A number is divided into two parts. In such a way that `80%` of first part exceeds 60% of second part by ` 3. 80%` of second part exceeds `90%` of first part by 6. Find the number.

Answer

To solve the problem step by step, we will define the two parts of the number and set up equations based on the information given in the question.

### Step 1: Define the Variables
Let the first part be \( a \) and the second part be \( b \). The total number can be expressed as:
\[ x = a + b \]

### Step 2: Set Up the First Equation
According to the problem, \( 80\% \) of the first part exceeds \( 60\% \) of the second part by \( 3 \). This can be written as:
\[ 0.8a = 0.6b + 3 \]
Rearranging gives us:
\[ 0.8a - 0.6b = 3 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that \( 80\% \) of the second part exceeds \( 90\% \) of the first part by \( 6 \). This can be expressed as:
\[ 0.8b = 0.9a + 6 \]
Rearranging gives us:
\[ 0.8b - 0.9a = 6 \quad \text{(Equation 2)} \]

### Step 4: Eliminate Decimals
To make calculations easier, we can multiply both equations by \( 10 \) to eliminate the decimals:
- For Equation 1:
\[ 8a - 6b = 30 \]
- For Equation 2:
\[ 8b - 9a = 60 \]

### Step 5: Rearrange Both Equations
Now we rearrange both equations:
- Equation 1:
\[ 8a - 6b = 30 \quad \text{(Equation 1)} \]
- Equation 2:
\[ -9a + 8b = 60 \quad \text{(Equation 2)} \]

### Step 6: Solve the System of Equations
To solve these equations, we can multiply Equation 1 by \( 9 \) and Equation 2 by \( 6 \) to align the coefficients of \( a \):
- Multiply Equation 1 by \( 9 \):
\[ 72a - 54b = 270 \]
- Multiply Equation 2 by \( 6 \):
\[ -54a + 48b = 360 \]

### Step 7: Add the Equations
Now we add the two equations:
\[ 72a - 54b - 54a + 48b = 270 + 360 \]
This simplifies to:
\[ 18a - 6b = 630 \]
Dividing the entire equation by \( 6 \):
\[ 3a - b = 105 \quad \text{(Equation 3)} \]

### Step 8: Substitute Back
Now we can use Equation 3 to express \( b \) in terms of \( a \):
\[ b = 3a - 105 \]

### Step 9: Substitute into One of the Original Equations
We can substitute \( b \) back into Equation 1:
\[ 8a - 6(3a - 105) = 30 \]
Expanding this gives:
\[ 8a - 18a + 630 = 30 \]
Combining like terms:
\[ -10a + 630 = 30 \]
Subtracting \( 630 \) from both sides:
\[ -10a = -600 \]
Dividing by \( -10 \):
\[ a = 60 \]

### Step 10: Find \( b \)
Now, substitute \( a \) back to find \( b \):
\[ b = 3(60) - 105 = 180 - 105 = 75 \]

### Step 11: Find the Total Number
Finally, we can find the total number:
\[ x = a + b = 60 + 75 = 135 \]

### Final Answer
The number is \( \boxed{135} \).

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