Mike and Lidia are 106 miles apart and will begin riding their bicycle

After 1 hour 13 miles per hour, L has covered 13 miles
After 2 hours, L has covered 26 miles and M has covered 18 miles
After 3 hours, L has covered 39 miles and M has covered 36 miles
Half way or mid point =106/2=53 miles
After 4 hours, L has covered 52 miles and M has covered 54 miles
Answer C

Lidia and Mike are 106 miles apart to begin with. Lidia rides for 1 hour before Mike begins and she rides at a constant rate of 13 mph. Mike rides at 18 mph. Once Mike begins riding riding, Lidia has already travelled 13 mile, thus there is 93 miles for them to cover before they meet. At a combined rate of 31 mph (Lidia's 13 and Mike's 18), they will cover the remaining 93 miles in 3 hours. Lidia will have ridden 39 miles in this 3 hours, plus the 13 in the initial hour, so she will have ridden a total of 52 miles when she and Mike meet.

Answer (C) 52 is correct

D=R*T

Lidia: D = 13 * (t+1)
Mike: 106-D = 18t

Their distances will equal 106, so you can add the two equations:

13t+13+18t = 106
31t=93
t=3

Lidia: 13* (3+1) = 52

C

Two things to remember in problems in which two objects approach each other.
1. The rates/speed can be added(Relative Speed)
2. When the two objects meet, the time taken by both the objects will be same.

Here, Since Lidia Starts 1 hour before, she would have covered 13 km before Mike starts. Hence Rem. Distance = 106 - 13 =93km

Now Time Taken for them to Meet, t = Distance/Relative Speed = 93/(13+18) = 3 hours.

In Total Lidia would have traveled => 1+3 hours => 4 hours. Hence Total Distance Covered = 4 * 13 = 52 km. Option C

total speed 18+13=31; Total distance - 106-13= 93. so they meet after 93/31=3 hours. The question is How many miles will Lidia have traveled and that is 13*3+ 13= 52 miles.

800score Official Solution:

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.

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