Session 4 SAT/ACT Homework Assignment

    • Full-length Practice SAT
      • Take SAT Practice Test 3 from the Official SAT Study Guide in full as a test simulation
        • Time yourself strictly on each section
        • Use the provided grid sheet to bubble your answers; when grading, grade your grid sheet rather than your test booklet (as will be the case on the real test).
        • Grade yourself using the answer key (immediately following the test in the Official Guide) and mark and review any questions you didn't answer correctly.
    • Math Challenge Problems - For ambitious mathematicians, here are some challenging problems to hone your skills to take out the boss:
      • SAT Averages
        • A group of high school juniors and seniors take a practice SAT. Each student's score is a multiple of ten between 400 and 1600. The juniors average 1312, the seniors average 1360, and the full group's average is 1332. What is the minimum number of students in the group?
        • Answer and Explanation

          Let's say there are j juniors. The sum of their scores must end in zero since each student's score is a multiple of ten. But the sum of their scores is 1312j, so j must be a multiple of 5 - only a multiple of 5 times a number ending in 2 generates a product ending in zero. Thus the number of juniors is a multiple of 5.

          Since the seniors' average is already a multiple of ten, we don't really know if there are more than one senior based on the average alone.

          Let's say there are s seniors. The average of the full group is the sum of all scores, which is expressible as 1312j + 1360s, divided by the total students, expressible as j + s. We can then create the equation (1312j+1360s)/(j+s) = 1332. In two steps we find that 7s = 5j, and in a couple more steps that j/s = 7/5, in other words, the ratio of juniors to seniors is 7 to 5. This means that the number of juniors is a multiple of 7.

          Since j is an integer divisible by both 5 and 7, it must be a multiple of their product, 35. Thus the minimum number of juniors is 35; in this case, the number of seniors is 25, and the total number of students is 60.
      • Relative Age
        • Lila says to Nate, "When I'm three times my current age, you'll be twice my age." How many times Lila's age is Nate at present?
        • Answer and Explanation
          Nate is currently 4 times Lila's age.

          If Lila is currently L years old, then she will be three times her current age, 3L, in 2L years. At that point, Nate will also be 2L years older than his current age, and based on Lila's statement, will then be 6L years old, twice Lila's age (3L) at that point. So Nate must be 4L years old at present, 4 times Lila's present age.
      • Billable Hours
        • The law firm Dewey Cheatem and Howe has 52 lawyers, each either a partner or an associate. Senior partners each work with four associates exclusively assigned to them; junior partners each work with two associates also exclusively assigned. Partners bill at $400/hour; associates bill at $250/hour. On a busy day, every lawyer at the firm works eight billable hours, earning the firm a billable total of $120,800. How many senior partners are there?
        • Answer and Explanation

          Let's say there are P partners and A associates, and P+A=52, so that A=52-P. The total cost for 8 hours of work for all partners and associates is 8 * (400P+250A), which we are told equals 120,800. We substitute 52-P for A, distribute the 250, combine like terms, and in two more steps we find that P=14. This means that A=38 since their sum is 52.

          Let's say there are S senior partners and J junior partners. S+J=14, the total number of partners, and thus J=14-S. The number of associates, 38, must be A = 4S+2J. We substitute 14-S for J, distribute, combine like terms, and find that S=5.
      • Double Donors
        • A group of twenty donors each contributed $12 to Kittens Without Borders or $18 to Recovering Politicians Anonymous or made both contributions. If $180 was collected for each charity, how many donors contributed to both?
        • Answer and Explanation

          $180 collected for Kittens Without Borders means 15 donations of $12 each, while $180 collected for Recovering Politicians Anonymous means 10 donations of $18 each. Thus, 25 total donations were made by only 20 donors, meaning 5 donors made both donations.