Problem 92 - 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. $$\frac{3}{4}+\left(-\frac{3}{5}\right)=-\frac{3}{20}$$

Step-by-Step Solution

Answer

The statement is false. The correct statement is \(\frac{3}{4} - \frac{3}{5} = \frac{3}{20}\).

Step 1: Find a Common Denominator

In order to add fractions, they must have the same denominator. In the given statement, the denominators are 4 and 5. The least common denominator (LCD) of 4 and 5 is their product, which is 20. This is because 4 and 5 have no common factors other than 1.

Step 2: Convert Fractions to Have Common Denominator

Now, convert both fractions to equivalent fractions with denominator 20. Multiply both the numerator and denominator of \(\frac{3}{4}\) by 5 to get \(\frac{15}{20}\). Multiply both the numerator and denominator of \(-\frac{3}{5}\) by 4 to get \(-\frac{12}{20}\). The equation becomes \(\frac{15}{20} - \frac{12}{20} = -\frac{3}{20}\)

Step 3: Perform Addition

After converting the fractions, we can now add them together. Subtract the numerators and keep the same denominator, \(\frac{15}{20}-\frac{12}{20} = \frac{3}{20}\)

Step 4: Check Against Given Result

The result of the addition was \(\frac{3}{20}\), not \(-\frac{3}{20}\) as given in the statement. Therefore, the statement was incorrect.

Step 5: Correct the Statement

Change the given statement from \(\frac{3}{4} + \(-\frac{3}{5}\) =\(-\frac{3}{20}\) to \(\frac{3}{4} + \(-\frac{3}{5}\) =\(\frac{3}{20}\) to make it a true statement.