Let, also, 72 be the decimal;

= .72;

then, x=.

also, 100x 72.72;

therefore, 100xx, or 99x=72;
and x=3}=f·

Again, let .3176 be the decimal;

then, x=3176;

also, 10000x=3176.3176,

and 9999x=3176;

therefore, x=ZIJ8.

From these several examples, we arrive at the same conclusion as in § 184.

187. If the decimal was .38, let a represent, as before, the fraction required;

then, x= =.38,

and 10x=3.8;

also, 100x=38.8.

Subtracting the second line from the third, we find :

By subtracting line two from line three, we have:

99900x9000,

and x%=744.

From which example the same inference is drawn as in § 185.

188. EXERCISES.

1. What are the least vulgar fractions equivalent to .6; .53; 5925?

2. Required, the least vulgar fractions equivalent to 4.7543; .000444: .0227.

3. Convert .027; .6761904; 9.37024 into vulgar fractions.

ADDITION OF DECIMALS.

189. The addition of decimals is performed like simple addition, taking care to set the figures in their respective places.

Suppose we are to determine the sum of 4852.791, 4.00745, 2.7, .049.

4852.79100

4.00745

2.70000

.04900

4859.54745

Having set the numbers under each other as mentioned, and reduced to the same denominator or not, we proceed to add up, beginning at the right, putting the point in the sum under those of the quantities proposed.

190. If there were recurring decimals in the quantities to be added up, we should repeat the longest period twice, and sometimes two or three places more, and bring all the others to the same denomination, or reduce into vulgar fractions the recurring decimals, and the sum of all these is reduced to a circulating decimal.

Thus, find the sum of 71.4+6.72+18.703+.472+618i +.54629.

71.444444444

6.722222222 Required, the sum of .3, .945, and 769230.

18.703703703

.472222222

.618118118

.546296296

98.507007005

or 98.50700

048508=2.04851004.

999999

SUBTRACTION

OF

DECIMALS.

191. We must observe the same order as in addition, in setting down the figures, and proceed as in simple subtraction. Find the difference of 100.011 and 2.07568:

100.01100

2.07568

97.93532

192. When the operation is to be performed with recurring decimals, we must carry out the periods sufficiently, viz., twice the longest period, and two or three places more.

Ex. From 5.8300

Take 4.1777

1.6523=1.652

From 3.34242

Take 1.75555

1.58687=1.586

Or we might convert the decimals into vulgar fractions, and having found the difference and reduced it to a decimal, it will be the answer required.

Ex. Find the value of 82.8546-8.72.

Here 82.8546=82.8818, and 8.72-873-87373; therefore, 828838-87873-743838-74.1274.

193. EXERCISES IN ADDITION AND SUBTRACTION OF DECIMALS. Find the value of :

1. 8400+5.2007+18.63752+.08+.000001+1.1+17.34. 2. 4+.07+273.14156+.357+6.53425+111.1.

3. 9.9+76320.0234+96.16+2372.334+1954.00075 +6.3997. 4. Five ten thousandths+seven thousandths+eight tenths+ twenty-five thousandths+four hundreths+two thousandths. 5. Three thousand and five hundredths+forty-five tenths+three hundred and fifty-five thousand and twenty-nine hundredths+ two hundred thousand and twelve thousandths+forty-nine thousand and forty-seven tenths.

6. Find the value of 3.4-.045; 496.02-1.0046; 1.2-.5673. 7. From .94501 take 527; from 725365 take .367473.

8. From 11.036 take 6.0575.

9. From 27 tenths take 7984 ten thousandths.

10. Take 4036.690 from 5658.605.

MULTIPLICATION

OF DECIMALS.

194. We shall have three cases to consider.

1st, when both fractions are terminating.

2nd, when one factor is finite and the other recurring.
3rd, when both factors are recurring.

195. First, to multiply 43.7 by 3.91, we must observe that those quantities are equivalent to 47 and 18; and we know they will be multiplied if the product of the numerators be divided by the product of the denominators;

then, 43.7 3.91=437 × 38}=178887=170.867. Therefore, the product of two decimals is found like the product of two integers, and point off from the right of the product as many places of decimals as there are decimal places in both factors. If the number of places in the product is not sufficient, as many ciphers must be placed at the beginning as will make up the deficiency. Thus:

196. Secondly, find the product of 7.35 and .54. Here we have 7.35x.54-7.35 x

0=

7.35x='47°=

4.009; or 7.35 X.54=7.35×.545454, &c. =4.00908690.

Thus, we may either reduce the recurring decimals to a vulgar fraction, and multiply as in (§ 141), then convert the product into a circulating decimal; or multiply as in the first case, taking care to take a sufficient number of periods to ascertain the period of the product.

197. Thirdly, find the value of 3.36×.03485.

3485

We have 3.36x.03485-338 × 38% = £7 × 18838= 25789 117234450718.

219978

The method in which we reduce both factors to vulgar fractions is shorter than the other, because it is necessary to repeat the periods many times, in order to obtain anything like an approximate value.

198. In the generality of problems, it is sufficient if the result be correct within a certain number of decimals, much less than those obtained by the multiplication of the factors. It would then be of a great advantage could we establish a method, by means of which the figures required, alone could be obtained.

199. Let it be proposed to multiply 34.253467 by 5.4637, retaining only three decimal places.

Then we have only to notice, in the several multiplications, the thousandths, the hundredths, the tenths, the units, &c.; but it will be well to mind the ten thousandths, or the fourth place of decimals, on account of their influence on the thousandths, .or third place.

Let us now reverse one of the factors, the multiplier, for instance, which becomes 7364.5; then place it under the multiplicand, so that 5, the units place, be under the ten thousandths of the multiplicand, and the tenths' place shall necessarily be under the thousandths' place of the multiplicand, the hundredths under the hundredths', and so on.

By this method, every figure of the multiplier is placed under a figure of the multiplicand, the product of which figures produces ten thousandths.

34.253467
7364.5

1712673

137014

20552

1027

240 187.1506

This being granted, we proceed to multiply as usual, beginning each figure of the multiplier with the one above it. We then