SAT Mathematics: Algebra And Geometry Quiz!

Explanation
Set up a simple equation: T = K + 4 K + 2 = 2 * (M + 2) M = 4 Substitute 4 for M. K + 2 = 2 * (4 + 2) K + 2 = 2 * 6 K + 2 = 12 K = 10 Then substitute 10 for K. T = K + 4 T = 10 + 4 T = 14

Explanation
2x = 3(x-2)
2x = 3x - 6
0 = x - 6
x = 6

Substitute 6 for x.

6*6 + 3 = 36 + 3 = 39

Explanation
All non-perfect square roots are irrational. These include the square root of 2, 3, 5, 6, 7, 8 and so forth.

Explanation
2(x-3) = 14, so (x-3) = 7.

x^2 - 6x + 9 factors into (x-3)^2.

Substitute 7 for (x-3).

(x-3)^2 = 7^2 = 49

Explanation
To solve this problem, plug each of the given values of x in. You will find that only the square root of 7/2 comes out to a clean 1/(integer) answer. 1 / (sqrt(7/2)^2 - 3) = 1 / (7/2 - 3) = 1 / (7/2 - 6/2) = 1 / (1/2) = 2

Explanation
An even number is always divisible by two, meaning two is always a factor in an even number. Therefore, the product of an even and an odd still includes two as a factor, making the product necessarily even. The only non-even answer is 15.

Explanation
There are a total of X + (X-4) + (3X + 2) marbles in the bag, which equals a total of 5X - 2 marbles. Furthermore, you are given the total number of marbles to be 18. So:

5X - 2 = 18
5X = 20
X = 4.

The chance of pulling a red or blue marble is equal to:

Probability(Red) + Probability(Blue) =

X/(5X-2) + (X-4)/(5X-2)

= (2X-4)/(5X-2)

Substituting 4 for X, you have the result:

(2X-4)/(5X-2)
(2*4 - 4)/(18)
4/18 = 2/9

Explanation
Thomas = 2J and Terry = Thomas - 40 = 2J-40.

If she receives an extra 20 dollars, it becomes:

2J-40 + 20 =
2J-20

Explanation
Break up x^6 into components that you understand.

x^3 * x^3 =
12 * 12 = 144

Explanation
First, simplify the left-hand side. a - (4 - a) = 3*a + 3 a - 4 + a = 2*a + 3 2a - 4 = 3a + 3 Now, solve for a by combining like terms: -4 = a + 3 a = -7

Explanation
Call the largest integer x. Then the other integer = (x-20). Also, the product between the two is equal to 96. So:

x*(x-20) = 96
x^2 - 20x - 96 = 0

Using the quadratic formula, you find that the positive root of this equation is 24. Thus, the large factor X = 24.

The other factor is equal to X - 20 = 24 - 20 = 4.

So, the sum of the factors is equal to 24 + 4 = 28

Explanation
Two lines can intersect at 0, 1, or many (infinite) points. Parallel lines (lines with the same slope) never intersect while all other lines intersect at exactly one point. If the two lines described are identical, they are everywhere intersecting. To determine if the lines are parallel, we find both of their equations in slope-intercept form:

2x + 2y = 3
2y = -2x + 3
y = -x + 3/2

The lines are:

y = -x + 3/2 and
y = -x + 1

Since they have the same slope but different y-intercepts, they never intersect.

Explanation
To determine the perimeter, we will first need the length of a single side. Since the area of the square is equal to the length of the side squared, we find:

s^2 = 121
s = 11

Then, since the perimeter of a square is 4 times the length of a side, we have the result:

4 * s =
4 * 11 =
44

Explanation
To find F(G(3))-G(F(3)), first find the values: F(3) = 2(3) + 2 = 6 + 2 = 8 G(3) = 3^2 - 1 = 8 Then, substitute 8 in each respective function: F(G(3)) - G(F(3)) = F(8) - G(8) = 2(8) + 2 - (8^2 - 1) = 16 + 2 - 65 = -45

Explanation
Only in C does the digit in the thousands place (1) equal the digit in the tenths place, or one place to the right of the decimal.

Explanation
X% of A means (X / 100) times A.

So,

X% of A = 20
A*(X/100) = 20
A = 20/(X/100)
A = 2000/X

A% of X means X(A/100)

X*(A/100) =
X*(2000/X)/100 =
20.

Explanation
Since F(x-3) = x, F(14) = F(17-3) = 17.

Explanation
If two integers are consecutive, and the larger is called X, then they can be said to be:

X and X-1

Since X(X-1) = A:

X-1 = A/X.

Explanation
Exactly 10 prime numbers are less than 31. To compute this, ignore even numbers other than 2(obviously not prime) and the number 1, which is neither prime nor composite. The primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Explanation
First, solve for x:

2x - 3 = 9
2x = 12
x = 6.

Next, substitute 6 for x.

x^2 - 4x + 2 =
6^2 - 4*6 + 2 =
36 - 24 + 2 =
12 + 2 =
14

Explanation
To determine for how many integer values of x the expression x^2 - 4 < 36 holds true, you can set up the inequality and solve it:
x^2 - 4 < 36
First, add 4 to both sides of the inequality:
x^2 - 4 + 4 < 36 + 4
x^2 < 40
Now, take the square root of both sides, but remember to consider both the positive and negative square roots:
√(x^2) < √40
x < ±√40
x < ±2√10
So, the inequality x^2 - 4 < 36 holds true for all integer values of x that are less than ±2√10. This means that x can be any integer between -6 and 6 (excluding -6 and 6), because ±2√10 is approximately ±6.32, and integers in the range -6 to 6 (excluding -6 and 6) satisfy the inequality. Therefore, there are 13 integer values of x that satisfy the inequality.

Explanation
Set up a simple equation:

A = 100t; B = 150t

You want to know when the distance traveled by A = dist traveled by B, so set A + (head start) = B

A + t(A) = B
100t + .5(100) = 150t
100t + 50 = 150t
50t = 50
t = 1

One hour = 60 minutes.

Explanation
To find the number of ways the top three runners can place in a race with five runners, you can use permutations. In this case, you want to calculate the number of permutations of 3 runners out of 5.
The formula for permutations is:
n P r = n! / (n - r)!
Where:
n is the total number of items to choose from (in this case, 5 runners).
r is the number of items to choose (in this case, 3 runners).
"!" denotes factorial, which means multiplying all positive integers from 1 to the given number.
So, for this problem:
5 P 3 = 5! / (5 - 3)! 5 P 3 = 5! / 2!
Calculating the factorials:
5! = 5 x 4 x 3 x 2 x 1 = 120 2! = 2 x 1 = 2
Now, divide 5! by 2!:
5 P 3 = 120 / 2 = 60
There are 60 different ways the top three runners can place in the race.