What should be added to `((1)/(3) + (1)/(4) - (1)/(5))` to get 4 ?

Options

A. `4 (23)/(60)`
B. `3 (37)/(60)`
C. `(23)/(60)`
D. `-3 (37)/(60)`

Answer

To solve the problem of what should be added to \(\left(\frac{1}{3} + \frac{1}{4} - \frac{1}{5}\right)\) to get 4, we can follow these steps:

### Step 1: Simplify the Expression
First, we need to simplify the expression \(\frac{1}{3} + \frac{1}{4} - \frac{1}{5}\).

To do this, we need to find a common denominator for the fractions. The least common multiple (LCM) of 3, 4, and 5 is 60.

### Step 2: Convert Each Fraction
Now, we convert each fraction to have a denominator of 60:
- \(\frac{1}{3} = \frac{20}{60}\) (because \(1 \times 20 = 20\) and \(3 \times 20 = 60\))
- \(\frac{1}{4} = \frac{15}{60}\) (because \(1 \times 15 = 15\) and \(4 \times 15 = 60\))
- \(\frac{1}{5} = \frac{12}{60}\) (because \(1 \times 12 = 12\) and \(5 \times 12 = 60\))

### Step 3: Combine the Fractions
Now we can combine the fractions:
\[ \frac{20}{60} + \frac{15}{60} - \frac{12}{60} = \frac{20 + 15 - 12}{60} = \frac{23}{60} \]

### Step 4: Set Up the Equation
Now we set up the equation to find \(x\):
\[ \frac{23}{60} + x = 4 \]

### Step 5: Isolate \(x\)
To isolate \(x\), we subtract \(\frac{23}{60}\) from both sides:
\[ x = 4 - \frac{23}{60} \]

### Step 6: Convert 4 to a Fraction
Next, we convert 4 into a fraction with a denominator of 60:
\[ 4 = \frac{240}{60} \]

### Step 7: Perform the Subtraction
Now we can perform the subtraction:
\[ x = \frac{240}{60} - \frac{23}{60} = \frac{240 - 23}{60} = \frac{217}{60} \]

### Conclusion
Thus, the value that should be added to \(\left(\frac{1}{3} + \frac{1}{4} - \frac{1}{5}\right)\) to get 4 is:
\[ \frac{217}{60} \]

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