Ex. 2. Subtract $2916 from 989583, so as to be correct to 5 places of decimals.

Note. This method may be advantageously applied in the Addition and Subtraction of circulating decimals. In the Multiplication and Division, however, of circulating decimals, it is always preferable to reduce the circulating decimals to Vulgar Fractions, and having found the product or quotient as a Vulgar Fraction, then, if necessary, to reduce the result to a decimal.

Ex. XXXI.

(1) Find the value (correct to 6 places of decimals) of

1. 2-418+1·16+3·009+7354+24′042.

2. 234.6+9·928+ 0123456789+0044+456.

3. 6·45-3; and 7·72–6·045; and 309–94724.

278
390

(2) Express the sum of 42, 338, and 12, and the difference of 1812 and 4, as recurring decimals.

3. 411-3519 by 58.7645; 2·16595 by '04; 6559903 by 48·76.

Ex. XXXII.

Miscellaneous Questions and Examples on Arts. (80-99).

I.

(1) Define a Decimal; and shew how its value is affected by affixing. and prefixing cyphers. Reduce '0625, and 3.14159 to fractions; and express the difference between 20 and 17 as a decimal.

(2) Find the value of 103+1+1+18 both by vulgar fractions, and by decimals; and shew that the results coincide.

(3) Find the sum, difference, product, and quotient of 573-005 and '000754; and of 1.015 and 01015, and prove the truth of each result.

(4) If a vulgar fraction, being converted into a decimal, do not terminate, prove that it must recur. What must be the limit to the number of figures in the recurring part? Is 64 convertible into a terminating decimal?

(5) Simplify 1. 23+72ğ+3161 +2·875.

6144

2. 026649÷218.

4. 18+009÷016.

; reduce the quotient to the form

1.0714285. Divide 91·863 by 87·56.

II.

32546

(1) Write down in a decimal form seven hundred thousand four hundred and nine billionths. Express 12·1345 as a fraction, and 1000000 as a decimal.

(2) State the effect as regards the decimal point of multiplying and dividing a decimal by any given power of 10. Write down in words the meaning of 397008-405009; multiply it by 1000, and also divide it by 1000; and write down the meaning of each result in words.

7 3

(3) What decimal multiplied by 125 will give the sum of §, 16, 09375 and 2:46?

(4) Multiply 1.05 by 10'5; and reduce the result to a fraction in its

lowest terms. Divide 8727588 by 1620; find the value of

13 625

reduce 1+180-2 to a decimal.

(5) Simplify, expressing each result in a decimal form, 1. 10000 of 21.

3.

44+3 7.375+

2. (21+6)+(31-1).

'0003 x *004

*006

4. 23000+11800+56000+2.000875.

(6) Find a number which multiplied into 3132-458 will give a product which differs only in the 7th decimal place from 7823-6572.

III.

(1) Divide 684 1197 by 1200-21, and also by '0120021; and 594.27 by ⚫047 to three places of decimals, and explain fully how the position of the decimal point is determined in each of the quotients.

(2) Simplify, expressing each result in a fractional and decimal form;

(3) What is meant by a 'Recurring Decimal'? What kind of vulgar fractions produce such decimals? State the rules for reducing any recurring decimal to a vulgar fraction. Multiply 5.81 by 4583, and divide 1.13 by 000132. Is reducible to a recurring decimal?

(4) Shew that if 11, 2, 3, 47 be added together, (1) as fractions, and (2) as decimals, the results coincide.

(5) A man walked in 4 days 60 miles; in each of the three first days he walked an equal distance, in the fourth day he walked 13.95 miles; find the amount of his daily walking.

(6) A person has 1875 part of a mine, he sells 17 part of his share; what fractional part of the mine has he still left?

IV.

(1) State the Rules for the Addition and Subtraction of Decimals. Add together 1.23, 123, 0123, 00123, and 123; and find the vulgar fraction corresponding to the result. Find the fraction equivalent to 31-457457, and subtract it from the fraction 494.

(2) Write down in figures the number, three millions six thousand and five. Also write down in words the signification of the same figures when the last is marked off as a decimal.

(3) Compare the values of 5 × 05, 1·5 × ·75, and 2·625÷5.

(4) Find the product of 0147147 by 333; and the quotients of 12693 by 19-39; of 132790 by 245: of '014904 by 32%; of 61061 by 3·05; and of 6106.1 by 305000.

5

(5) Shew that the decimal 90437532 is more nearly represented by •90438 than by 90437; and find the value of

(6) A person sold 15 of an estate to one person, and then of the remainder to another person. What part of the estate did he still retain ?

V.

3125

(1) Express (61+23-3), $178, and also the product of 35 and (31-3) of 4 as decimals.

3. (of 351-31) + (2·5625 +71).

4. 593÷178 × 36÷072.

(3) State at length the advantages which decimals possess over vulgar fractions; what disadvantages have they?

Shew whether 2 or 338 is nearer to the number 3.14159.

106

expressing it (1) as a decimal, and (2) as a fraction.

106

(5) Find the Earth's equatoreal diameter in miles, supposing the Sun's diameter, which is 111.454 times as great as the equatoreal diameter of the Earth, to be 883345 miles.

(6) In what sense is a vulgar fraction said to be the value of a recurring decimal? Explain how a sufficient degree of accuracy may be obtained in the addition and subtraction of circulating decimals to any given number of decimal places, without converting the decimals into fractions. Ex. Find the sum of 125, 4·163, and 9·457, correct to 5 places of decimals.

VI.

(1) Prove the Rule for Multiplication of decimals by means of the example 404-04 multiplied by 030303. Multiply 345 by

divide '04813489963 by 6593, and by '006593.

•111

4:3

; and

(2) Explain the meaning of 72, and 7o; and find what vulgar fraction is equivalent to the sum of 20.5 and 2:05 divided by the difference.

Reduce 1293131 to its equivalent vulgar fraction.

(5) What decimal added to the sum of 1,, §, and 13 will make the sum total equal to 3?

(6) The quotient being 211 and the divisor '15, find the dividend.

-

CONCRETE NUMBERS.

TABLES.

100. OUR operations hitherto have been carried on with regard only to abstract numbers, or concrete numbers of one denomination. It is evident that if concrete numbers were all of one denomination; if, for instance, shillings were the only units of money, yards of length, years of time, and so on, such numbers would be subject to the common rules for abstract numbers. Again, if the concrete numbers were of different denominations, and those denominations differed from each other by 10 or multiples of 10, then all operations with such concrete numbers could be carried on by the rules which have been given for Decimals. But generally with concrete numbers such a relation does not hold between the different denominations, and therefore it is necessary to commit to memory tables, which connect the different units of money together, the different units of length together, the different units of time together, and so on.

We shall now put down some of the most useful of these tables, with a few brief remarks on each,

Pounds, shillings, pence, and farthings were formerly denoted by £, s, d, and q respectively, these letters being the first letters of the Latin words libra, solidus, denarius, and quadrans, the Latin names of certain Roman coins or sums of money. £, s, d are still the abbreviated forms for pounds, shillings, and pence respectively; but annexed to pence denotes 1 farthing, denotes a half-penny, denotes three farthings; shewing that one farthing, two farthings, and three farthings are respectively 4th, 4ths or 1, and ths of the concrete unit, one penny.