Kevin says it will take him less than 1 hour to do homework, play soccer, and practice piano because these tasks will take a total of $\frac{8}{12}$ hour. Is he correct? Explain.
To determine if Kevin is correct, we compare the total time required for his tasks to the reference duration of 1 hour. We are given that the total time for his homework, soccer, and piano practice is \(\frac{8}{12}\) of an hour. We need to evaluate whether this duration is strictly less than 1 hour, which can be expressed as the inequality \(\frac{8}{12} < 1\).
First, we simplify the fraction \(\frac{8}{12}\) to make it easier to compare with 1. To do this, we find the greatest common divisor of the numerator (8) and the denominator (12), which is 4. By dividing both the numerator and the denominator by 4, we get:
\[\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}.\]
Thus, the total time required for the tasks is \(\frac{2}{3}\) of an hour.
We now compare the simplified time, \(\frac{2}{3}\) hour, with the reference time of 1 hour. Since any fraction where the numerator is smaller than the denominator (and both are positive) represents a value less than 1, we can see that:
\[\frac{2}{3} < 1.\]
This confirms that the total time for the tasks is indeed less than 1 hour. Therefore, Kevin's statement is correct.
Understanding how to compare fractions to whole numbers is vital for efficient time management. For example, if you are planning your study schedule and calculate that your assignments take \(\frac{3}{4}\) of an hour to complete, you know that this is less than 1 hour, allowing you to fit other activities into your day. This basic skill helps in budgeting time and ensuring that your daily goals are realistic and achievable.