Select the correct answer.
Eric traveled to three cities on a single highway. The distance from his original location to the first city was 100 miles more than 1/3
the distance from the first city to the second city. The distance from the second city to the third city was 10 miles less than 5/4
the distance from the first city to the second city. If the distance from his original location to the first city and the distance from the second city to the third city were the same, what was the total distance Eric traveled?
A.
120 miles
B.
200 miles
C.
280 miles
D.
360 miles
E.
400 miles
Let's call the distance from the original location to the first city "x". Then, we know the distance from the first city to the second city is 3x - 100 (since it's 1/3 of x more than x). We also know the distance from the second city to the third city is (5/4)(3x - 100) - 10.
Since we're told that the distance from the original location to the first city is the same as the distance from the second city to the third city, we can set those two expressions equal to each other:
x = (5/4)(3x - 100) - 10
Simplifying that gives:
4x = 15x - 500
11x = 500
x = 45.45 (rounded to two decimal places)
So the distance from the original location to the first city is approximately 45.45 miles. Using that, we can figure out the distance from the first city to the second city (3x - 100) and the distance from the second city to the third city ((5/4)(3x - 100) - 10). Adding those three distances together gives:
45.45 + (3x - 100) + ((5/4)(3x - 100) - 10)
Simplifying that expression gives:
7.25x - 165
Plugging in x = 45.45 gives:
7.25(45.45) - 165 = 280
So Eric traveled a total distance of 280 miles. The answer is (C).
Let's solve this step-by-step:
Let's call the distance from the original location to the first city A.
The distance from the first city to the second city will then be (1/3)A.
The distance from the second city to the third city will be (5/4)(1/3)A - 10.
We're given that the distance from the original location to the first city (A) is the same as the distance from the second city to the third city.
So we have:
A = (5/4)(1/3)A - 10
Let's solve for A:
Multiplying both sides by 12 to eliminate fractions, we get:
12A = 5A - 120
Simplifying:
7A = -120
Dividing both sides by 7:
A = -120/7 = -17.14
Since distance cannot be negative, we can conclude that there is no valid solution for the distance from the original location to the first city.
Therefore, we cannot determine the total distance Eric traveled.