SECTION TWO

FRACTIONS - DECIMALS

THE logarithmic scales of slide rules are, with a few exceptions, subdivided into decimal fractions, or, as we more often say, into decimals. It is impossible to take practical advantage of the slide rule without a working knowledge of the decimal system. We believe a brief note of explanation may assist those readers who think that the slide rule is useless to them because they cannot easily work in decimals. This section is not intended for readers who are familiar with the decimal system, and can use it without difficulty.

We propose to start with a short reference to ordinary fractions. The word fraction means "a part". Thus when we speak of ½ an inch - which is sometimes called an ordinary fraction, as distinct from a decimal fraction - we think of a length of 1" being divided into two equal parts, of which we take one part. When we mention ¾ as a fraction, we think of something, say, a yard, or an hour, or a shilling, being divided into four equal parts, of which we take three. The upper figure of an ordinary fraction is called the numerator, and the lower figure the denominator. A fraction in which the numerator is smaller than the denominator is always less than 1, and is sometimes called a proper fraction. A fraction such as 7/5, in which the numerator is larger than the denominator, is called an "improper" fraction. These terms proper and improper, when referred to fractions, are of no practical importance.

The value of a fraction is not altered if we multiply or divide numerator and denominator by any number. For instance,

 3 = 3 x 4 = 12 = 12 x 2 = 24 5 5 x 4 20 20 x 2 40

The fractions 3/5, 12/20 and 24/40 are all exactly equal to one another but we may say that the 3/5 is the simplest form, and in general this is the way it is written. You will see that the fraction can be reduced to 3/5 by dividing both numerator and denominator by 8. This kind of simplification is called cancelling.

To add together two or more fractions, we express them in terms of a common denominator, and then add together the numerators.

Example: Add together 2/3 and 4/5.

 2 + 4 = (2 x 5) + (4 x 3) = 10 + 12 = 22 3 5 (3 x 5) (5 x 3) 15 15 15

Result is 22/16 or 1 7/15

Problem 1. Add together 1/6 + 1/4   2/5

Subtraction of one fraction from another is effected in a similar manner.

Example: Find the result of taking 1/6 from 3/8
The smallest number which is a multiple of 6 and 8 is 24.

 3 - 1 = (3 x 3) - (1 x 4) = 9 - 4 = 5 8 6 (8 x 3) (6 x 4) 24 24 24

Multiplication and Division

To multiply together two or more fractions it is only necessary to multiply together all the numerators to form the numerator of the result and to multiply all the denominators to obtain the denominator of the result.

 Example: Evaluate 2 x 4 x 2 = (2 x 4 x 2) = 16 3 5 7 (3 x 5 x 7) 105

Cancellation of numbers common to both numerator and denominator should be effected whenever possible since this leads to simplification.

Example: Evaluate 1/3 x 3/5 x  1 1/4
This may be written 1/3 x 3/5 x 5/4 = 1/4 the 3's and 5's cancelling out leaving only the 1 in the numerator and the 4 in the denominator.

Problem 2. Evaluate 1 3/5 x 3/4 x 2 1/6

Division may be regarded as a special case of multiplication. To divide a number by a fraction you may interchange the numerator and denominator of the divisor, and then multiply by the inverted factor. An easy example is that of dividing by 2, which is exactly the same as multiplying by a ½.

Example: Divide 3/4 by 2/5
This should be written 3/4 x 5/2 = 15/8 = 1 7/8

Problem 3. Divide the product of 3/8 and   2 1/5 by 3/4.

Decimals

The word decimal is derived from the Latin word meaning ten, and the decimal system is based on 10. Consider, for example, the number 8888, it is built up of 8000 + 800 + 80 + 8. It is evident that the 8's are not all of equal value and importance. The first 8 expresses the number of thousands, the second the number of hundreds, the third the number of tens, and the last the number of units.

We have used the number consisting of the same figure 8 used four times; this was done because we wished to emphasise that the same figure can have different values attached to it, depending upon its position in the group. The number might have included any or all of the figures from 0 to 9 arranged in an infinite number of ways.

Let us consider a simpler number, say 15. In this the 1 actually means 10 units, and the 5 represents 5 units. Now we might desire to add a fraction to the 15 making it, say, 15½, and it seems feasible to do so by extending beyond the units figure this system of numbering by 10's. To indicate the end of a whole number we write a dot, called the decimal point, and any figures on the right-hand side of it represent a part or fraction of a unit.

We have seen that any figure in the fourth place to the left, counting from the units figure, represents so many thousands, the next to the right so many hundreds, and next so many tens, and the next so many units. If we continue we shall here pass the decimal point, and the next figure to the right must represent so many tenths of a unit. Still moving to the right the next figure will represent so many hundredths, the next so many thousandths, and so on indefinitely.

Now 1/2 , is 5/10 ,and remembering that the figure immediately to the right of the decimal point represents so many tenths, we can express 15 1/2 by 15·5. Instead of saying fifteen and a half, we should say fifteen decimal five, or as is more usual, fifteen point five. It would not be incorrect to express 15·5 by 15·50, or by 15·5000; the final noughts in both these cases are unnecessary but not actually wrong. In the form of a common fraction, the ·50 means 50/100 which cancels to 5/10, and finally to 1/2, and similarly ·5000 as a common fraction becomes 5000/10000 which also cancels to 1/2.

We sometimes see one or more noughts preceding a whole number, e.g. 018. The nought has no significance, and is only used when for some reason we wish to have the same number of figures in a series of numbers. 018 means 18, and 002 means 2. We must understand that one or more noughts at the beginning of a whole number, and noughts following the decimal part of a number do not alter the value of the number.

Conversion of Decimal Fractions into Ordinary Fractions

It is easy to convert a decimal fraction into an ordinary fraction. Take as an example the number 46·823, which means 46 units and a fraction of a unit. Earlier we have said that the first figure to the right of the decimal point indicates so many tenths of a unit, the next to the right so many hundredths, and the next so many thousandths of a unit. We have, therefore,

46  + 8/10 + 2/100 + 3/1000 which may be written 46 + 800/1000 + 20/1000+ 3/1000 which reduces to 46 823/1000.

From this we deduce the simple rule for converting a decimal fraction into an ordinary fraction. As the numerator of the fraction write all the figures following the decimal point, and for the denominator write a 1, followed by as many noughts as there are figures in the numerator.

Example: 152·61 = 152 + 61/100.        9·903 = 9 903/1000

Problem 4. Convert the following into numbers and common fractions expressed in the simplest forms: 6·8, 13·08, 19·080, 20·125, 41·0125, 86·625.

There is a different rule for recurring decimals which will be given later.

When numbers include fractions it is easy to effect addition or subtraction in the decimal notation. It is only necessary to write down the numbers so that their decimal points are in a vertical line, then add or subtract in the usual manner, and insert the decimal point in the answer immediately below the decimal points of the original figures.

Example:
Add together 16·26, 8·041 and 186·902.

 16·26  8·041 186·902 211·203

Subtract 108·694 from 423·47.

 423·470 108·694 314·776

Problem 5. Add together 12·801, ·92, 5·002 and 11·0. Subtract 82·607 from 96·2.

Multiplication and Division

A number expressed in the decimal system is very easily multiplied by or divided by 10 or 100, etc. To multiply by 10, move the decimal point one place to the right; to multiply by 100, move the decimal point two places to the right, and so on. When dividing move the decimal point to the left one place for each division by 10.

Examples:

 61·24 x 10 = 612.4 61·24 x 100 = 6124 61·24 x 1000 = 61240 61·24 ¸ 10 = 6.124 61·24 ¸ 100 = 0.6124 61·24 ¸ 1000 = 0.06124

Multiplication, when neither of the factors is 10 (or an integral power of 10, i.e. 100, 1000, etc.) should be carried out in the usual way, and the position of the decimal point ignored until the product is obtained. The number of decimal figures in the answer is easily obtained; it is equal to the sum of the numbers of figures after the decimal points of the factors.

Example: Multiply 62·743 by 8·6.

 62·743 Here there are 3 + 1 = 4 decimal   figures in the two factors. Starting  from the last figure in the product   we count off 4 decimal figures and insert the decimal point. 8·6 501·944 37·6458 539·5898

Problem 6. Multiply 9·274 by 82·6.

When dividing in the decimal notation it is advisable to convert the divisor into a whole number by moving the decimal point. If the decimal point of the dividend is moved the same number of places and in the same direction, the result will not be affected by these changes.

The following example will make this procedure clear.

Example: Divide 896·41 by 22·5.

 Here the result is 39·8. The next figure in the answer would be a 4, so that if the result is required to only one decimal place it is 39·8. 39·8 225) 8964. 1 675 2214 2025 1891 1800 910

Had the next figure been 5 or over 5, the result would then be given as 39·9, since this result would have been nearer to the exact answer than 39·8. When a numerical result which does not divide out exactly is to be expressed to a stated number of places of decimals, the division should be carried to one further decimal place. If this additional figure is less than 5 the figure preceding it should be left unaltered, but if the additional figure is 5 or over the preceding figure should be increased by 1.

Contracted Methods

When the factors which enter into the operations of multiplication or division are large, contracted methods should be used. This section is not intended to deal with all arithmetical rules and processes, but the reader will find a chapter dealing with contracted methods in books on elementary mathematics.

Conversion of Ordinary Fractions into Decimal Fractions

An ordinary fraction can be converted into a decimal expression by dividing the numerator by denominator. If we desire to change into decimals we divide 3·00 by 4. We generally add noughts to the 3 as shown. This is a case of simple division which we should often work mentally, but for the sake of clarity we will write it out in full.

4)3.00
·75
Now 4 will not divide into 3 so we include with the 3 the 0 which follows it, and divide 4 into 30. This gives 7 with 2 over and the 2 with the next 0 makes 20, which divides by 4 and gives 5 with no remainder. We insert the decimal point immediately below the decimal point in the original number and so obtain ·75 as the decimal equivalent of 3/4.

Examples: Express as decimals 5/8 and 13/16.

8)5.000
.625

.68
25)17.00
150
200
200

Problem 7. Express as decimals 7/8 and 13/16

The reader will see that we can convert an ordinary fraction into a decimal fraction by converting the fraction into a form in which the denominator is 10 or 100 or 1000, as the mathematicians say, into a positive integral power of 10.

Reverting to the 17/25 considered a little earlier, we can convert the 25 into 100 by multiplying by 4, but to maintain the value of the fraction unaltered we must also multiply the 17 by 4. We have, therefore,

 17 = (17 x 4) = 68 =.68 25 (25 x 4) 100

This method of conversion is sometimes quicker and easier than dividing denominator into numerator.

Recurring Decimals

If we attempt to convert the fraction 1/3 into decimals by division, we obtain a result which is unending.

3)1·0000
.3333 . . .

 . This result is said to be a recurring decimal and is often written ·3. The dot over the 3 indicates that the 3 is repeated indefinitely.
 . . A number such as 24·8216 , means 24·82161616 - the 16 being repeated indefinitely.

The rule to which we referred earlier is:
Subtract the figures which do not repeat from the whole of the decimal expression and divide by a number made up of a 9 for each recurring figure, and a 0 for each non recurring figure.

 . . Example: Convert 14·642. , into an ordinary fraction

642
6
636  Result 14 636/990 = 14 106/165

 . . Problem 8. Convert 2·8313 into an ordinary fraction.
 . . Check the result by dividing denominator into numerator to see if 2·8313 results..

© Hodder Stoughton, reproduced with permission.