Three times the difference of one - third of a number and one - fifth of a number. When added to the sum of one - third of the number and one - fifth of the number is one less than the number . Find the number.

Answer

To solve the problem step by step, let's define the unknown number as \( x \).

### Step 1: Set up the equation
We need to express the given information mathematically.

1. **One-third of the number**: \( \frac{x}{3} \)
2. **One-fifth of the number**: \( \frac{x}{5} \)

Now, we need to find:
- The **difference** of one-third and one-fifth of the number:
\[ \frac{x}{3} - \frac{x}{5} \]

To subtract these fractions, we need a common denominator, which is 15:
\[ \frac{x}{3} = \frac{5x}{15}, \quad \frac{x}{5} = \frac{3x}{15} \]
Thus, the difference is:
\[ \frac{5x}{15} - \frac{3x}{15} = \frac{2x}{15} \]

Now, we multiply this difference by 3:
\[ 3 \left( \frac{2x}{15} \right) = \frac{6x}{15} \]

Next, we find the **sum** of one-third and one-fifth of the number:
\[ \frac{x}{3} + \frac{x}{5} = \frac{5x}{15} + \frac{3x}{15} = \frac{8x}{15} \]

### Step 2: Write the complete equation
According to the problem, when we add the result of the multiplication to the sum, it equals one less than the number:
\[ \frac{6x}{15} + \frac{8x}{15} = x - 1 \]

### Step 3: Combine the left-hand side
Combining the fractions on the left side:
\[ \frac{6x + 8x}{15} = \frac{14x}{15} \]
So, we have:
\[ \frac{14x}{15} = x - 1 \]

### Step 4: Clear the fraction
To eliminate the fraction, multiply both sides by 15:
\[ 14x = 15(x - 1) \]

### Step 5: Distribute on the right-hand side
Distributing gives:
\[ 14x = 15x - 15 \]

### Step 6: Rearrange the equation
Now, we can rearrange the equation to isolate \( x \):
\[ 14x - 15x = -15 \]
This simplifies to:
\[ -x = -15 \]

### Step 7: Solve for \( x \)
Multiplying both sides by -1 gives:
\[ x = 15 \]

### Conclusion
The number is \( 15 \).

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