Verified Answer :

  • Identify the denominators of the fractions, which are 5 and 7, and find their least common multiple, 35.
  • Convert \(\frac{3}{5}\) to \(\frac{21}{35}\) and \(\frac{2}{7}\) to \(\frac{10}{35}\) to establish a common denominator.
  • Sum the numerators, 21 and 10, to get 31, while maintaining the common denominator of 35.
  • The final sum is \(\boxed{\frac{31}{35}}\).

Explanation

  1. Analyzing the Problem and Finding a Common Denominator

To find the sum of \(\frac{3}{5}\) and \(\frac{2}{7}\), we must first recognize that these two fractions have different denominators. In order to add fractions, they must be expressed with a common denominator. The denominators are 5 and 7. Since 5 and 7 are both prime numbers, their least common multiple is simply their product, \(5 \times 7 = 35\).

  1. Converting Fractions to a Common Denominator

Now, we convert each fraction to an equivalent one that has 35 as its denominator. For the first fraction, \(\frac{3}{5}\), we multiply both the numerator and the denominator by 7:
\[\frac{3 \times 7}{5 \times 7} = \frac{21}{35}.\]
For the second fraction, \(\frac{2}{7}\), we multiply both the numerator and the denominator by 5:
\[\frac{2 \times 5}{7 \times 5} = \frac{10}{35}.\]

  1. Adding and Simplifying the Resulting Fraction

With both fractions now having the same denominator, we can add them by summing their numerators and keeping the denominator constant:
\[\frac{21}{35} + \frac{10}{35} = \frac{21 + 10}{35} = \frac{31}{35}.\]
Since 31 is a prime number and does not divide 35, the fraction is already in its simplest form. Thus, the sum is \(\frac{31}{35}\).

Examples

Adding fractions is a fundamental skill in many real-life situations, such as cooking or DIY projects. For instance, if a recipe calls for \(\frac{3}{5}\) of a cup of flour for one batch and \(\frac{2}{7}\) of a cup of flour for another, you can calculate the total amount of flour needed by finding the sum \(\frac{3}{5} + \frac{2}{7} = \frac{31}{35}\) of a cup. Understanding how to find common denominators allows you to combine measurements accurately, ensuring your projects or culinary creations turn out exactly as planned.

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