Working With Arithmetic

Frontmatter

1. Integral Numbers

Abstract

The invention of numbers ranks as one of the oldest inventions of civilized man. Right back to the earliest times of recorded history man has found it necessary to find some means for counting. Many early civilizations were led in this direction in their need to find some means for numbering the days and seasons, or in their need to have some standard for bartering foods and clothing. In many cases there was a need for numbers as an aid to measurement, for the construction of buildings, or the surveying of land.

D. M. Neal, D. J. New

2. Fractions

Abstract

When solving problems in the previous chapter we were able to find solutions using only whole numbers, i.e. 1, 2, 3, 4, 5, …, called integers.

D. M. Neal, D. J. New

3. Decimals

Abstract

The number system which we use today is based on the idea that the value of a figure within a number depends on two things (i) the figure itself, and (ii) the position of the figure within the number.

D. M. Neal, D. J. New

4. Averages and Unitary Methods

Abstract

On a recent hiking holiday I walked 12 miles on the first day, 15 on the next, then 10, 0, 20 and 9 miles. Altogether I walked 66 miles in 6 days or an average of 11 miles each day. This number 11, called the average, is not the distance I walked on any day but rather a number to give some fair representation of my average performance.

D. M. Neal, D. J. New

5. The Common Systems of Measurement

Abstract

Measuring is the process of finding out the extent or quantity of an object, by comparison with another object, or unit, of known size.

D. M. Neal, D. J. New

6. Measurement of Area

Abstract

From a piece of cardboard (or paper) cut two squares, each side 4 in., and weigh them on a physical balance. They weigh the same because they are of the same material and shape.

D. M. Neal, D. J. New

7. Manipulation of Compound Quantities

Abstract

Quantities such as £ 3. 5s. 8d. or 5 ft 7 in. are called compound quantities because, in each case, more than one unit of measurement is involved. Multiplication of such quantities by any number greater than 12 is generally rather long and cumbersome if carried out by the direct method. The methods explained in this chapter provide shorter ways of arriving at the solution.

D. M. Neal, D. J. New

8. Volumes

Abstract

Volume is a measure of the quantity of space occupied by some body or substance. A liquid such as water, petrol, or oil is measured and sold by volume.

D. M. Neal, D. J. New

9. Ratio and Proportion

Abstract

A ratio is used to compare the sizes of two different quantities.

D. M. Neal, D. J. New

10. Percentage

Abstract

The words ‘per cent.’ originate from the Latin phrase per centum which means ‘ by the hundred’, that is a certain part of each hundred. Hence, if twenty books in each hundred are non-fiction we say 20 per cent. are non-fiction. The words per cent. are usually denoted by the sign % and thus 20 per cent., or 20 percent., is written 20%.

D. M. Neal, D. J. New

11. Graphs

Abstract

A graph is a pictorial representation of a set of facts, or experimental readings, which enables us quickly to build up a mental picture of all the information given and to interpret it.

D. M. Neal, D. J. New

12. Use of Tables

Abstract

Since (1·245)² = (1·245) × (1·245) we can find the answer to the problem by multiplying out long hand.

D. M. Neal, D. J. New

13. Logarithms

Abstract

Logarithms provide a rapid method for finding the product of two numbers without undertaking the lengthy procedure of long multiplication.

D. M. Neal, D. J. New

14. The Circle

Abstract

Suppose we are carrying out a scientific experiment which requires the use of a circular hoop of wire whose circumference is 10 in. We could construct this hoop by cutting a piece of wire 10 in. long, bending it round a suitable former and then soldering the ends together. (See Fig. 14.1).

D. M. Neal, D. J. New

15. Rates and Percentages

Abstract

We now consider some problems, of a more difficult nature, the solutions of which involve the use of principles explained earlier in the book, e.g. unitary methods, rate and proportion, the method of mixtures and percentage. Many of the problems use a combination of these ideas which results in longer solutions; thus the student is liable to make more careless mistakes. If each line is thought out then these difficulties are easily overcome.

D. M. Neal, D. J. New

16. Further Volumes

Abstract

Fig. 16.1 illustrates three of the more common objects having curved surfaces. The volumes of the cylinder, cone and sphere are frequently required but it is impossible at this stage of mathematics to give any proof for the volumes of the cone and sphere. The student must wait until he has learned the calculus before he is able to derive the formulae for the volumes of these two.

D. M. Neal, D. J. New

17. Shares and Stocks

Abstract

In order to form a business money is required, called the capital, with which to trade. Nowadays the cost of buildings, equipment, materials, etc. are so great that it is often impossible for one man to subscribe all the capital, so a company is formed.

D. M. Neal, D. J. New

18. Miscellaneous Examples

Abstract

This chapter consists of harder problems on ideas introduced earlier in the book. The examples are designed to give practice both in the solution of harder problems and in the orderly presentation of these solutions. Many of the questions are taken from public examination papers.

D. M. Neal, D. J. New

Backmatter