SAT Reasoning Student-Produced Responses

Question 1: In the x,y plane, what is the distance between origin and the (4,3) point?

Answer: 5
Explanation: We apply Pythagoras' Theorem to the triangle created by the following points: (4,3), (4,0) and (0,0).
The distance between origin and the (4,3) point is equal to 5, as the other 2 sides of this right triangle are 4 and 3.

Question 2:   The sum of 4 consecutive integers is 26. What is the product of the 4 numbers?

Answer: 1680
Explanation: If m is the least of the integers, m + (m + 1) + (m + 2) + (m + 3) = 26
4m + 6 = 26
m = 5, so the 4 consecutive numbers are 5, 6, 7, 8.
Their product is 5·6·7·8 = 1680

Question 3:   60% of the players of a team score 5 points each during a basketball game, 30% of players score 1 point each, and 10% score 7 points each. What is the average number of points scored by the players of the team?

Answer: 4
Explanation: n being the number of players, 60% of the players is .6·n, 30% of the players is .3·n and 10% of the players is .1·n.
The numbers of points scored by players that scored 5 points is (.6·n)·5.
The numbers of points scored by players that scored 1 point is (.3·n)·1.
The numbers of points scored by players that scored 7 points is (.1·n)·7.
The total number of points will be (.6·n)·5 + (.3·n)·1 + (.1·n)·7 = 4·n and the average number of points = ( total number of points)/number of players = 4·n/n = 4

Question 4:   If f(x) = x - 1, what is the value of x that satisfies the equality f(x) - 1 = f(x²) - x²

Answer: 1
Explanation:
f(x) - 1 = f(x²) - x²
(x - 1) - 1 = (x - 1)² - x²
x - 2 = x² - 2x + 1 - x²
x - 2 = -2x + 1
x = 1

Question 5:   The amount of time taken to build a house is inversely proportional to the number of construction workers on the job. If it takes 6 workers 6 days to complete such a job, how many days will it take if the job is done by only 4 workers?

Answer: 9
Explanation: The amount of time taken to build a house is inversely proportional to the number of construction workers on the job, so N_workers = k/N_days
k = N_workers·N_days
k = 6·6, as it takes 6 workers 6 days to build the house
k = 36 worker·days
k = 4 workers · N_{days_4workers}
N_{days_4workers} = 36/4 = 9 days

Question 6:   If side a of triangle ABO is tangent to the circle of center O, what is the value of the radius of the circle in inches? Side a of the ABO triangle is 4 inches and side b of the same triangle is 5 inches.

Answer: 3
Explanation: Side a is tangent to the circle, so the triangle created by the sides a, b and c is right.
c² = b² - a²
c² = 5² - 4²
c² = 9
c = 3

Question 7:   A jar contains a nickel, a dime, and six quarters. Tom draws a nickel from the jar, and then Mary draws a dime from those remaining. If the probability that Tom draws the nickel and Mary draws the dime is 1/x, what is the value of x?

Answer: 56
Explanation: The probability that Tom draws the nickel is (number of nickels)/(number of coins in the jar) = 1/8
After Tom draws a coin, the number of coins in the jar is 8 - 1 = 7
The probability that Mary draws the dime is (number of dimes)/(number of coins in the jar) = 1/7
The probability that Tom draws the nickel AND Mary draws the dime is the product of the 2 probabilities calculated above.
1/8 · 1/7 = 1/56 so x = 56

Question 8:   What is the length of the sides of an equilateral triangle, if the lengths of the altitudes of the triangle are √3?

Answer: 2
Explanation: In the triangle formed by the the altitude a, one side l and the base of the altitude, l² = a² + (l/2)²
(3/4)·l² = a²
(3/4)·l² = (√3)²
(3/4)·l² = 3
l² = 4 and l = 2.

Question 9:   If the population of virus doubles every 90 minutes, the population after 9 hours will be how many times the population at the start?

Answer: 64
Explanation:
After 90 minutes the virus population will be 2·x, where x is the population at the start.
After 3 hours the virus population will be 2·(2·x) = 2²·x.
After 4 hours and 30 minutes, the population will be 2³·x.
After 6 hours, the population will be 2⁴·x.
After 7 hours and 30 minutes, the population will be 2⁵·x.
After 9 hours, the population will be 2⁶·x.

Question 10:   f(x) = |x| and g(x) = x - 1. How many solutions x satisfy f(x) = g(x)?

Answer: 0
Explanation:
The simplest way to solve this problem is to draw the 2 functions in the x,y plane.
We notice that for any x < 0, f(x) is positive and g(x) is negative and they don't intersect.
For any x > 0, f(x) is parallel with g(x) and they do not intersect
in conclusion there are no x that satisfy f(x) = g(x).