Verified Answer :

  • Identify the least common multiple (LCM) of the denominators 5 and 3, which is 15.
  • Convert both fractions to equivalent forms with the denominator 15: \(\frac{9}{15}\) and \(\frac{5}{15}\).
  • Sum the numerators while keeping the denominator, resulting in \(\frac{14}{15}\).
  • Confirm the result is in simplest form as the greatest common divisor of 14 and 15 is 1. The final result is \(\boxed{\frac{14}{15}}\).

Explanation

  1. Analysis of the Problem

We are given two fractions to add: \(\frac{3}{5}\) and \(\frac{1}{3}\). These fractions have different denominators, 5 and 3. To add them, we need to find a common denominator, which is the least common multiple (LCM) of the current denominators. The least common multiple of 5 and 3 is 15.

  1. Converting to Common Denominator

To express both fractions with a common denominator of 15, we multiply the numerator and denominator of each fraction by the necessary factor:

  • For \(\frac{3}{5}\), we multiply by \(\frac{3}{3}\): \[\frac{3}{5} \times \frac{3}{3} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}\]
  • For \(\frac{1}{3}\), we multiply by \(\frac{5}{5}\): \[\frac{1}{3} \times \frac{5}{5} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\]
  1. Adding and Simplifying Fractions

Now that the fractions share the same denominator, we can add their numerators while keeping the denominator the same:
\[\frac{9}{15} + \frac{5}{15} = \frac{9 + 5}{15} = \frac{14}{15}\]
To determine if \(\frac{14}{15}\) is in simplest form, we check the greatest common divisor (GCD) of the numerator (14) and the denominator (15). Since the divisors of 14 are 1, 2, 7, and 14, and the divisors of 15 are 1, 3, 5, and 15, the only common divisor is 1. Therefore, the fraction \(\frac{14}{15}\) is already in its simplest form.

Examples

Understanding how to add fractions is essential in daily life, such as when following a cooking recipe. If a recipe calls for \(\frac{3}{5}\) of a cup of flour and then adds \(\frac{1}{3}\) of a cup of sugar for a specific mix, you would need to add these fractions to find the total amount of ingredients needed (\(\frac{3}{5} + \frac{1}{3} = \frac{14}{15}\) cups). This skill ensures accuracy in measurements for cooking, construction, or even financial calculations involving parts of a whole.

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