Mike and Lidia are 106 miles apart and will begin riding their bicycles toward each other on the same straight road. Lidia will begin riding 1 hour before Mike does, and Lidia and Mike will travel at constant rates of 13 and 18 miles per hour, respectively. How many miles will Lidia have traveled by the time she and Mike meet on the road? A. 39 B. 48 C. 52 D. 54 E. 65

(C) For this problem we will need the distance formula: Distance = Rate × Time.

We must also realize that since Mike and Lidia are traveling toward each other until they meet, the sum of their individual distances must equal the total distance.

Keeping these things in mind, we must: 1. Use the given information to determine the time that Lidia will have traveled before meeting Mike. 2. Use this time to determine the distance that Lidia will have traveled.

Let’s define some variables: Dm = distance Mike travels. Dl = distance Lidia travels. t = time Lidia travels. t – 1 = time Mike travels.

We can then write the equation: Dm + Dl = 106 18(t – 1) + 13t = 106 18t – 18 + 13t = 106 31t = 124 t = 4. Lidia will travel for 4 hours and Mike will travel for 3 hours.

We can now determine the number of miles Lidia will travel in 4 hours: Distance = Rate × Time = 13 × 4 = 52 miles. The correct answer is choice (C).

Alternate Method (Backsolving): Here we could start with the answer choices and determine which one makes all of the information in the question true. Lidia’s speed is 13 miles per hour, so her total distance will likely be a multiple of 13 (since the test writer will make the numbers easy to deal with if you understand how to set up the problem). So we should start with the choices that are multiples of 13: (A), (C), and (E).

Let’s start with choice (A): If Lidia travels 39 miles at 13 miles per hour, it will take her 3 hours. Therefore, Mike will travel for 2 hours. Since his speed is 18 miles per hour, he will travel 36 miles in this time. Together, the number of miles covered by Mike and Lidia is: 39 + 36 = 75. This is incorrect, since they must travel 106 miles in order to meet.

Let’s try choice (C): If Lidia travels 52 miles at 13 miles per hour, it will take her 4 hours. Therefore, Mike will travel for 3 hours. Since his speed it 18 miles per hour, he will travel 54 miles in this time. Together, the number of miles covered by Mike and Lidia is: 52 + 54 = 106. All of the information in the question stem is satisfied if Lidia travels 52 miles, so choice (C) must be correct. ---------- This seems to indicate how long it would take Lidia to travel the entire distance, but not how long it would take for her to cover the distance before running into Mike. Can you clarify?

If by entire distance you mean 106 miles, then NO. Considering Lidia's speed, 13 mph, it will take her 106/13 = 8 2/13 hours to travel that distance.

But actually the question statement is about their meeting only, and it does NOT specify were they head afterwards. So when we set up the variables, we consider situation of their meeting, and do NOT consider their path after the meeting.

Quote:

Let’s define some variables: Dm = distance Mike travels. Dl = distance Lidia travels. t = time Lidia travels. t – 1 = time Mike travels.

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