The document provides examples for calculating averages (arithmetic means) of sets of numbers. It gives 10 examples of calculating averages for various scenarios involving test scores, speeds, angles of shapes, and variables in equations. The examples demonstrate calculating averages by adding all values and dividing by the total number of values. They also show finding individual values before calculating an average.

1. SAT Prep Tutoring – AveragesSeptember 9th, 2010

2. Measures of Central TendencyMean = Average of a set of NumbersMedian = The number in the middle…don’t forget to place them in order from smallest to largest first!!Mode = The number that occurs most often

3. Example 1What is the average (arithmetic mean) mean of 211and 213 ?211 +213 (you can’t add those!!!)Divide by 2:Split it up: Simplify:

4. Example 2If x + y = 10, and y + z = 14 and z + x = 12, what is the average of x, y and z?Let’s solve for one of the variables x = 10 – yAnd again: z = 14 – yNow, plug both of those into the last equation:14 – y + 10 – y = 12Solve for y: 24 – 2y = 12-2y = -12y = 6So, that means x = 4, z = 8, y = 6The question asks for the AVERAGE of them:4 + 8 + 6 = 18Divide by 2 = 9.

5. Example 3Mark's average (arithmetic mean) on four tests is 70. What grade does he need on his fifth test to raise his average to an 75? Mean on 4 tests = 70 so assume each test was 70.Now, average 5 tests: (70 + 70 + 70 + 70 + x)/5 = 75Solve for x: (280 + x)/5 = 75280 + x = 375x = 95

6. Example 4If a + b = 2(c + d), which of the following is the average (arithmetic mean) of a, b, c, and d?Solve for each variable separately:a = 2c + 2d – bb = 2c + 2d – aa + b – 2d = 2cc = ½ a + ½ b –da + b – 2c = 2dd = ½ a + ½ b – cAverage those:(2c + 2d – b + 2c + 2d – a + ½ a + ½ b –d +½ a + ½ b – c)/4(3c + 3d)/4 (can’t simplify any further)

7. Example 5If 6a + 6b = 48, what is the average (arithmetic mean) of a and b?Divide both sides by 6 and you’ll get: a + b = 8Well, if a + b = 8, then divide by 2 to finish the average and get 4.

8. Example 6What is the average (arithmetic mean) of the measures of the five angles in a pentagon?We don’t even have to know what the degrees of the angles are…just their average!The sum of the interior angles in a polygon: (n-2)*180So, in a pentagon, that’s: (5-2)*180 = 3*180 = 540.540/5 = 108 degrees.

9. Example 7If the average (arithmetic mean) of 7, 8, 9, and x is 12, what is the value of x?(7 + 8 + 9 + x)/4 = 127+8+9+x = 4824 + x = 48x = 24

10. Example 8Ten students took their Math test on Tuesday and their average (arithmetic mean) was 80. Five students took a make up test and their average (arithmetic mean) was 85. What was the average for the entire class?If 10 had an average of 80, then we can figure in 80 for all 10 of those scores. If 5 had an average of 85, then we’ll figure in 85 for those 5 scores…and average all of them.(80 + 80 + 80 + 80 + 80 + 80 + 80 + 80 + 80 + 80 + 85 + 85 + 85 + 85 + 85)/15 = 81.66666

11. Example 9For the first 120 miles of the trip Dan drove at 60 miles per hour. Then because of a wreck on the Interstate slowing down traffic, he drove at 55 mph for the next two hours. What was his average speed , in miles per hour, for the entire trip?120 miles @ 60 mph (isn’t that 2 hours??)110 miles @ 55 mph (this is also 2 hours!)Since he went the same amount of time at both speeds, then average them to get 57.5mph

12. Example 10Jane's grades on her 10 tests in Science class are 60, 70, 80, 82, 84, 88, 88, 90, 95, 98. What is the average (arithmetic mean) of the median and mode of this set of data?Be VERY careful of the wording!!Median = 86 (I had to average the two in the middle)Mode = 88AVERAGE of the Median and Mode = 87. 

13. Questions?Study this and then go back and take 3.3 Averages Practice Quiz.If you’ve already taken the quiz twice and scored poorly, then nicely ask your teacher for a third chance. 