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\title{Consumer arithmetic }

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\begin{document}
\maketitle
This chapter deals with some important practical financial topics such as investing and borrowing money, income tax and GST, inflation, depreciation, profits and losses, discounts and commissions. Mention is also made of topics such as population, rainfall and the composition of materials.

When buying a car, using a credit card, or deciding how good a particular 'bargain' is, we all need to be competent in our understanding of consumer arithmetic. Otherwise, we run the serious risk of wasting our money or, even worse, being at the mercy of unscrupulous individuals. This is particularly important when we come to the 'big ticket items' such as purchasing a car or a home, or planning superannuation. It is also important when completing our tax returns.

\section*{Using a calculator and approximations}
Everything in this chapter involves calculations with percentages. We are now assuming that you are using a calculator, so we have made little attempt to set questions where the numbers work out nicely. However, some of the problems can be done mentally. This chapter gives you the opportunity to learn how to efficiently use a calculator.

Nevertheless, you should always look over your work and check that the answers to your calculations are reasonable and sensible.

When the calculator displays numbers with many decimal places, you will need to round the answer in some way that is appropriate in the context of the question.

Whenever you round a number, you should be careful to use the symbol \(\approx\), which means approximately equals, rather than the symbol \(=\), which means exactly equals.

For example, if we are told that a grocer received \(\$ 1000\) in cash payments for goods that he had sold and that he banked one-third of this, we would write:

\[
\text { amount banked }=1000 \div 3
\]

\[
\approx \$ 333.33
\]

not

\[
\begin{aligned}
\text { amount banked } & =1000 \div 3 \\
& =\$ 333.3333 \ldots
\end{aligned}
\]

because the grocer could not have banked fractions of a cent.

\section*{Review of percentages}
A percentage such as \(35 \%\) is a rational number and can be rewritten as a decimal or as a fraction, as follows:

\[
\begin{aligned}
& \begin{aligned}
35 \% & =35 \div 100 & \text { or } & 35 \%=\frac{35}{100} \\
& =0.35 & &
\end{aligned} \\
& =\frac{7}{20}
\end{aligned}
\]

In this chapter, we encourage you to write percentages as decimals rather than fractions. Decimals are more commonly used than fractions when dealing with money.

\section*{Converting a percentage to a decimal or a fraction}
To convert a percentage to a decimal or a fraction, divide by 100 .

For example: \(42 \%=42 \div 100\)

\[
\begin{aligned}
12 \frac{1}{4} \% & =\frac{49}{4} \% \\
& =\frac{49}{4} \times \frac{1}{100} \\
& =\frac{49}{400}
\end{aligned}
\]

Conversely, to convert a decimal such as 1.24 , or a fraction such as \(\frac{3}{8}\), to a percentage, multiply
by \(100 \%\).

\[
\begin{aligned}
& 1.24=1.24 \times 100 \% \\
& \frac{3}{8}=\frac{3}{8} \times \frac{100}{1} \% \\
& =124 \% \\
& =37 \frac{1}{2} \%
\end{aligned}
\]

\section*{Converting a decimal or a fraction to a percentage}
To convert a decimal or a fraction to a percentage, multiply by \(100 \%\).

\section*{Percentages of a quantity}
To calculate a percentage of a quantity:

\begin{itemize}
  \item convert the percentage to a decimal or a fraction

  \item multiply the quantity by the fraction or decimal.

\end{itemize}

\section*{Example 1}
\section*{Calculating a percentage}
We can express one quantity as a percentage of another. Remember that both quantities must be expressed in the same unit of measurement before calculating a percentage.

\section*{Example 2}

\section*{Calculating a percentage}
To calculate the percentage that one quantity, \(a\), is of another quantity, \(b\) :

\begin{itemize}
  \item first convert both quantities to the same unit of measurement

  \item then form the fraction \(\frac{a}{b}\) and multiply it by \(100 \%\).

\end{itemize}

\section*{Exercise 3A}



4 Copy and complete this table.

\begin{center}
\includegraphics[max width=\textwidth]{2023_11_16_3bbfc34a770529e9f285g-05}
\end{center}



9 Find what percentage the first quantity is of the second quantity, correct to 4 decimal places where necessary.
a \(48 \mathrm{~mm}, 1 \mathrm{~km}\)
b 1.5 hours, 3 years
c \(7.8 \mathrm{~g}, 60 \mathrm{~kg}\)
d 3.5 cents, \(\$ 1400\)



14 The label on a Sunnyvale tomato paste bottle says that in every \(25 \mathrm{~g}\) serving, there are \(3.6 \mathrm{~g}\) of carbohydrate, \(0.1 \mathrm{~g}\) of fat, and \(105 \mathrm{mg}\) of sodium.

a Express as a percentage of the \(25 \mathrm{~g}\) serving:

i the mass of carbohydrate

ii the mass of fat

iii the mass of sodium

b The Sunnyvale website claims that the percentage of protein is \(3.2 \%\). What mass of protein is that per \(25 \mathrm{~g}\) serving?

15 Mt Kosciusko has a height of \(2228 \mathrm{~m}\), while the height of Mt Everest is \(8848 \mathrm{~m}\). Calculate your answers to this question correct to 3 decimal places.

a What percentage is the height of Mt Everest of the height of Mt Kosciusko?

b The Earth's radius is about \(6400 \mathrm{~km}\). What percentage of the radius of the Earth is the height of Mt Everest?

16 In the Federal Parliament, there are 150 members in the House of Representatives, of whom 37 are from Victoria.

a Correct to 1 decimal place, what percentage of members are from Victoria?

b The population of Australia is about 22.6 million. What percentage of Australians are members of the House of Representatives?

17 The distance by air from Melbourne to Darwin is \(3346 \mathrm{~km}\), and from Melbourne to Singapore it is \(6021 \mathrm{~km}\). What percentage, correct to the nearest percent, is:

a the Melbourne-Darwin distance of the Melbourne-Singapore distance?

b the Melbourne-Singapore distance of the Melbourne-Darwin distance?

\section*{Using percentages}
Percentages are used extensively in finance, and are common in many other practical situations. The rest of this chapter will give examples of a few well-known situations, concentrating on financial applications. If you read a newspaper for a few days, you will find a great variety of further interesting uses of percentages.

First, however, we will introduce another important method that will be used with percentages throughout this chapter.

\section*{Reversing the process to find the original amount}
Suppose that \(15 \%\) of the total mass of a chicken roll is actually chicken. What mass of chicken rolls can be made with \(600 \mathrm{~g}\) of chicken?

\[
\text { Mass of chicken }=\text { mass of rolls } \times 15 \%
\]

Reversing this:

\[
\begin{aligned}
\text { mass of rolls } & =\text { mass of chicken } \div 15 \% \\
& =600 \div 0.15 \\
& =4000 \mathrm{~g}
\end{aligned}
\]

Hence \(4 \mathrm{~kg}\) of chicken rolls can be made with \(600 \mathrm{~g}\) of chicken.

\section*{Finding the original amount}
\begin{itemize}
  \item To find, for example, \(15 \%\) of a given amount, multiply by \(15 \%\).

  \item Conversely, to find the original amount given \(15 \%\) of it, divide by \(15 \%\).

\end{itemize}

\section*{Example 3}

\section*{Example 4}

Note: One reason for the choice of rounding to 3 decimal places is that there are \(1000 \mathrm{~g}\) in a kilogram. That is, we are calculating to the nearest gram.

\section*{Commission}
If a person takes a painting to a gallery to be sold, the gallery will usually charge the vendor or seller a percentage of the selling price as the fee for exhibiting, advertising and selling the painting. Such a fee is called a commission, and it applies whenever an agent sells goods or services such as a house or a car on behalf of someone else.

\section*{Example 5}

\section*{Profit}
Businesses aim to make a profit on their investments. A profit equation can be formulated as: profit \(=\) total revenue \((\) sales \()-\) total costs

\section*{Profit and loss as percentages}
Is an annual profit of \(\$ 20000\) a great performance or a poor performance? For a business with annual turnover of \(\$ 100000\), such a profit would be considered very large. For a business with annual turnover of \(\$ 100000000\), however, it would be considered a very poor performance.

For this reason, it is often relevant to express profit or loss as a percentage of the total costs or the annual turnover.


\section*{Example 7}

\section*{Solution}

\section*{Income tax}
Income tax rates are often progressive. This means that the more you earn, the higher the rate of tax on each extra dollar you earn.

\section*{Example 8}

\section*{Exercise 3B}
Note that some of the questions can be done using the unitary method. For example, Question \(\mathbf{5}\) is one such question.

1 a Five per cent of a particular amount is \(\$ 12\). Find the amount by dividing

\(\$ 12\) by \(5 \%=0.05\).

b Check your answer to part a by taking \(5 \%\) of it.

2 a Twenty-two per cent of a sand pile is \(284 \mathrm{~kg}\). Find the mass of the sand pile.

b Check your answer to part a by taking \(22 \%\) of it.



8 A book dealer sells rare books and charges a commission of \(8 \%\) on the selling price. Find, correct to the nearest cent, the commission charged on a book that sells for these prices, and the amount that the seller eventually receives.
a \(\$ 400\)
b \(\$ 1300\)
c \(\$ 575\)
d \(\$ 142.50\)

9 Shara the stockbroker charges \(0.15 \%\) commission on all shares that she sells for clients. In each case, find the price at which a parcel of shares was sold if her commission was:
a \(\$ 30.00\)
b \(\$ 67.35\)
c \(\$ 384.75\)
d \(\$ 36.51\)

10 Jeff works as a salesman selling second-hand tractors. He is paid a salary of \(\$ 35000\) a year, together with a \(6 \%\) commission on all the sales he makes. Find his total annual income if his sales for the year were:
a \(\$ 20000\)
b \(\$ 1000000\)
c \(\$ 126000\)
d \(\$ 3458000\)

11 A salesperson is paid a commission on her monthly sales. What is the percentage commission if she receives a payment of:
a \(\$ 168\) on sales of \(\$ 1200\)
b \(\$ 540\) on sales of \(\$ 6000\)
c \(\$ 1530\) on sales of \(\$ 18000\)
d \(\$ 1596\) on sales of \(\$ 42000\)


Example 6 14 The Secure Locksmith Company had sales last year of \$568000 and costs of \$521000.

a What was their profit?

b What was the profit as a percentage of the cost price? (Calculate the percentage correct to 2 decimal places.)

c In the previous year, they made a loss of \(4 \%\) of their costs of \(\$ 250000\). Find their loss and their sales.

15 a A company made a profit of \(\$ 18000\), which was a \(2.4 \%\) profit on its costs. Find the costs and the total sales.

b A company made a loss of \(\$ 657000\), which was a \(4.5 \%\) loss on its costs. Find the costs and the total sales.

c A company made a loss of \(\$ 250800\), which was a \(3.8 \%\) loss on its costs. Find the costs and the total sales.

16 This question uses the income tax rates in the nation of Immutatia, which are as follows.

\begin{itemize}
  \item There is no tax on the first \(\$ 12000\) that a person earns in any one year.

  \item From \(\$ 12001\) to \(\$ 30000\), the tax rate is 15 c for each dollar over \(\$ 12000\).

  \item From \(\$ 30001\) to \(\$ 75000\), the tax rate is 25 c for each dollar over \(\$ 30000\).

  \item Over \(\$ 75000\), the tax rate is 35 c for each dollar over \(\$ 75000\).

\end{itemize}

a Find the income tax payable on:
i \(\$ 8000\)
ii \(\$ 14000\)
iii \(\$ 36000\)
iv \(\$ 200000\)

b What percentage, to 2 decimal places, of each person's income was paid in income tax in parts i-iv of part a?

c Find the income if the income tax on it was:
i \(\$ 1260\)
ii \(\$ 3420\)
iii \(\$ 13950\)
iv \(\$ 14650\)

\section*{Simple interest}
When money is lent by a bank, whoever borrows the money normally makes a payment, called interest, for the use of the money.

The amount of interest paid depends on:

\begin{itemize}
  \item the principal, which is the amount of money borrowed

  \item the rate at which interest is charged

  \item the time for which the money is borrowed.

\end{itemize}

Conversely, if a person invests money in a bank or elsewhere, the bank pays the person interest because the bank uses the money to finance its own investments.

This section will deal only with simple interest. In simple interest transactions, interest is paid on the original amount borrowed.

\section*{Formula for simple interest}
How much simple interest will I pay altogether if I borrow \(\$ 4000\) for 10 years at an interest rate of \(7 \%\) per annum? (The phrase per annum is Latin for 'for each year'; it is often abbreviated to p.a.)

Last year you probably learned to set out the working for simple interest in two successive steps, something like this:

\[
\begin{aligned}
\text { Interest paid at the end of each year } & =4000 \times 7 \% \\
& =4000 \times 0.07 \\
& =\$ 280
\end{aligned}
\]

Total interest paid over 10 years \(=280 \times 10\)

\[
=\$ 2800
\]

This working can all be done in one step if we can develop a suitable formula.

Suppose I borrow \(\$ P\) for \(T\) years at an interest rate \(R\). Using the same two-step approach as before:

Interest paid at the end of each year \(=P \times R\)

Total interest paid over T years \(=P \times R \times T\)

\[
=P R T
\]

This gives us the well-known simple interest formula:

\[
I=P R T \quad(\text { Interest }=\text { principal } \times \text { rate } \times \text { time })
\]

Using this formula, the calculation can now be set out in one step:

\[
\begin{aligned}
I & =P R T \\
& =4000 \times 7 \% \times 10 \quad(\text { Note }: \text { The interest rate } R \text { is } 7 \%, \text { not } 7 .) \\
& =4000 \times 0.07 \times 10 \\
& =\$ 2800
\end{aligned}
\]

Note: The interest rate is given per year, so the time must also be written in years. In some books \(R\) is written as \(r \%\).

\section*{Example 9}

\section*{Reverse use of the simple interest formula}
There are four pronumerals in the formula \(I=P R T\). If the values of any three are known, then substituting into the simple interest formula allows the fourth value to be found.

\section*{Example 10}

(Interest rates are normally written as percentages.)


\section*{Exercise 3C}


\section*{Percentage increase and decrease}
When a quantity is increased or decreased, the change is often expressed as a percentage of the original amount.

This section introduces a concise method of solving problems about percentage increase and decrease. This method will be applied in various ways throughout the remaining sections of the chapter.

\section*{Percentage increase}
This evening's news reported that shares in the Consolidated Nail Factory Pty Ltd were selling at \(\$ 12.00\) yesterday, but rose \(14.5 \%\) today.

Rather than calculating the price increase and adding it on, the calculation can be done in one step by using the fact that the new price is \(100 \%+14.5 \%=114.5 \%\) of the old price.

\[
\begin{aligned}
\text { New price } & =\text { old price } \times 114.5 \% \\
& =12 \times 114.5 \% \\
& =12 \times 1.145 \\
& =\$ 13.74
\end{aligned}
\]

\section*{Example 11}
The number of patients admitted to St Spyridon's Hospital this year suffering from pneumonia is \(56 \%\) greater than the number admitted for this condition last year. If 245 pneumonia patients were admitted last year, how many were admitted this year?

\section*{Solution}
This year's total is \(100 \%+56 \%=156 \%\) of last year's total.

This year's total \(=245 \times 156 \%\)

\(=245 \times 1.56\)

\(\approx 382\) (correct to the nearest whole number)

\section*{Inflation}
The prices of goods and services in Australia usually increase by a small amount every year.

This gradual rise in prices is called inflation, and is measured by taking the average percentage increase in the prices of a large range of goods and services.

Other things such as salaries and pensions are often adjusted automatically every year to take account of inflation.

\section*{Example 12}

\section*{Percentage decrease}
The same method can be used to calculate percentage decreases. For example, suppose that \(35 \%\) of a farmer's sheep station, which has an area of 7500 hectares, went under water during the recent floods.

We can calculate how much land remained above the water for his stock to graze:

\(100 \%-35 \%=65 \%\) of his land remained above water.

Area remaining above water \(=7500 \times 65 \%\)

\[
\begin{aligned}
& =7500 \times 0.65 \\
& =4875 \text { hectares }
\end{aligned}
\]

\section*{Example 13}

Percentage increase and decrease

\begin{itemize}
  \item To increase an amount by, for example, \(15 \%\), multiply by \(1+0.15=1.15\).

  \item To decrease an amount by, for example, \(15 \%\), multiply by \(1-0.15=0.85\).

\end{itemize}

\section*{Finding the rate of increase or decrease}
\section*{Example 14}
Suppose that the rainfall has increased from \(480 \mathrm{~mm}\) p.a. to \(690 \mathrm{~mm}\) p.a. What rate of increase is this?

\section*{Solution}
\section*{Method 1}
Find the actual increase by subtraction, and then express the increase as a percentage of the original rainfall.

Increase \(=210 \mathrm{~mm}\)

\[
\begin{aligned}
\text { Percentage increase } & =\frac{\text { increase }}{\text { original rainfall }} \times 100 \% \\
& =\frac{210}{480} \times 100 \% \\
& =43.75 \%
\end{aligned}
\]

\section*{Method 2}
Express the new value as a percentage of the original value, and then subtract \(100 \%\).

\[
\begin{aligned}
\frac{\text { new rainfall }}{\text { old rainfall }} & =\frac{690}{480} \times 100 \% \\
& =143.75 \%
\end{aligned}
\]

So the rainfall has increased by \(43.75 \%\).

Note: Percentage decrease is sometimes represented as a negative percentage increase or, in other contexts, as a negative percentage change. This understanding is consistent with the formula, amount of change = new amount - old amount, where the new amount is less than the old amount in situations of decrease.

\section*{Reversing the process to find the original amount}
\section*{Example 15}

\section*{Example 16}
The price of shares in the Fountain Water Company has decreased by \(15 \%\) over the last month to \(\$ 52.70\). What was the price a month ago?

\section*{Solution}

\section*{Finding the original amount}
\begin{itemize}
  \item To find the original amount after an increase of, for example, 15\%, divide by 1.15 .

  \item To find the original amount after a decrease of, for example, \(15 \%\), divide by 0.85 .

\end{itemize}

\section*{Discounts}
It is very common for a shop to discount the price of an item. This is done to sell stock of a slowmoving item more quickly, or simply to attract customers into the shop.

Discounts are normally expressed as a percentage of the original price.

\section*{Example 17}
The Elegant Shirt Shop is closing down and has discounted all its prices by \(35 \%\).

a What is the discounted price of a shirt whose original price is:
i \(\$ 120\) ?
ii \(\$ 75\) ?

b What was the original price of a shirt whose discounted price is \(\$ 92.30\) ?

\section*{Solution}
a The discounted price of each item is \(100 \%-35 \%=65 \%\) of the old price.

i Hence discounted price \(=\) old price \(\times 0.65\)

\[
\begin{aligned}
& =120 \times 0.65 \\
& =\$ 78
\end{aligned}
\]

ii discounted price \(=75 \times 0.65\)

\[
=\$ 48.75
\]

b From part a, discounted price \(=\) old price \(\times 0.65\)

Reversing this, \(\quad\) old price \(=\) discounted price \(\div 0.65\)

\[
\begin{aligned}
& =92.30 \div 0.65 \\
& =\$ 142
\end{aligned}
\]

\section*{The GST}
In 1999, the Australian Government introduced a Goods and Services Tax, or GST for short. This tax applies to nearly all goods and services in Australia.

The current rate is \(10 \%\) on the pre-tax price of the goods or service.

\begin{itemize}
  \item When GST applies, it is added to the pre-tax price. This is done by multiplying the pre-tax price by 1.10 .

  \item Conversely, if a quoted price already includes GST, the pre-tax price is obtained by dividing the quoted price by 1.10 .

\end{itemize}

\section*{Example 18}
The current GST rate is \(10 \%\) of the pre-tax price.

a If a domestic plumbing job costs \(\$ 630\) before GST, how much will it cost after adding GST, and how much tax is paid to the Government?

b I paid \(\$ 70\) for petrol recently. What was the price before adding GST, and what tax was paid to the Government?

\section*{Solution}
The after-tax price is \(110 \%\) of the pre-tax price.


\section*{Exercise 3D}


5 In another state, the percentage decrease in rainfall over the last five years has been quite variable, and in some cases, rainfall has actually increased. Find the rate of decrease or increase if the annual rainfall five years ago and now are, respectively:
a \(500 \mathrm{~mm}\) and \(410 \mathrm{~mm}\)
b \(920 \mathrm{~mm}\) and \(960 \mathrm{~mm}\)
c \(140 \mathrm{~mm}\) and \(155 \mathrm{~mm}\)
d \(420 \mathrm{~mm}\) and \(530 \mathrm{~mm}\)

7 A clothing store is offering a 15\% discount on all its summer stock. What is the discounted price of an item with original price:
a \(\$ 80\) ?
b \(\$ 48\) ?
c \(\$ 680\) ?
d \(\$ 1.60\) ?

Example 17 8 A shoe store is offering a 35\% discount at its end-of-year sale. Find the original price of an item whose discounted price is:
a \(\$ 1820\)
b \(\$ 279.50\)
c \(\$ 1.56\)
d \(\$ 20.80\)


10 Mr Brown has a spreadsheet showing the value at which he bought his various parcels of shares, the value at 31 December last year, and the percentage increase or decrease in their value. (Decreases are shown with a negative sign.) Unfortunately, a virus has corrupted one entry in each row of his spreadsheet. Help him by calculating the missing values, correct to the nearest cent, and the missing percentages, correct to 2 decimal places.

\begin{center}
\begin{tabular}{r|c|c|c|}
\hline
 & Value at purchase & Value at 31 December & Percentage increase \\
\hline
\(\mathbf{a}\) & \(\$ 12000\) &  & \(30 \%\) \\
\hline
\(\mathbf{b}\) & \(\$ 28679.26\) &  & \(-62 \%\) \\
\hline
\(\mathbf{c}\) & \(\$ 5267.70\) &  & \(289.14 \%\) \\
\hline
\(\mathbf{d}\) &  & \(\$ 72000\) & \(20 \%\) \\
\hline
\(\mathbf{e}\) &  & \(\$ 26000\) & \(-22 \%\) \\
\hline
\(\mathbf{f}\) &  & \(\$ 112000\) & \(346.5 \%\) \\
\hline
\(\mathbf{g}\) &  & \(\$ 15934\) & \(-91.38 \%\) \\
\hline
\(\mathbf{h}\) & \(\$ 60000\) & \(\$ 81000\) &  \\
\hline
\(\mathbf{i}\) & \(\$ 98356.68\) & \(\$ 14321.57\) &  \\
\(\mathbf{j}\) & \(\$ 14294.12\) & \(\$ 2314.65\) &  \\
\hline
\end{tabular}
\end{center}

11 The GST is a tax on most goods and services at the rate of \(10 \%\) of the pre-tax price.

a Find the after-tax price on goods or services whose pre-tax price is:
i \(\$ 170\)
ii \(\$ 4624\)
iii \(\$ 68920\)
iv \(\$ 6.80\)

b Find the pre-tax price on goods or services whose after-tax price is:
i \(\$ 550\)
ii \(\$ 7821\)
iii \(\$ 192819\)
iv \(\$ 5.28\)

c Find the after-tax price on goods or services on which the GST is:
i \(\$ 60\)
ii \(\$ 678.20\)
iii \(\$ 54000\)
iv \(\$ 0.93\)

12 a A shirt originally priced at \(\$ 45\) was increased in price by \(100 \%\). What percentage discount will restore it to its original price?

b The daily passenger total of the Route 58 bus was 460 , and in one year, it increased by \(24 \%\). What percentage decrease next year would restore it to its original passenger total?

c Shafqat had savings of \(\$ 6000\), but he spent \(35 \%\) of this last year. By what percentage of the new amount must he increase his savings to restore them to their original value?
d The profit of the Arborville Gelatine Factory was \(\$ 86400\), but it then decreased by \(42 \%\). By what percentage must the profit increase to restore it to its original value?

13 a Find, correct to 2 decimal places, the percentage decrease necessary to restore a quantity to its original value if it has been increased by:
i \(10 \%\)
ii \(22 \%\)
iii \(240 \%\)
iv \(2.3 \%\)

b Find, correct to 2 decimal places, the percentage increase necessary to restore a quantity to its original value if it has been decreased by:
i \(10 \%\)
ii \(22 \%\)
iii \(75 \%\)
iv \(2.3 \%\)

\section*{Repeated increase and decrease}
The method introduced in the last section becomes very useful when two or more successive increases or decreases are applied, because the original amount can simply be multiplied successively by two or more factors. Here is a typical example.

\section*{Repeated increase}
\section*{Example 19}

\section*{Solution}


(continued over page)

d Population three years afterwards

\section*{Repeated decrease}
The same method can be applied just as easily to percentage decreases, as demonstrated in the next example.

\section*{Example 20}

\section*{Solution}

\section*{Example 21}

\section*{Repeated increases and decreases}
\begin{itemize}
  \item To apply successive increases of, for example, \(15 \%, 24 \%\) and \(38 \%\) to a quantity, multiply the quantity by \(1.15 \times 1.24 \times 1.38\).

  \item To apply successive decreases of, for example, 15\%, \(24 \%\) and \(38 \%\) to a quantity, multiply the quantity by \(0.85 \times 0.76 \times 0.62\).

\end{itemize}

\section*{Reversing the process to find the original amount}
As shown before, division reverses the process and allows us to find the original amount, as in the following example.

\section*{Example 22}
A clothing shop discounted a shirt by \(45 \%\) a month ago, and has now discounted the reduced price by \(20 \%\).

a What was the total discount on the shirt?

b If the shirt is now selling for \(\$ 61.60\), what was the original price of the shirt?

\section*{Solution}
a After the first discount, the price was \(100 \%-45 \%=55 \%\) of the original price.

After the second discount, the price was \(100 \%-20 \%=80 \%\) of the reduced price.

Thus final price \(=\) original price \(\times 0.55 \times 0.80\)

\[
=\text { original price } \times 0.44
\]

So the total discount is \(100 \%-44 \%=56 \%\).

b Reversing this, original price \(=\) final price \(\div 0.44\)

\[
=61.60 \div 0.44
\]

\[
=\$ 140
\]

\section*{Reversing repeated increases and decreases}
\begin{itemize}
  \item To find the original quantity after successive increases of, for example, \(15 \%, 24 \%\) and \(38 \%\), divide the final quantity by \((1.15 \times 1.24 \times 1.38)\).

  \item To find the original quantity after successive decreases of, for example, \(15 \%, 24 \%\) and \(38 \%\), divide the final quantity by \((0.85 \times 0.76 \times 0.62)\).

\end{itemize}

\section*{Successive divisions}
Calculations involving brackets can be tricky to handle when using the calculator.

For example, working with the figures in the summary box above, suppose that a population has increased by \(15 \%, 24 \%\) and \(38 \%\) in three successive years and is now 50000 . Then:

\[
\begin{aligned}
\text { original population } & =50000 \div(1.15 \times 1.24 \times 1.38) \\
& \approx 25408
\end{aligned}
\]

We suggest that it is easier to avoid brackets and divide 50000 successively by 1.15, 1.24 and 1.38. In effect, the working then goes like this:

\[
\begin{aligned}
\text { original population } & =50000 \div(1.15 \times 1.24 \times 1.38) \\
& =50000 \div 1.15 \div 1.24 \div 1.38 \\
& \approx 25408
\end{aligned}
\]

Try the calculation both ways and see which you find more natural.

\section*{Using the power button on the calculator}
When a quantity is repeatedly increased or decreased by the same percentage, the power button makes calculations quicker. Make sure that you can use it correctly by experimenting with simple calculations like \(3^{4}=81\) and \(2^{5}=32\).

\section*{Example 23}
The drought in Paradise Valley has been getting worse for years. Each year for the last five years, the rainfall has been \(8 \%\) less than the previous year's rainfall.

a What is the percentage decrease in rainfall over the five years?

b If the rainfall this year was \(458 \mathrm{~mm}\), what was the rainfall five years ago?

\section*{Solution}
Each year the rainfall is \(92 \%\) of the previous year's rainfall.

a Final rainfall \(=\) original rainfall \(\times 0.92 \times 0.92 \times 0.92 \times 0.92 \times 0.92\)

\[
\begin{aligned}
& =\text { original rainfall } \times(0.92)^{5} \\
& \approx \text { original rainfall } \times 0.659
\end{aligned}
\]

So rainfall has decreased by about \(100 \%-65.9 \%=34.1 \%\) over the five years.

b From part a, final rainfall \(=\) original rainfall \(\times(0.92)^{5}\)

Reversing this, original rainfall \(=\) final rainfall \(\div(0.92)^{5}\)

\[
\begin{aligned}
& =458 \div(0.92)^{5} \\
& \approx 695 \mathrm{~mm}
\end{aligned}
\]

\section*{Exercise 3E}


8 Calculate the total increase or decrease in a quantity when:

a it is increased by \(20 \%\) and then decreased by \(20 \%\)

b it is increased by \(80 \%\) and then decreased by \(80 \%\)

c it is increased by \(10 \%\) and then decreased by \(10 \%\)

d it is increased by \(30 \%\) and then decreased by \(30 \%\)

9 The price of gemfish has been rising. The price has risen by \(10 \%, 15 \%\) and \(35 \%\) in three successive years, and they now cost \(\$ 24\) per \(\mathrm{kg}\). Find:
a the price one year ago
b the price two years ago
c the original price three years ago

10 The crime rate in Gotham City has been rising each decade. In the last four decades the number of robberies has risen by \(64 \%, 223 \%, 75 \%\) and 12\%. If there are now 958 robberies per year, find how many robberies per year there were:
a one decade ago
b two decades ago
c three decades ago
d four decades ago


12 Flash Jim is desperate to attract customers to his used car yard. He has cut prices recently by \(5 \%\), then by \(10 \%\), then by \(24 \%\). Find, correct to the nearest \(\$ 100\), the original price of a used car now priced at:
a \(\$ 10000\)
b \(\$ 35000\)
c \(\$ 4600\)
d \(\$ 76800\)

13 The radioactivity of any sample of the element strontium- 90 decreases by \(90.75 \%\) every century. Find the percentage reduction in radioactivity over each of the periods given below. (Calculate percentages correct to 3 decimal places.)
a Two centuries
b Three centuries
c Five centuries

14 Here is a table of the annual inflation rate in Australia in the years ending 30 June 2001 to 30 June 2006 (from the Reserve Bank of Australia website).

\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\begin{tabular}{l}
Year \\
\begin{tabular}{l}
Inflation \\
rate \\
\end{tabular} \\
\end{tabular} & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\
\hline
\end{tabular}
\end{center}

Suppose the salary for certain jobs at Company \(\mathrm{X}\) rises on the 1 July every year, in line with Australia's inflation rate for the financial year just past (ending 30 June).
a A junior secretary earned \(\$ 40000\) from 1 July 2000 to 30 June 2001. Determine how much someone in that position would earn from:

i 1 July 2001 to 30 June 2002

ii 1 July 2006 to 30 June 2007

b A team manager was on an annual salary of \(\$ 100000\) from 1 July 2006 to 30 June 2007. Determine how much someone in that position would earn:

i in the previous financial year

ii from 1 July 2003 to 30 June 2004

15 At the start of the trading day, shares of a particular stock decrease in value by \(20 \%\).

However, by the end of the day the shares 'recover' and record a 15\% increase from its lowest value. Determine the percentage decrease in the value of the shares over the course of the day.

\section*{Compound interest}
In all the examples in this section, the interest is compounded annually. This means that at the end of each year, the interest earned is added to the principal invested or borrowed. That increased amount then becomes the amount on which interest is earned in the following year. This is called compound interest.

\section*{Example 24}

\section*{Compound interest on a loan}
Exactly the same principles apply when someone borrows money from a bank and the bank charges compound interest on the loan. If no repayments are made, the amount owing compounds in the same way, and can grow quite rapidly.

This is shown in the following example.

\section*{Example 25}

\section*{Compound interest}
Suppose that an amount \(P\) is invested at an interest rate, say \(7 \%\), compounded annually. The interest is calculated each year on the new balance. The new balance is obtained by adding on \(7 \%\) of the balance of the previous year. Thus:

\[
\begin{aligned}
\text { amount after four years } & =P \times 1.07 \times 1.07 \times 1.07 \times 1.07 \\
& =P \times(1.07)^{4}
\end{aligned}
\]

\section*{Reversing the process to find the original amount}
As always, division reverses the process to find the original amount, as in the following example.

\section*{Example 26}

\section*{Exercise 3F}





\(11 \mathrm{Mr}\) Brown has had further difficulties with the virus that attacks his spreadsheet entries. Here is the remains of a spreadsheet that he prepared in answer to questions from business friends. The spreadsheet calculated interest compounded annually, on various amounts, at various interest rates, for various periods of time. Help him reconstruct the missing entries.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
 & Principal & Rate \(\mathbf{p} . \mathbf{a}\). & Time in years & Final amount & Total interest \\
\hline
\(\mathbf{a y y y y y}\) & \(\$ 4000\) & \(6 \%\) & 20 &  &  \\
\hline
b & \(\$ 10000\) & \(8.2 \%\) & 15 &  &  \\
\hline
c & \(\$ 2000000\) & \(4.8 \%\) & 10 &  &  \\
d &  & \(6 \%\) & 20 & \(\$ 4000\) &  \\
\hline
e &  & \(8.2 \%\) & 15 & \(\$ 10000\) &  \\
f &  & \(4.8 \%\) & 10 & \(\$ 2000000\) &  \\
\hline
\end{tabular}
\end{center}

12 Ms Smith invested \(\$ 50000\) at \(6 \%\) p.a. interest, compounded annually, for four years. The tax department wants to know exactly how much interest she earned each year. Calculate these figures for Ms Smith.

13 Mrs Robinson has taken out a loan of \(\$ 300000\) at \(8 \%\) p.a. interest, compounded annually, for four years. She wants to know exactly how much interest she will be charged each year so that she can include it as a tax deduction in her income tax return. Calculate these figures for Mrs Robinson.

14 a Find the total percentage growth in a compound interest investment:
i at \(15 \%\) for two years
ii at \(10 \%\) for three years
iii at \(6 \%\) for five years
iv at \(5 \%\) for six years
\(\mathbf{v}\) at \(3 \%\) for 10 years
vi at \(2 \%\) for 15 years

b What do you observe about these results?

15 A couple took out a six-year loan to start a business. For the first three years, compound interest of \(8 \%\) p.a. was charged. For the second three years, compound interest of \(12 \%\) p.a. was charged. Find the total percentage increase in the amount owing.

16 One six-year loan attracts compound interest calculated at 2\%, 4\%, 6\%, 8\%, 10\% and \(12 \%\) in successive years. Another six-year loan attracts compound interest calculated at \(12 \%, 10 \%, 8 \%, 6 \%, 4 \%\) and \(2 \%\) in successive years. Find the total percentage increase in money owing in both cases, compare the two results, and explain what has happened.

17 An investment at an interest rate of \(10 \%\) p.a., compounded annually, returned interest of \(\$ 40000\) after five years. Calculate the original amount invested.

\section*{Depreciation}
Depreciation occurs when the value of an asset reduces as time passes. For example, a company may buy a car for \(\$ 40000\), but after four years the car will be worth a lot less, because the motor will be worn, the car will be out of date, the body and interior may have a few scratches, and so forth.

Accountants usually make the assumption that an asset such as a car depreciates at the same rate every year. This rate is called the depreciation rate. It works like compound interest. The depreciation is applied each year to the current value. In the following example, the depreciation rate is taken to be \(20 \%\).

\section*{Example 27}

\section*{Solution}

\section*{Depreciation}
Suppose that an asset with original value \(P\) depreciates at, for example, \(7 \%\) every year. To find the depreciated value, decrease the current value by \(7 \%\) each year. Thus:

\[
\begin{aligned}
\text { value after four years } & =P \times 0.93 \times 0.93 \times 0.93 \times 0.93 \\
& =P \times(0.93)^{4}
\end{aligned}
\]

\section*{Reversing the process to find the original amount}
If we are given a depreciated value and the rate of depreciation, we can find the original value by division.

\section*{Example 28}

\section*{Solution}
\section*{Exercise 3G}
Note: The depreciation rates in this exercise are taken from the Australian Taxation Office's

Schedule of Depreciation. These are intended for income tax purposes. A company may have reasons to use different rates.





11 St Scholasticus Grammar School bought photocopying machines six years ago, which it then depreciated at \(25 \%\) p.a. They are now worth \(\$ 72000\).

a How much were they worth one year ago?

b How much were they worth two years ago?

c How much were they worth six years ago?

d What is the total percentage depreciation on them over the six-year period?

e What was the average depreciation in dollars per year over the six-year period?

12 Ms Wu's seven-year-old car is worth \(\$ 5600\), and has been depreciating at \(22.5 \%\) p.a. Calculate your answers to the nearest dollar.

a How much was it worth four years ago?

b How much was it worth seven years ago?

c Ms Wu, however, only bought the car four years ago, at its depreciated value at that time. What has been Ms Wu's average depreciation in dollars over the four years she has owned the car?

d What was the average depreciation in dollars over the first three years of the car's life?

13 I take \(900 \mathrm{~mL}\) of a liquid and dilute it with \(100 \mathrm{~mL}\) of water. Then I take \(900 \mathrm{~mL}\) of the mixture and again dilute it with \(100 \mathrm{~mL}\) of water. I repeat this process 20 times.

a What proportion of the original liquid remains in the mixture at the end?

b How much mixture should I take if I want it to contain \(20 \mathrm{~mL}\) of the original liquid?

14 I take a sealed glass container and remove \(80 \%\) of the air. Then I remove \(80 \%\) of the remaining air. I do this process six times altogether. What percentage of the original air is left in the container?


\end{document}
