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1999 CEMC Gauss (Grade 8) Problems/Problem 15

Problem

The graph shown at the right indicates the time taken by five people to travel various distances. On average, which person travelled the fastest?

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$\text{(A)}\ \text{Allison} \qquad \text{(B)}\ \text{Bina} \qquad \text{(C)}\ \text{Curtis} \qquad \text{(D)}\ \text{Daniel} \qquad \text{(E)}\ \text{Emily}$

Solution 1

We know that $s = d/t$, where $s$ is the average speed, $d$ is the distance, and $t$ is the time. We want to look for the highest value of $s$ among the people.

According to the graph, Allison traveled $1$ kilometer in $20$ minutes. This means that her speed was $\frac{1}{20} = 0.05$ kilometers per minute.

Using the same method, we see that Bina traveled $1$ kilometer in $50$ minutes. This means that her speed was $\frac{1}{50} = 0.02$ kilometers per minute. This means that she was not the fastest, as her rate was lower than Allison's rate.

Curtis traveled $3$ kilometers in $30$ minutes. His rate was $\frac{3}{30} = \frac{1}{10} = 0.1$ kilometers per minute, which was higher than Allison's rate. This means that he is the fastest so far.

Daniel traveled $5$ kilometers in $50$ minutes. His rate was $\frac{5}{50} = \frac{1}{10} = 0.1$ kilometers per minute. This was the same as the rate at which Curtis traveled, so neither of them were the fastest (according to the rules of the test, none of the problems had more than one answer).

Because all of the other answer choices have been eliminated, the answer must be $\boxed {\textbf {(E) } \text{Emily}}$. We can also verify this by calculating Emily's rate, which was $\frac{5}{20} = \frac{1}{4} = 0.25$.

~anabel.disher

Solution 1.5

Similar to solution 1, we can use $s = d/t$.

Without calculating the rates, we know that Bina could not have traveled the fastest because Bina took longer to travel $1$ kilometer than Allison did. Similarly, we know that Daniel also couldn't have been the fastest because Daniel took longer than Emily to travel the same distance.

We also know that Allison was slower than Emily because she traveled a smaller distance in the same amount of time.

This means that we only need to compare the speed of Emily and Curtis. Using the same process as solution 1, we see that Daniel's rate was $\frac{5}{50} = \frac{1}{10} = 0.1$ kilometers per minute, and that Emily's rate was $\frac{5}{20} = \frac{1}{4} = 0.25$ kilometers per minute.

Of these two, $\boxed {\textbf {(E) } \text{Emily}}$ was faster.

~anabel.disher

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