Today, I'm going to walk you through a math problem that comes up frequently on the SAT and ACT. It's an averages question, but rather than asking you to find the average of 5 numbers, the problem gives you the average of known data and asks you to find the value of a missing data point. For example: The Sharks basketball team scored an average of 52 points in their first four games. How many points would they need to score in their fifth game to raise their average score to 55? To solve this sort of averages problem, you must first understand the basic formula to find the average of a set of numbers:
Sum = Average # of Items Most people know this formula. If I told you to find the average of 5, 8, 12, and 19, you could do that by finding the sum (44) and dividing it by the number of items (4) to find the average (11). In averages problems with missing data points, the first step isn't to find the average (they've given you the average in the problem). Rather, it is to find the sum of the known values—in this case, zxthe team's scores in its first four games. Why does finding the sum help us? Because we are looking for a missing data point, and the individual data points are added together in the "sum" part of the formula. To find the sum of the known data points, we enter the information we've been given into the formula: Sum = 52 4 Since we don't know the sum, we can treat it like an unknown, x. Now the formula looks like this: x = 52 4 We can treat this like a basic pre-algebra problem. Find x, the sum of known values, by multiplying 4 and 52. x = 4 * 52 x = 208 Now, let's move to the next stage of the problem, finding the missing data point. Again, we know the average—this is the goal that the problem set out for the team. The team wants to raise their average score to 55. We know the number of games, now 5, since the missing game counts as an extra data point. We don't know the sum, but we do know the sum of all of the points scored in previous games. By plugging in the known information, we arrive at an equation that looks like this: 208 + x = 55 5 In the above equation, x is the value of our missing data point, the points needed to raise the team's overall average score to 55, and 208 + x is the sum of the points scored in all 5 games. Now that all of the values are plugged in, we can solve it like a basic pre-algebra problem. 208 + x = 55 * 5 208 + x = 275 x = 275 - 208 x = 67 Good luck, Sharks! You need to score 67 points in your fifth game to reach your desired average. Sometimes, rather than asking for a single missing value, the question asks you to find the sum or average of multiple missing values. You can use this same methodology. This time, in the second equation, x will represent the sum of the missing values rather than a single value. For example: Mrs. Thompson's math class took a test on Monday, but 6 of the 18 students were absent. The students who took the test on Monday averaged 43 out of 50 points. The students who were absent on Monday will make up the test on Wednesday. What will these remaining students need to average to raise the class average to 45? Note that there is some superfluous information in this problem. It doesn't really matter to us, for the purposes of solving the question, when the students take the make-up exam. It also doesn't matter how many possible points were on the exam. This is why underlining is so important. By underlining, you can make the relevant information stand out. Also note that they've told us how many students were absent, so we need to deduce how many students took the test on Monday by subtracting 6 from 18. Mrs. Thompson's math class took a test on Monday, but 6 of the 18 students were absent. The students who took the test on Monday averaged 43 out of 50 points. The students who were absent on Monday will make up the test on Wednesday. What will these remaining students need to average to raise the class average to 45? Okay, let's solve it. Sum = Average # of Items x = 43 12 x = 12 * 43 x = 516 The 12 students who took the test on Monday scored 516 points, total. Now, let's find the total points the 6 students taking the make-up exam will need to score to raise the class average to 45. 516 + x = 45 18 516 + x = 810 x = 810 - 516 x = 294 294 is the sum of the scores of the remaining 6 students. To find their average score, simply divide 294 by 6. 294 = 49 6 The remaining students must average 49 out of 50 to raise the class average to 45. Possible, but I hope they've been studying! This is a great problem style to master before the SAT and ACT. Are there additional problem types that you would like explained in the blog? Contact me!
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