Let's solve a classic meeting problem. Two cars start simultaneously from cities A and B, driving toward each other. Car B travels at four-fifths the speed of car A. They meet at a point that is 12 kilometers away from the midpoint between the two cities. We need to find the total distance between cities A and B.
Now let's set up our variables. We'll call the total distance between cities A and B as d kilometers. Car A travels at speed v, while car B travels at four-fifths of that speed. The meeting point is 12 kilometers away from the midpoint, which means it's at a distance of d/2 plus 12 from city A.
Here's the key insight: since both cars start simultaneously and meet at the same point, they must have traveled for exactly the same amount of time. Car A travels a distance of d/2 plus 12 kilometers, while car B travels d/2 minus 12 kilometers. Using the formula time equals distance divided by speed, we can set up an equation where both travel times are equal.
Now let's solve the equation step by step. First, we simplify the denominator on the right side by flipping the fraction. Then we cross multiply to eliminate the fractions. After canceling out v from both sides, we expand the parentheses. Multiplying both sides by 2 to clear the remaining fraction, we get 4d plus 96 equals 5d minus 120. Solving for d, we find that d equals 216 kilometers.