The smallest fraction, which should be added to the sum of `2(1)/(2),3(1)/(3),4(1)/(4) and 5(1)/(5)` to make the result a whole number, is

Options

A. `(13)/(60)`
B. `(1)/(4)`
C. `(17)/(60)`
D. `(43)/(60)`

Answer

To solve the problem of finding the smallest fraction that should be added to the sum of \( 2\frac{1}{2}, 3\frac{1}{3}, 4\frac{1}{4}, \) and \( 5\frac{1}{5} \) to make it a whole number, we can follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert each mixed number to an improper fraction:
- \( 2\frac{1}{2} = \frac{5}{2} \)
- \( 3\frac{1}{3} = \frac{10}{3} \)
- \( 4\frac{1}{4} = \frac{17}{4} \)
- \( 5\frac{1}{5} = \frac{26}{5} \)

### Step 2: Find the Sum of the Fractions
Next, we need to find the sum of these fractions:
\[ \frac{5}{2} + \frac{10}{3} + \frac{17}{4} + \frac{26}{5} \]

### Step 3: Find the Least Common Multiple (LCM)
To add these fractions, we need to find the least common multiple (LCM) of the denominators \( 2, 3, 4, \) and \( 5 \). The LCM of these numbers is \( 60 \).

### Step 4: Convert Each Fraction to Have the Common Denominator
Now we convert each fraction to have the denominator of \( 60 \):
- \( \frac{5}{2} = \frac{5 \times 30}{2 \times 30} = \frac{150}{60} \)
- \( \frac{10}{3} = \frac{10 \times 20}{3 \times 20} = \frac{200}{60} \)
- \( \frac{17}{4} = \frac{17 \times 15}{4 \times 15} = \frac{255}{60} \)
- \( \frac{26}{5} = \frac{26 \times 12}{5 \times 12} = \frac{312}{60} \)

### Step 5: Add the Converted Fractions
Now we can add these fractions:
\[ \frac{150}{60} + \frac{200}{60} + \frac{255}{60} + \frac{312}{60} = \frac{150 + 200 + 255 + 312}{60} = \frac{917}{60} \]

### Step 6: Separate the Whole Number and Fractional Part
Next, we separate the whole number and the fractional part:
\[ \frac{917}{60} = 15 + \frac{17}{60} \]
Here, \( 15 \) is the whole number and \( \frac{17}{60} \) is the fractional part.

### Step 7: Determine the Smallest Fraction to Add
To make the result a whole number, we need to add a fraction that will convert \( \frac{17}{60} \) to a whole number. The smallest fraction we need to add is:
\[ 1 - \frac{17}{60} = \frac{60 - 17}{60} = \frac{43}{60} \]

### Final Answer
Thus, the smallest fraction that should be added is:
\[ \frac{43}{60} \]

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