# How fast does the athlete run?

#### onesun0000

##### New member
Two athletes at the same position on a circular 4-mile race track start running. If they run in opposite directions, they meet in 12 minutes. How ever, it takes one hour for the faster runner to gain a lap if they run in the same direction. How fast, in miles per hour, does the faster athlete run?

This is the explanation to the answer provided by the book:

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Let $$\displaystyle x$$ and $$\displaystyle y$$ be the speed of the faster athlete and slower athlete in mules per hour, respectively. Two athletes meet in 12 minutes or ($$\displaystyle \frac{1}{5}$$hr) is 4 miles. This can be written as

$$\displaystyle \frac{x}{5}+\frac{y}{5}=4\Rightarrow x+y=20$$

It takes one hour for the faster athlete to gain a lap if both athletes run n the same directions, which means that the difference of the distances that they run is 4 miles. This can be written as

$$\displaystyle x-y=4$$
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The explanation continues to solving the system and eventually get $$\displaystyle x=12.$$ I don't understand the part "...which means the difference of the distances they run is 4 miles...". Why is that? and why the same variables when it says those represent the speed of the two athletes?

#### Subhotosh Khan

##### Super Moderator
Staff member
Two athletes at the same position on a circular 4-mile race track start running. If they run in opposite directions, they meet in 12 minutes. How ever, it takes one hour for the faster runner to gain a lap if they run in the same direction. How fast, in miles per hour, does the faster athlete run?

This is the explanation to the answer provided by the book:

---
Let $$\displaystyle x$$ and $$\displaystyle y$$ be the speed of the faster athlete and slower athlete in mules per hour, respectively. Two athletes meet in 12 minutes or ($$\displaystyle \frac{1}{5}$$hr) is 4 miles. This can be written as

$$\displaystyle \frac{x}{5}+\frac{y}{5}=4\Rightarrow x+y=20$$

It takes one hour for the faster athlete to gain a lap if both athletes run n the same directions, which means that the difference of the distances that they run is 4 miles. This can be written as

$$\displaystyle x-y=4$$
----

The explanation continues to solving the system and eventually get $$\displaystyle x=12.$$ I don't understand the part "...which means the difference of the distances they run is 4 miles...". Why is that? and why the same variables when it says those represent the speed of the two athletes?
".... it takes one hour for the faster runner to gain a lap...."

In ONE hour the first athlete would have run "x mile" at the rate of x miles/hour

In ONE hour the second athlete would have run "y mile" at the rate of y miles/hour

#### lev888

##### Full Member
1. When they run in opposite directions their speeds combine: x+y
Speed * Time = Distance: (x+y)*(1/5) = 4 or x+y = 20

2. Same direction. Their combined speed now is x-y. In one hour this difference in speed accounts for one lap (4 miles): (x-y)*1=4.

#### Jomo

##### Elite Member
In the equation x+y=20 why are you not saying that x+y, which is the sum of two speeds, can't equal 20 which is a number of miles. The reason is x+y = 4 comes from x*(1/5) +y*(1/5) =4 where the left side is the sum of distances.

In x-y = 4. it is really x*1hour - y*1hour, which is a distance and equal 4 miles which is too a distance.

It is excellent that you are looking at units but sometimes you need to be careful!