Finding the Smallest Fraction to Make a Sum a Whole Number

The problem asks us to find the smallest fraction that, when added to the sum of the given mixed numbers \(2\frac{1}{2}\), \(3\frac{1}{3}\), \(4\frac{1}{4}\), and \(5\frac{1}{5}\), results in a whole number.

Step-by-Step Calculation of the Sum

First, let's calculate the sum of the four given mixed numbers. We can separate the whole number parts and the fractional parts.

• Sum of the whole parts: \(2 + 3 + 4 + 5 = 14\)
• Sum of the fractional parts: \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}\)

To add the fractional parts, we need to find a common denominator. The least common multiple (LCM) of the denominators 2, 3, 4, and 5 is 60.

Now, we convert each fraction to an equivalent fraction with a denominator of 60:

• \(\frac{1}{2} = \frac{1 \times 30}{2 \times 30} = \frac{30}{60}\)
• \(\frac{1}{3} = \frac{1 \times 20}{3 \times 20} = \frac{20}{60}\)
• \(\frac{1}{4} = \frac{1 \times 15}{4 \times 15} = \frac{15}{60}\)
• \(\frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60}\)

Now, add the equivalent fractions:

\(\frac{30}{60} + \frac{20}{60} + \frac{15}{60} + \frac{12}{60} = \frac{30 + 20 + 15 + 12}{60} = \frac{77}{60}\)

The total sum of the four mixed numbers is the sum of the whole parts plus the sum of the fractional parts:

\(\text{Total Sum} = 14 + \frac{77}{60}\)

The fractional part \(\frac{77}{60}\) is an improper fraction. Let's convert it to a mixed number:

\(\frac{77}{60} = 1\frac{17}{60}\)

So, the total sum is:

\(\text{Total Sum} = 14 + 1\frac{17}{60} = 15\frac{17}{60}\)

Finding the Smallest Fraction to Reach a Whole Number

The sum is \(15\frac{17}{60}\). This means the sum is between the whole numbers 15 and 16. To make the result a whole number, we need to add a fraction to \(15\frac{17}{60}\) such that it becomes the next whole number, which is 16.

The fractional part of the sum is \(\frac{17}{60}\). To reach the next whole number (by completing the fraction to 1), we need to add a fraction that, when combined with \(\frac{17}{60}\), equals 1 (\(\frac{60}{60}\)).

The required smallest fraction is \(1 - \text{fractional part of the sum}\).

Required fraction \( = 1 - \frac{17}{60}\)

To subtract, we write 1 as \(\frac{60}{60}\):

Required fraction \( = \frac{60}{60} - \frac{17}{60} = \frac{60 - 17}{60} = \frac{43}{60}\)

Therefore, 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 the result a whole number is \(\frac{43}{60}\).

Summary of Calculation

Conclusion

The sum of the given mixed numbers is \(15\frac{17}{60}\). To make this a whole number, we need to add the difference between 1 and the fractional part \(\frac{17}{60}\), which is \(\frac{43}{60}\). Adding \(\frac{43}{60}\) to \(15\frac{17}{60}\) results in \(15 + \frac{17}{60} + \frac{43}{60} = 15 + \frac{60}{60} = 15 + 1 = 16\), which is a whole number.

Revision Table: Fraction Concepts

Additional Information: Adding Mixed Numbers

There are typically two ways to add mixed numbers:

• Method 1: Separate whole and fractional parts (as done in the solution). Add the whole numbers together. Add the fractional parts together (finding a common denominator). If the sum of fractions is an improper fraction, convert it to a mixed number and add the whole part of this mixed number to the sum of the initial whole numbers.
• Method 2: Convert mixed numbers to improper fractions. Convert each mixed number into an improper fraction. Find a common denominator for the improper fractions. Add the improper fractions. Convert the resulting improper fraction back to a mixed number or simplify as needed. This method can sometimes involve larger numbers, making the first method often simpler for addition.

In this specific problem, finding the fraction needed to reach a whole number requires focusing on the fractional part of the final sum, making Method 1 more direct for the calculation steps shown.