Task 1: (10 marks)

1. A class has 45 students where the ratio of girls to boys is 5:4. How many girls and how many boys are there? You must show all your workings.
2. Melissa wants to share out a bonus of £2,500 between her top two employees in the ratio 3:2. What is the amount for the highest bonus? Show all your workings.
3. A construction project requires 15 workers and takes 6 days to complete. How many workers are required to complete the same project in 5 days? Explain your answer with calculations in details.
4. A bakery sells 900 cakes in a month generating revenue of £3,000. How much revenue is generated the following month, with the same unit price if only 600 cakes are sold? You must show all your workings.

Task 2: (10 marks)

In the diagram below:

1. Find the gradient of the line.
2. Find the intercept of the line.
3. Write the equation of the line in the form y = ax + b
4. Find the exact value of y when x is 3
5. Find the exact value of y when x is 0

Explain your methods clearly.

Task 3: (10 marks)

For this task, you need to show all your calculations, step by step.

1. Tony received an electricity bill for £445 last month. The bill includes a fixed monthly charge of £75 and the cost per unit is 40 pence. Calculate to the nearest whole number, the quantity of units he has used.
2. The financial report of Zebra Company stated:Net profit of £325,750 for the year 2022 (an increase of 7%).What was the value of Net profit in the previous year? (show to the nearest whole number)
3. A landlord currently receives £600 monthly rental payments from an existing tenant. The rental is increased by 5% per month. What is the total annual rental received after the increase? (show all workings)
4. A laptop was advertised for sale at £1,400, which was only 80% of original price. What was the original price before the discount? (show all workings)

Task 4: (10 marks)

In this task, you are asked to formulate the linear programming problem below and then solve it.

A carpenter produces Tables (T) and Chairs (C).

Each Table unit requires 6kg of wood and 1kg of plastic.

Each Chair unit requires 4kg of wood and 4kg of plastic.

The carpenter has only 100kg of wood and only 50kg of plastic in stock.

On each sale, the carpenter makes a profit of £15 per Table unit sold and a profit of £30 per Chair unit sold.

Required:

Formulate the Linear Programming problem above by defining the variables, stating the object function and the constraints.

You are required to produce the graph for all inequalities (copy image to Word) Find the quantity to make of Tables and Chairs to maximise profit.

Task 5: (10 marks)

1. The data in the table below shows the height of athletes in metres and their respective weights in kilograms.
2. You consider entering a Lottery, where you choose three different numbers between 1 to 50 inclusive. Explain in words and show how you would calculate the possible combinations of winning the jackpot (i.e., obtaining all three numbers). Show all the steps in your calculation.