Division is probably the most awkward of the four arithmetic operations. Since you may have a calculator, you do not need to be able to carry out complicated divisions by hand, but you do need to carry out simple divisions in order to check your calculator calculations. Division is the reverse process of multiplication. The quantity 12 Ã· 3 tells us how many times 3 goes into 12. Since 4 Ã— 3 = 12, 12 Ã· 3 = 4.

In order to compare quantities, it is best to express them in the same units.

## Example 10

Three children have just measured their own heights in metric units. Isaac says â€˜My height is 1098â€™, Jasmine says â€˜My height is 112â€™ and Kim says â€˜Mine is 1.1â€™. What units were

Now that youâ€™ve learned the principles of doing division on paper, you may want to practise your new skills. If so, go to the Dividing decimals page of the math.com website and follow the instructions. You will need a pen and paper to carry out each calculation. You can then enter your answer on the website to check if it is correct.

3.4 Cylinders and shapes with a uniform cross-section

An important idea when calculating volumes of simple shapes is that of a **cross-section**. In the case of the rectangular box considered above, it is possible to slice through the box horizontally so that the sliced area is exactly the same as the area of the base or top; in other words, the areas of the horizontal cross-sections are equal.

The sum of the angles of any triangle is 180Â°. This property can be demonstrated in several ways. One way is to draw a triangle on a piece of paper, mark each angle with a different symbol, and then cut out the angles and arrange them side by side, touching one another as illustrated.

## Question 1

Draw a line of symmetry on each of the shapes below.

There is another kind of symmetry which is often used in designs. It can be seen, for instance, in a car wheel trim.

Look at the trim on the left. It does not have line symmetry but

Two straight lines that do not intersect, no matter how far they are extended, are said to be **parallel**. Arrows are used to indicate parallel lines.

1.3.4 Vertically opposite angles

When two straight lines cross, they form four angles. In the diagram below, these angles are labelled *Î±*, *Î²*, *Î¸* and *Ï†* and referred to as alpha, beta, theta and phi. The angles opposite each other are equal. They are called **vertically opposite** angles. Here *Î±* and *Î²* are a pair of vertically opposite angles, as are *Î¸* and *Ï†*. Although such angles are called â€˜vertically oppositeâ€™, they do not need to be vertically above and bel

## Question 1

A company carried out a survey, recording how staff in a particular office spent their working time. The table shows the average number of minutes spent in each hour on various activities.

You can use the fact that the sum of angles at a point is 360Â° to draw a pie chart.

## Example 4

Over a five-year period a mathematics tutor found that 16 of her students gained distinctions, 32 gained pass grades and 12 failed to complete the course. Draw a pie chart to re

## Activity 14

A piece of computer software is to be developed by a team of programmers. It is estimated that a team of four people would take a year. Which of the following times is the length of time taken by three program

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