Unit 9

Calculate Rafik's annual income tax bill.

The equation of a straight line is given by . Rafik's income after tax, as a function of his pre-tax salary , is linear above $50,000.

Write down an equation for the income after tax for all , using the figures from part (a).

On a set of axes, sketch the graph of against for using your answer from (b). Clearly indicate the point at which the tax band changes.

Toffee A: 1.35 kg for $9.99. Toffee B: 2.1 kg for $15.82.

Work out the cost per kg for each and decide which is better value for money.

The cost per kg, , for a pack weighing kg with total price dollars is given by .

If a special almond pack has weight and price dollars, show, by algebraic reasoning, that the cost per kg increases as decreases.

A third product, caramel fudge, is priced so that for a pack of kg, the price is dollars.

The company claims it is never better value per kg than Toffee B for packs up to 4 kg. Is their claim correct? Justify your answer with calculations and reasoning.

Write the piecewise function for after-tax salary , for all .

A company wishes to set annual after-tax salaries between $40,000 and $80,000 for its staff.

If the company can only offer salaries in increments of $5,000, determine all possible gross salaries that would give after-tax salaries in the required range.

Explain, using algebraic reasoning, why there are more possible gross salaries in the upper range ($>50,000) compared to the lower range ($\leq 50,000), for the same after-tax interval.

Sergio receives a salary of £30,000 per year. He also receives a bonus of £12,000 in that year. The first £30,000 of his income is taxed at 20%, and any amount above £30,000 is taxed at 40%. Work out how much income tax Sergio should pay in total for the year.

Let x be the salary (in £) and y be the bonus (in £) that produces exactly £8,600 tax for this tax system, given that x + y = £41,500. Using the rules above and your answer from part (a), show how to set up and solve a pair of simultaneous equations for x and y, and find the values of x and y.

Calculate how many dollars Amira receives for her £480. Then, work out how many pounds she will receive when changing $216 back into pounds at the same rate.

Amira notices that the total number of pounds she gets back for her $216 is not in the same ratio as the pounds to dollars in the original exchange. Prove algebraically that, if no commission or rounding is involved and the rate is the same both ways, exchanging money and exchanging back always gives the original currency value (neglecting leftover cents or pence).

Popcorn is sold in tubs of two sizes: - Small Tub: 150 g for $2.85 - Large Tub: 420 g for $7.20 Work out which tub offers the better value for money by calculating the cost per gram for both tubs. Give your answer correct to 3 decimal places.

A student buys one of the better-value tubs and some nachos. The total weight of food bought is 420 g and the total cost is $8.50.

The cost of Nachos (in dollars) can be modelled as a quadratic function of their mass (in grams):

where is the mass of nachos in grams, and is the cost in dollars of grams of nachos.

Set up simultaneous equations using your result from (a) and solve them to find the mass of nachos and the cost of nachos bought by the student. Give both answers correct to 2 decimal places.

Convert the UK price of the calculator into dollars. Give your answer to 2 decimal places.

Hence, prove that buying the calculator in the UK is a better deal, and explain your reasoning.

Let be the price in pounds, and be the price in dollars, with the exchange rate .

Write a general proof that shows for any UK price and other country price , the UK is the better deal if .