The following solutions are for my PSAT Math Grid-In Practice Questions. Once you've completed the problems on that page, then read through this solutions page to see how well you've done!
Solution for PSAT Math Practice Question 1
First figure out how many hours a week Karen works. She works 3 days of 10 hours (so 10 * 3 = 30 hours) and 2 days of 6 hours (so 2*6 = 12 hours). Now that we know that Karen works a total of 42 hours a week, we can divide that into the total amount of money she made, and that will give us her wages per hour. So 378/42 = 9. Therefore, Karen makes $9 per hour.
General Tip: When problems use the word “per” that means that the following measure (hours, meters, grams, etc.) will be in the denominator of the total fraction. So if a problem like the one here looks for how much a person makes per hour, you know (in most cases) that you need to divide by hours to get the answer.
Solution for PSAT Math Practice Question 2
Since a negative times a negative makes a positive, any even number of negative integers will produce a positive. In our case, 7 negative integers would give us 3 pairs of negatives that produce positives, with a negative left over to make the final product negative. Therefore, at most 6 negative numbers will produce a positive product.
General Tip: Go over your rules for addition, subtraction, multiplication, and division of negatives. There will always be a question on the SAT or PSAT that tests this area.
Solution for PSAT Math Practice Question 3
This cube will be cut into three parts per side. So there will be three sorts of smaller painted cubes. The corners will have paint on 3 sides. The inner pieces will have pain on one side. The cubes on an edge, but not on a corner will be painted on two sides. Since there are 12 edges and only one of these cubes will emerge painted on two sides per edge (since we cut it into 3’s) our answer is 12.
General Tip: Draw a picture when the question is asking a geometrical problem. It can greatly help, even if you are sure of the math.
Solution for PSAT Math Practice Question 4
¼ = 0.25, so 0.27 – 0.25 = 0.02. 0.02 = 2/100 and reducing, we know it’s 1/50.
General Tip: If you don’t like fractions, then turn them into decimals. Just make sure that when you are converting, you always put a multiple of 10 in the denominator of the fraction. So don’t try to convert .25 into anything but 25/100. After that, you can reduce to ¼.
Solution for PSAT Math Practice Question 5
Factor: x squared – y squared = (x + y) * (x – y). Then we have [(x + y) * (x – y)] / (x + y). So we can cancel out the (x + y) in the numerator and denominator and we are left with x – y.
General Tip: Keep that rule handy! The difference of two squares will always produce the result above. So if we have 144 * a squared – 169 * b squared, that equals (12a + 13b) * (12a – 13b). Always.
Solution for PSAT Math Practice Question 6
Catherine runs a total of 26 + 30 = 56 miles per hour. And since she ran the course twice, and the course is 4 miles long, her total distance is 4 * 2 = 8 miles. To find the miles per hour, we divide 56 by 8 and get 7 miles per hour.
General Tip: Don't assume the numbers in the problem will directly lead you to the answer. Rather, figure out what the question is asking you to do first, and then figure out how you will use the numbers.
Solution for PSAT Math Practice Question 7
Solution by substitution: x + 2y = 10 => x = 10 – 2y. 2x – y = 25 => 2(10 – 2y) – y = 25 => 20 – 4y – y = 25 => 20 – 5y = 25 => - 5y = 5 => y = -1. x = 10 – 2y => x = 10 – 2 (-1) = 10 + 2 = 12. So x = 12 and y = -1. Then 3 (12) + 3 (-1) = 36 – 3 = 33.
General Tip: If you have a way you like to solve something, use it! This could be solved many different ways—multiplying an equation by a number and then adding or subtracting both equations together, using matrices, even using calculus. You should always pick a way to solve linear equations that makes sense to you.
Solution for PSAT Math Practice Question 8
A reduction of 30% is the same thing as multiplying an item by 70% (which is 100% - 30%). The same thing is true with 20% and 80%. Then .7 * .8 = .56. But 56% is not a reduction, so we need to turn it back. 100% - 56% = 44%. That is our answer.
General Tip: When dealing with percentages, always differentiate between when you are reducing (like in this problem) or when you are finding a “percentage”. Reducing a number by 25% is the same thing as finding 75% of that number.
Solution for PSAT Math Practice Question 9
Draw a picture of the situation. The formula for the circumference of a circle is C = dπ where d is the diameter of the circle. Notice that you have a right triangle if you connect any opposite ends of the rectangle with a straight line and that line you just drew is also the diameter of the circle. Then you just need to use Pythagorean's theorem using the two sides of the rectangle to give you that diameter. So 62 + 82 = 36 + 64 = 100. And the square root of 100 is 10. So the diameter is 10 and our answer is 10π.
General Tip: If you need to use a diameter or a radius that isn't in the original picture, start drawing extra lines until you find a line that helps you solve the problem.
Solution for PSAT Math Practice Question 10
The first bounce is 343 inches, so we need to find 2/7 of 343. To do that, just multiply 343 * (2/7) to get 98. That is the second bounce. Multiplying again to get the third bounce, we have 98 * (2/7) = 28. For our fourth and final bounce, we need to multiply one more time. 28 * (2/7) = 8.
General Tip: When you are looking for a fraction of a given amount, you can just multiply by the whole fraction. The denominator of the fraction is a built-in function that takes care of the necessary division.
