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Maria rode her bike 2 miles to the library. She took a shortcut on the way home that was $\frac{2}{3}$ of a mile shorter.



Which expression represents the total distance Maria biked?

A$2+\left(2+\frac{2}{3}\right)$
B$2+\left(2-\frac{2}{3}\right)$
C$2-\left(2+\frac{2}{3}\right)$
D$2-\left(2-\frac{2}{3}\right)$



Evaluate the expression to find the total distance Maria biked. Write your answer as a whole or mixed number.

Verified Answer :

  • The outbound trip is \(2\) miles, and the shortcut inbound trip is \(2 - \frac{2}{3}\) miles.
  • The total distance is the sum of these, represented by the expression: \(2 + (2 - \frac{2}{3})\), which corresponds to option (b).
  • Evaluating the expression: \(2 + (2 - \frac{2}{3}) = 2 + \frac{4}{3} = \frac{10}{3} = 3 \frac{1}{3}\) miles.
  • The final answer is \(\boxed{3 \frac{1}{3} \text{ miles}}\).

Explanation

  1. Analyzing the Trip and Selecting the Expression

To find the total distance Maria biked, we first identify the components of her journey. The outbound trip to the library is given as \(2\) miles. The return trip (the shortcut) is described as being \(\frac{2}{3}\) of a mile shorter than the outbound trip. Therefore, the distance of the shortcut is \(\left(2 - \frac{2}{3}\right)\) miles. The total distance is the sum of these two parts: the outbound distance and the inbound distance. We represent this as \(2 + \left(2 - \frac{2}{3}\right)\). Comparing this to the given options, we can see that this matches option (b).

  1. Evaluating the Total Distance

Now, we calculate the total distance represented by the expression \(2 + \left(2 - \frac{2}{3}\right)\). First, we evaluate the term inside the parentheses:
\[2 - \frac{2}{3} = \frac{6}{3} - \frac{2}{3} = \frac{4}{3}.\]
Next, we add this to the initial outbound distance:
\[2 + \frac{4}{3} = \frac{6}{3} + \frac{4}{3} = \frac{10}{3}.\]
Finally, we convert the improper fraction \(\frac{10}{3}\) into a mixed number. Since \(10 \div 3 = 3\) with a remainder of \(1\), we have:
\[\frac{10}{3} = 3 \frac{1}{3}.\]
So, the total distance Maria biked is \(3 \frac{1}{3}\) miles.

Examples

This problem demonstrates how to set up and solve real-world distance expressions. For example, if you are planning a round trip to school, knowing your base distance allows you to calculate the total mileage when factoring in shortcuts or detours. Understanding these algebraic expressions helps in accurately tracking distance for fitness goals, planning fuel consumption, or simply managing travel time effectively.

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