and Equation 2 by \( 8 \) to make the coefficients of \( x \) the same:
1. Multiply Equation 1 by \( 9 \):
\[ 7.2x - 5.4y = 27 \quad \text{(Equation 3)} \]

2. Multiply Equation 2 by \( 8 \):
\[ 7.2x - 6.4y = -48 \quad \text{(Equation 4)} \]

### Step 5: Subtract the Equations
Now, subtract Equation 4 from Equation 3:
\[ (7.2x - 5.4y) - (7.2x - 6.4y) = 27 - (-48) \]
This simplifies to:
\[ (6.4y - 5.4y) = 27 + 48 \]
\[ 1y = 75 \]
Thus, \( y = 75 \).

### Step 6: Substitute Back to Find \( x \)
Now substitute \( y = 75 \) back into Equation 1:
\[ 0.8x - 0.6(75) = 3 \]
\[ 0.8x - 45 = 3 \]
\[ 0.8x = 48 \]
\[ x = \frac{48}{0.8} = 60 \]

### Step 7: Find the Total Number
Now we find the total number:
\[ \text{Total number} = x + y = 60 + 75 = 135 \]

### Final Answer
The number is \( 135 \).

---