Section 3.5
Averages
Examples:
a. Judy has grades of 75%, 78%, and 86% on three tests. What grade must she get on the final exam (which counts as two tests) in order to pull her grade average up to 85%.
Solution:
Let $x$ be Judy's grade on the final exam. Then the average of the five grades 75, 78, 86, $x$, and $x$ (since the final exam counts as two tests) must be 85.
\begin{align}
\frac{75+78+86+x+x}{5}&=85\\
\frac{239+2x}{5}&=85\\
239+2x&=425&&\textit{Mulitply both sides by 5}\\
2x&=186\\
x&=93
\end{align}
Judy needs to get a 93% on the final exam.
b. A baseball team has won 24 of its first 30 games. It then goes into a tailspin and loses several games in a row, bringing its winning percentage down to .600. How many games does the team lose during its slump?
Solution:
Let $x$ be the number of games the team loses in a row. After those losses, it will have won 24 games out of $30+x$ games.
\begin{align}
\frac{24}{30+x}&=.600\\
\frac{24}{30+x}&=\frac{3}{5}\\
90+3x&=120&&\textit{Cross multiply}\\
3x&=30\\
x&=10
\end{align}
The team loses 10 straight games.
Averages: Introduction to averages and algebra problems involving averages.
Exercises
1. Sara has test grades of 95, 96, 93, and 98 going into the final exam. The exam counts as two tests. How badly can she do on the final and still have a 93 average for the term?
2. The average weight of five boys is 58 kilos. Three of the boys weigh in at 55, 59, and 64 kilos, respectively. The other two boys have the same weight. What is that weight?
3. Brady's average on the first three tests of the term is 75%. What does he need to score on the fourth test to bring his average up to 80%?
4. A student has test scores of 75, 82, and 87 on the first three tests of the term. She has two tests, plus a final exam that counts as two tests, still to take. Her hope is to do well enough to bring her average up to 85. What must she average on the remaining work to make that happen?
5. To get a satisfactory grade, a student must have an average of 70% on five tests. The average of the first four tests was 68.5%. What grade on the fifth test would guarantee a satisfactory grade?
6. A basketball player has made 80% of his free throws. He then makes his next 10 free throws in a row, bringing his average up to 84%. How many free throws had he taken before going on the great run?
7. At the end of May, a baseball player had 36 hits in 135 at-bats. During the month of June, he hit .400, and thereby brought his average for the season up to .320. How many at-bats and how many hits did the player have in June?
8. A baseball team played at a .333 clip over the first one-third of their season (18 wins in 54 games). They played at a .667 clip over the remainder of the season (72 wins in 108 games). What was their winning percentage for the season?
9. A soccer team has won 5 of their first 20 games in the season. If they win half of their remaining games, their winning percentage will climb to .400. How many games remain in the season?
10. Juanita had five tests this term with grades of 72, 80, 86, 75, and 83. Her quiz average is 77. Her homework grade is 91. The final exam counts for two tests while the quiz grade and homework grade each count for half a test.
a. What is the highest average Juanita can get for the term?
b. What grade does Juanita need on the final exam to have a term average of 83?
11. A man drove at 40 miles per hour for the first part of a trip and then at 60 miles per hour for the second part of the trip.
a. What was his average speed for the trip if each part was 2 hours long?
b. What was his average speed for the trip if each part was 120 miles long?
c. If the first part of the trip lasted two hours, then how long should the second part of the trip last if the average speed for the whole trip is 45 mph?
12. Alex takes part in a mini-triathlon. He swims 2 miles, then bikes 18 miles, and finally runs 10 miles. He swims at 4 miles per hour and bikes at 27 miles per hour. How fast does he need to run in order that his average speed for the whole mini-triathlon is 12 miles per hour?
13. The average of the numbers a and b is 10. The average of the numbers c and d is also 10. What is the average of the four numbers $a$, $b$, $c$, and $d$?
14. There are 320 seniors, 290 juniors, 280 sophomores, and 210 freshmen in a certain high school. On January 1, the average age of the seniors was 17.8, the average age of the juniors was 16.9, and the average age of the sophomores was 15.4. The average age of the whole student body was 16.3. What was the average age of the freshman class?