Solve the simultaneous equations: x + 2y = 7, 2x − 2y = 2. Consider the sets P = x: x is a prime number less than 15 and Q = x: x is an even number less than 15. List elements of P and Q, find P U Q,

The solution is $x=3$ and $y=2$.
$\boxed{x=3, y=2}$

b) Consider the sets:
$P = \{x: x \text{ is a prime number less than 15}\}$
$Q = \{x: x \text{ is an even number less than 15}\}$

i) List all the elements of sets P and Q.
Prime numbers less than 15 are numbers greater than 1 that are only divisible by 1 and themselves.

Even numbers less than 15 are integers divisible by 2.

ii) Find $P \cup Q$.
The union of sets $P$ and $Q$ includes all elements that are in $P$, or in $Q$, or in both.

iii) State whether $P$ and $Q$ are disjoint sets.
Disjoint sets have no common elements (their intersection is an empty set).
The intersection of $P$ and $Q$ is $P \cap Q = \{2\}$.
Since $P \cap Q \neq \emptyset$, the sets are not disjoint.
$\boxed{\text{No, P and Q are not disjoint sets.}}$

Question 2

a) Solve $2(x + 4) \ge 5x - 1$ and graph the solution on a number line.

This can also be written as $x \le 3$.
$\boxed{x \le 3}$

Graph on a number line:
A closed circle at 3 with an arrow extending to the left, indicating all numbers less than or equal to 3.

<-----|-----|-----|-----|-----|-----|-----|----->
     -2    -1     0     1     2     3     4
                        [-----------

b) Change $11101_2$ to base ten numeral.

$\boxed{29_{10}}$

Question 3

a) Three machines produce packages at regular intervals: Machine A every 12 minutes, Machine B every 18 minutes, and Machine C every 24 minutes. Find the first time (in minutes) when all three machines will produce a package at the same time.

To find when all three machines will produce a package at the same time, we need to find the Least Common Multiple (LCM) of their intervals: 12, 18, and 24 minutes.

The highest power of 2 is $2^3$.
The highest power of 3 is $3^2$.

All three machines will produce a package at the same time after 72 minutes.
$\boxed{72 \text{ minutes}}$

b) If an item is sold at a 20% profit for GH¢60, find its original cost price.

Let the original cost price be $C$.
The profit is 20% of the cost price, which is $0.20C$.
The selling price (SP) is the cost price plus the profit.

We are given that the selling price is GH¢60.

The original cost price is GH¢50.
$\boxed{\text{GH¢}50}$

c) Arrange the numbers $-20, 15, -2, 0, 56, 4$ in ascending order.

Ascending order means arranging numbers from the smallest to the largest.

$-20, -2$.

$4, 15, 56$.

Combining these in order:

$\boxed{-20, -2, 0, 4, 15, 56}$