Question

A student spends \(\frac{1}{3}\) of a typical day sleeping, \(\frac{1}{6}\) of the day in classes, and \(\frac{1}{8}\) of the day watching TV. a. If \(x\) represents the fraction of the rest of the day available for study, write an equation describing the situation. b. Solve the equation for \(x .\)

Short answer

Answer: The student spends \(\frac{3}{8}\) of their day studying.

Step-by-step solution

Identify the fractions given in the problem

The student spends \(\frac{1}{3}\) of their day sleeping, \(\frac{1}{6}\) in classes and \(\frac{1}{8}\) of the day watching TV.

Assign a variable for the fraction of the day spent studying

Let \(x\) represent the fraction of the rest of the day available for study.

Set up the equation

Since the sum of all fractions should be equal to 1, we can set up the following equation: \(\frac{1}{3}+\frac{1}{6}+\frac{1}{8}+x=1\)

Solve for x

Begin by finding a common denominator, which in this case is 24. \(\frac{8}{24}+\frac{4}{24}+\frac{3}{24}+x=1\) Now, add the fractions on the left side: \(\frac{15}{24}+x=1\) Next, subtract \(\frac{15}{24}\) from both sides of the equation: \(x=1-\frac{15}{24}\) Now simplify the right side of the equation: \(x=\frac{9}{24}\) Finally, reduce the fraction: \(x=\frac{3}{8}\) So, the student spends \(\frac{3}{8}\) of their day studying.