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Problem 21
Steph scored 15 baskets out of 20 attempts in the first half of a game, and 10 baskets out of 10 attempts in the second half. Candace took 12 attempts in the first half and 18 attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
Solution:
Let \(x\) be the number of baskets that Candace scored in the first half, and let \(y\) be the number of baskets Candace scored in the second half.
Since Candace and Steph took the same number of attempts and they ended with the same overall percentage of baskets scored.
Each scored \(x+y=10+15=25\).
We have the following inequalities:
\[
\frac{x}{12}<\frac{15}{20} \Longrightarrow x<9,
\]
and
\[
\frac{y}{18}<\frac{10}{10} \Longrightarrow y<18 .
\]We know that, \(x+y=25\)
So, the only possible solution is \((x, y)=(8,17)\).
Problem 22
A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus 3 stops behind. After how many minutes will Zia board the bus?
Solution:
Suppose the bus is at Stop 0 (starting point) and Zia is at Stop 3 initially.
We cab construct a table of 5-minute intervals:
After 15 minutes the bus will leave stop 5 and then Zia will wait for the bus at stop 6. The bus takes 2 minutes to reach stop 6. Zia will board the bus after 15+2=17 minutes.
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