Problem 145 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{1}{2}+\frac{1}{5}=\frac{2}{7}$$

Step-by-Step Solution

Answer

The statement is false. The correct statement is \(\frac{1}{2} + \frac{1}{5} = \frac{7}{10}\).

Step 1 - Calculate the sum of the fractions

Add the two fractions \(\frac{1}{2}\) and \(\frac{1}{5}\). To do this, find the least common denominator (LCD), which is the least common multiple of 2 and 5. The LCD is 10. Then convert each fraction to an equivalent fraction using this LCD: \(\frac{1}{2} = \frac{5}{10}\) and \(\frac{1}{5} = \frac{2}{10}\). Now the fractions can be added as: \(\frac{5}{10} + \frac{2}{10} = \frac{7}{10}\)

Step 2 - Compare the calculated sum with the given sum

The calculated sum is \(\frac{7}{10}\), while the given sum in the problem is \(\frac{2}{7}\). Since the two sums are not equal, the statement is false.

Step 3 - Correct the false statement

To make the statement true, replace the false sum \(\frac{2}{7}\) with the correct sum \(\frac{7}{10}\). Thus, the corrected statement would be \(\frac{1}{2} + \frac{1}{5} = \frac{7}{10}\).