Practice Problem Set 1

$1$.
Suppose John has only green and red apples in his grocery bag. If the ratio of the number of green to red apples in his bag is $(k - 5)$ to $(k - 3)$ where $k$ is an integer greater than $4$, what's the probability John will randomly pick a red apple from the bag in terms of $k$?

(a) $\frac{k - 3}{2(k - 4)}$

(b) $\frac{k - 3}{k - 4}$

(c) $\frac{k-3}{4}$

(d) $\frac{k - 5}{2k - 8}$

$2$.
If $(2n + 1) > (5n + 1)$, then what can we say about $n$?

(a) $n < 0$

(b) $n > 0$

(c) $n = 0$

(d) None of the above

$3$.
Suppose $k$ is an integer such that $x = 2k$ and $y = 2k + 1$. What might $k$ be such that $\frac{y}{x}$ is an integer?

(a) $\frac{1}{2}$

(b) $1$

(c) $0$

(d) None of the above

$4$.
For what value(s) of $n$ is $n^2$ not equal to $2n$?

(a) $n = 0$

(b) $n = 2$

(c) both (a) and (b)

(d) None of the above

$5$.
Given the expressions below, what is the numerical value of $x$?
   (i)   $4$ of $x$ is $5$ more than $y$

   (ii)  $z$ divided by $y$ is $2$ less than $2$ of $y$

   (iii) $y$ divides $z$ is $2$ fewer than the second largest factor of $12$

(a) $x = 1$

(b) $x = 2$

(c) $x = 3$

(d) $x = 4$

For numbers $6$ and $7$ please refer to the description below:

There are $500$ students at the prestigious Redacted University. Recently the flu has been circulating around the area causing many people to get sick. Suppose that $\frac{1}{5}$ of all students are sick due to the flu at school today and the rest are healthy. The next day, a staggering $\frac{3}{5}$ of healthy students become sick and only $\frac{1}{5}$ of sick students become healthy.

$6$.
What is the proportion of all students who are healthy the next day?

(a) $\frac{1}{25}$

(b) $\frac{8}{25}$

(c) $\frac{9}{25}$

(d) $\frac{16}{25}$

$7$.
Exactly how many students are healthy the next day?

(a) $20$ students

(b) $160$ students

(c) $180$ students

(d) $320$ students

$8$.
What answer choice is equivalent to the expression $x^3 - 64$?

(a) $(x - 4)(x^2 + 4x + 16)$

(b) $(x - 4)(x^2 - 4x + 16)$

(c) $(x - 4)^3$

(d) None of the above

$9$.
What answer choice is equivalent to the expression $x^3+ 2x^2 - 4x - 8$?

(a) $(x - 2)(x^2 + 4)$

(b) $(x^2 + 4)(x + 2)$

(c) $(x^2 - 4)(x^2 + 4)$

(d) $(x-2)(x + 2)^2$

Challenge Problem:

$10$.
Suppose k is an integer such that $x = 2k$ and $y = 2k + 1$. What can we say about $x + y$?

(a) $x + y$ is always even

(b) $x + y$ is always odd

(c) Can be both even or odd

(d) Neither even nor odd

Answer key:

1.   a
2.   a
3.   d
4.   d
5.   b
6.   c
7.   c
8.   a
9.   d
10. b