Verified Answer :

  • Identify the least common multiple of the denominators 10 and 4, which is 20.
  • Rewrite the fractions as \(\frac{2}{20}\) and \(\frac{15}{20}\).
  • Add the numerators while maintaining the denominator to get \(\frac{17}{20}\).
  • Verify the simplest form of the result, which is \(\boxed{\frac{17}{20}}\).

Explanation

  1. Analyze the Problem

The problem asks us to calculate the sum of two fractions, \(\frac{1}{10}\) and \(\frac{3}{4}\), and express the result in its simplest form. To add these, we need a common denominator. The denominators are 10 and 4. We look for the least common multiple (LCM) of 10 and 4, which is 20.

  1. Rewrite Fractions with a Common Denominator

To rewrite each fraction with the common denominator of 20, we perform the following:

  • For \(\frac{1}{10}\): Multiply the numerator and denominator by \(\frac{20}{10} = 2\). We get \(\frac{1 \times 2}{10 \times 2} = \frac{2}{20}\).
  • For \(\frac{3}{4}\): Multiply the numerator and denominator by \(\frac{20}{4} = 5\). We get \(\frac{3 \times 5}{4 \times 5} = \frac{15}{20}\).
  1. Add and Simplify the Result

Now we add the two rewritten fractions:
\(\frac{2}{20} + \frac{15}{20} = \frac{2 + 15}{20} = \frac{17}{20}\).
To ensure the answer is in simplest form, we check if 17 and 20 have any common factors other than 1. Since 17 is a prime number and is not a factor of 20, the fraction \(\frac{17}{20}\) is already in its simplest form.

Examples

When cooking or working with measurements, you often need to combine fractional parts of a whole. For example, if a recipe calls for \(\frac{1}{10}\) of a cup of water for one mixture and \(\frac{3}{4}\) of a cup for another, you can determine the total volume by adding these fractions, resulting in \(\frac{17}{20}\) of a cup. This skill is essential for everyday tasks involving precise quantities, such as home repairs, DIY projects, or culinary arts.

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