Math SAT problem simplified!
Article published in Longmeadow Neighbors
(February 2022)
EXPERT CONTRIBUTOR
ALISON MCDONOUGH
Math Tutor
Mathnasium
413 612 2877
longmeadow@mathnasium.com
Mathnasium.com/longmeadow
It’s the test will be offered seven times between now and next year, March is a popular choice, particularly for first-time juniors. Whether you’re taking the test in a few weeks or a few months, for the first time or the fifth, the following tips applied to an example of a real-life problem will help transform the SAT Math sections from onerous to effortless. Let’s begin!
In a class of p students, the average (arithmetic mean) of the test scores is 70. In another class of n students, the average of the scores for the same test is 92. When the scores of the two classes are combined, the average of the test scores is 86. What is the value of p/n?
There are two key things to keep in mind when approaching this problem. The first: Never do more work than you must! The question asks for the value of p/n, not the individual values of either p or n. Finding individual values is not always necessary, and in this example is not actually possible with the information given. Don’t fall into this trap and waste precious minutes attempting the impossible! Instead, let’s translate the information we have into simple equations and see if we can solve for p/n directly.
The second key thing is to always use the easiest version of any formula. This isn’t always the version we learn first in math class! In this problem, students will recognize that they need to use the formula for arithmetic mean, but they may only be familiar with: “The sum of the set divided by the number of elements equals the mean”. However, if we multiply both sides of this formula by the number of elements, we get an equivalent formula that is extremely helpful on the SATs: “The sum of the set equals the mean multiplied by the number of elements”. This is the version we will use to solve the above problem.
“In a class of p students, the average of the scores is 70”. The sum of these scores equals 70p, the mean times the number of scores.
“In another class of n students, the average is 92”. The sum of the second class equals 92n.
“When the scores of both classes are combined, the average of the est scores is 86”. There are p students in the first class and n in the second, so there are p+n scores total and the sum of scores from both classes equals 86(p+n). Since the sum of the combined scores is simply the sum of the first class plus the sum of the second, this also equals 70p + 92n. Setting these equal, we get the equation:
86(p+n)=70p+92n
86p+86n = 70p + 92n
86p-70p = 92n-86n 16p=6n
At this point, keep in mind that we don’t need p or n, we just need p/n. Solving for a quotient can be just as straightforward as solving for a variable if we isolate p first and then divide by n:
p=6n/16
p/n=6/16
p/n=3/8.
Best of luck to all Longmeadow students taking the SATs!
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