High school math Trivia Quiz

Answer: -3/4

Because the line y = mx goes through the point (0,0), it must also go through the midpoint of (3,2) and (4m,-1) in order to split the triangle in half. To find the midpoint, simply find the average of the x- and y-coordinates, which would be ((4m + 3)/2, 1/2). Substituting that point into the line equation, we get that 1/2 = m(4m + 3)/2. Simplified, we get that 4m^2 + 3m - 1 = 0, which factors to (m + 1)(4m -1) = 0.
Our two answers for m are -1 and 1/4, the sum of which is -3/4.

Answer: 16

Let n represent the highest number (not integer) that can meet these three conditions. In order to meet the first condition, n must equal 1,000,000. In order to meet the second condition, n^3 must equal 1,000,000. In order to meet the last condition, we must find the lowest perfect cube that is a multiple of 72. Since 72 breaks down to 2^3 times 3^2, we must multiply it by 3 to get a perfect cube of 216. So, the final equation is:

216n^3 = 1,000,000

This, when you find the cubic root of each side, simplifies to:

6n = 100
n = 16.666

Since n is equal to 16.6666, there are 16 numbers that meet all three conditions.

Answer: 14

The way that the system works is that the value of coins increases exponentially. So, if a white coin has a value of 1, then a yellow coin has a value of n, a green coin has a value of n^2, and so on and so forth.

So, the transaction tells us that n^4 + 33n^2 + 38n = 12n^3 + 168. Moving it all to one side, you get that n^4 - 12n^3 + 33n^2 + 38n - 168 = 0. When factored, you get that (n - 3)(n - 4)(n - 7)(n + 2) = 0, so the only possible positive values of n are 3, 4, and 7, the sum of which is 14.

I have no idea how this system would help a struggling economy, but oh well.

Answer: 24

The first cog revolves at a rate of 80 revolutions per 60 seconds, or 4 revolutions every 3 seconds. The second cog revolves at a rate of 67 1/2 revolutions per 60 seconds, or 9 revolutions per 8 seconds. The lowest common multiple of the 3 seconds it takes the first cog to point in the same direction and the 8 seconds it takes the second cog is 24 seconds.

Answer: 7

If l represents any whole number, then N(z) can be broken down to 10^l + 64, where l depends on the value of z. First, factor out the highest possible power of two from the equation. This leaves us with 2^6 * (10^l/64 + 1). To get the highest possible value of n, we must find what the greatest possible value of n is in 10^l/64 + 1. To get this to be an even number, 10^l/64 must be odd, and the only possible number it could be ends in 25. Once the 1 is added, the number ends in 26, leaving us with only one more factor of 2.

Therefore, the highest possible value of n is 6 + 1, or 7.

Answer: 11

First, draw three circles tangent to one another so that they resemble a triangle. It is more convenient if one is placed above the other two. Next, draw the fourth circle around the smaller circles. Connect all of the centers of the circles to one another, and you'll notice that an equilateral triangle is formed, with each side having a length of 2. This is the main diagram used to find the radius.

Draw a dot to represent the center of the larger circle and draw a radius to the circumference of the larger circle through one of the centers of the smaller circles, preferably the one above it. You'll notice that the radius is equal to the radius of the smaller circle plus the distance from the center of the larger circle to the center of the smaller circle. This distance can be calculated by finding the location of the point in relation to the endpoints of the equilateral triangle.

The easiest way to do this is to find where the center is the same distance from all 3 endpoints. In order to find this, treat the points as if they were on the coordinate plane (This where drawing the circle above the center comes in handy). To find the height of the triangle, simply draw in an altitude and use Pythagorean Theorem. You end up with points (0,0), (2,0), and (1,sqrt(3)). Next, find the midpoints of all three lines. You end up with (1,0), (1/2,sqrt(3)/2), and (3/2,sqrt(3)/2). Next, connect each endpoint to the opposite midpoint, and find the equations of those lines. You get x = 1, y = sqrt(3)x/3, and y = (2 - x)sqrt(3)/3. If you find the point where all three lines intersect, you get the point (1,sqrt(3)). If you check, you will notice that the distance from this point to each of the triangle's endpoints is sqrt(4/3). Since the point lies on the triangle's altitude, simply subtract the y-coordinate of the point by the altitude of the triangle, and you get 2sqrt(3)/3.

Add this to the radius of the smaller circle and you get 1 + 2sqrt(3)/3, or (3 + 2sqrt(3))/3. 3 + 2 + 3 + 3 is 11. If you find the decimal value of this solution, you'll notice that it is slightly higher than 2.

Answer: False

The difference of two squares can be represented by a^2 - b^2, which factors out to (a - b)(a + b). If both a and b are even or both a and b are odd, then both (a - b) and (a + b) must be even, meaning a^2 - b^2 must be a multiple of 4. If a and b are not both even or both odd, then (a - b) and (a + b) both must be odd, meaning that a^2 - b^2 must be odd.

Therefore, a^2 - b^2 cannot be an even integer that is not a multiple of 4, proving the statement to be false.

Answer: 1/2

To solve the equation, you first need to simplify 7^(1 - 2n). First, find the inverse of the term, so you get 1/7^(2n - 1). Then, multiply the top and bottom by 7^1 (or 7), and you get 7/7^(2n). Then, square the 7 in the denominator to get 7/49^n. The equation now looks like the following:

49^n + 7/49^n = 8

Multiply each side by 49^n, and move the term on the right over, and you get:

(49^n)^2 - 8(49^n) + 7 = 0

Factor this and you get:

(49^n - 7)(49^n - 1) = 0

This means that 49^n equals either 1 or 7, meaning n equals either 0 or 1/2, the sum of which is 1/2.

Answer: 499

If the numbers are in an arithmetic sequence, then the difference between the first and second term must be the same as the difference between the second and third term. Therefore:

(2x + 3) - (3x - 5) = (5x - 1) - (2x + 3)
-x + 8 = 3x - 4
12 = 4x
x = 3

By substituting, we know that the first three terms are 4, 9, and 14, which represent the sequence a(n) = 5n - 1. Therefore, the 100th term is 5(100) - 1, or 499.

Answer: About 70%

One way to solve this is to find the probability that no one will have the same birthday, and then subtract it from one. The first person has to avoid no birthdays, so the first probability is 365/365. The next person has to avoid one birthday, so that probability is 364/365.

The next person has to avoid two, so that probability is 363/365. This continues, so the equation is 1 - (365 * 364 * 363 * ... / 365 * 365 * 365 * ...), or 1 - ((365 P n) / (365 ^ n)), where n represents the number of people in the room. Substituting 30 for n comes out to just over .7, or 70%.