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All materials included in this unit are derived from content originated at the Open University.

Author(s): The Open University

There is no single right way of calculating percentage increases or decreases. The next examples show two different approaches to the same problems.

## Example 1

A railway season ticket from my local station to London costs (at current prices) £825. Calculate the cost of a new ticket, if pric
Author(s): The Open University

Now return to the old till receipt and look at the section which deals with VAT.

 VAT-CODE NET-VAL VAT-VAL S=17.5% £8.04 £1.41

These three pieces of information might be read as follows:

• <
Author(s): The Open University

There is some information about Value Added Tax (VAT) on the receipt. VAT is charged on many goods purchased in the UK. At the time of the purchase on this receipt, the VAT rate was 17.5%. This means that on every £100 net cost, £17.50 would be charged in VAT, bringing the total to £117.50. In other words:

Net cost + VAT = Gross cost

£100.00 + £17.50 = £117.50

So the shop would charge £100, the tax office £17.50 and the customer would pay £117.50.

Because you
Author(s): The Open University

Multiplying a number by itself is called squaring it and there is a key on scientific and graphics calculators which does this. On the TI-84 the key is marked .

Author(s): The Open University

If you would like some more calculator practice, try your hand at the following puzzles. No answers are given because most of these activities have no single numerical answer. You may like to try them out with your friends – or make up some of your own.

## Brain stretcher: Doing it with your eyes closed!

Author(s): The Open University

A way of forcing a calculator to perform a calculation in a different order to that given in Section 2.3 is to use the bracket keys. For example the following sequence, on a scientific or graphics calculator:

7 2 Author(s): The Open University

Your calculator will interpret the order in which you press the keys, in a particular way. For example if you press the key sequence:

2 3 [Image_Link]http://www.open.edu/openlearn/ocw/pluginf
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The Open University

Here are a few points from the Exercise 1:

• The negative or minus sign for the answer 2 maybe slightly smaller and higher than the one used for subtraction in 5 − 7. There maybe two minus keys on your calculator keypad, as there are on the TI-84. The one which means do the operation subtract is [Image_Link]http://www.open.edu/openlearn/ocw/pluginfile.php/94692
Author(s):
The Open University

Some calculators, like the TI-84, provide you with several different screens for menus, drawing graphs, writing programs and so on. The most important screen, where calculations are carried out, is called the Home Screen. If you should find yourself trapped on another screen, the ‘panic’ buttons to return ‘home’ are usually one or other of the following:

Author(s): The Open University

First have a look at your calculator keyboard. Some of the main features are described below.

The screen (also called the display) is at the top. This is where calculations and so on are displayed. The remainder of the calculator, where the various keys are located, is called the keyboard.

The number keys are usually in the bottom part of the keyboard and there are also the four operation keys: Author(s): The Open University

As you have been working through this unit, have you thought about how you are studying, and what this process involves? Do you feel confident or concerned about whether you will be able to learn mathematics and use it in the future? Put your study methods under the spotlight now, before moving on with your studies.

Learning rarely happens passively. A number of aspects of this unit have been designed to encourage your more active participation and involvement. However, even that
Author(s): The Open University

Given any number, you now know how to find its square. But, given the squared number, how do you find the original number?

## Example 3

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 1 MB).

## Exercise 1

A vector a has magnitude |a| = 7 and direction θ = −70°. Calculate the component form of a, giving the components correct to two decimal places.

<
Author(s): The Open University

Another vector quantity which crops up frequently in applied mathematics is velocity. In everyday English, the words ‘speed’ and ‘velocity’ mean much the same as each other, but in scientific parlance there is a significant difference between them.

## Velocity and speed

Author(s): The Open University

The displacement from a point P to a point Q is the change of position between the two points, as described by the displacement vector If P and Q represent places on the ground, then it is natural to use a bearing to describe the direct
Author(s): The Open University

In the following subsections, we apply the vector ideas introduced so far to displacements and velocities. The examples will feature directions referred to points of the compass, known as bearings.

The direction of Leeds relative to Bristol can be described as ‘15° to the East of due North’, or N 15° E. This is an instance of a bearing. Directions on the ground are typically given like this, in terms of the directions North (N), South (S), East (E)
Author(s): The Open University

The following activity illustrates how the conversion processes outlined in the preceding sections may come in useful. If two vectors are given in geometric form, and their sum is sought in the same form, one approach is to convert each of the vectors into component form, add their corresponding components, and then convert the sum back to geometric form.

Author(s): The Open University

You have seen how any vector given in geometric form, in terms of magnitude and direction, can be written in component form. You will now see how conversion in the opposite sense may be achieved, starting from component form. In other words, given a vector a = a 1 i + a 2 j, what are its magnitude |a| and direction θ?

The first part of this question is dealt with using Pythagoras’ Theorem: the magnitude of a v
Author(s): The Open University