column and carry 1 hundredth to the next figure, as in common subtraction; and so on.

129. Rule.-Write the less number under the greater, so that the decimal points may fall under each other. Imagine a cipher to be placed in any vacant place in either number, and proceed as in common subtraction.

EXERCISE 34.

Find the value of―

A. (1) 958206-94′9854; (2) 769′0432505 7641; (3) 179'584-90 967; (4) 45 18439 2796; (5) 9'5986—8·799; (6) ·96 – ·8987; (7) *3046-29974; (8) 543026-539086; (9) 4'735646976; (10) 1003-0083.

Find the difference between

B. (1) 01 and 0083; (2) 919 and 101; (3) 0029 and 003; (4) 10'01 and 9'009; (5) 14·6058 and 145958; (6) one hundredth and one tenth ; (7) one hundred and seven ten-thousandths and seven hundred and nine millionths; (8) 93 hundred-thousandths and 1 thousandth; (9) 1043 tenthousandths and 9435 hundred-thousandths; (10) 908 thousandths and 154 hundredths.

MULTIPLICATION.

130. In multiplication of vulgar fractions the product of two or more fractions may be found by multiplying together separately all the numerators and all the denominators, and then dividing the former result by the latter.

Thus, in multiplying,, and, the product of the numerators is 2 X 5 X 4 = 40, and the product of the denominators is 3 × 7 × 11 = 231. Dividing the product of the numerators by the product of

the denominators, we get 40÷231, or

; and this is evidently the same result as we should obtain by the ordinary rule.

131. Applying this principle to find the product of 4.39 by 23'4 we first express these numbers by the corresponding fractions 38 and 23. Then the product of the numerators is 439 × 234 or 102726; and the product of the denominators 100

X 10 or 1000.

Therefore the required product = 102726 ÷ 1000. = 102'726.

Hence the following

132. Rule. Omit the decimal points, and multiply the numbers together as in common multiplication, and from the right of the product point off as many decimal places as there are in the given numbers together, prefixing ciphers, if necessary, to the left of the product.

Examples.-(1) Multiply 378.56 dy ‘047.

37856

47

264992

151424

1779232

The required product is 17.79232.

(2) Multiply 0435 by '036.

435

36

2610

1305

15660

There must be 7 decimal places in the product, and since there are only five figures we must prefix two ciphers, and place a point before them, thus 0015660; or, omitting the cipher after the decimal, the product is '001566.

EXERCISE 35.

Multiply together

A. (1) 3'47 and 2; (2) 46'8 and 3; (3) 489 and 4; (4) 65 and 5; (5) ·876 and 5; (6) 5.68 and 20; (7) 687 and 40; (8) 7.689 and 70; (9) 47625 and 08; (10) 5875 and 600.

B. (1) 3'46 and 2'3 ; (2) 74°3 and 34; (3) 2·637 and 25; (4) 3468 and 46; (5) 46'39 and 64; (6) 3864 and 29; (7) 4836 and 67; (8) 3467 and 038; (9) 0879 and 087; (10) 0867 and 095.

C. (1) 74893 and ·587; (2) 587.69 and 0843; (3) 97 865 and 0789; (4) 7·9486 and 3874; (5) 83749 and 6807; (6) 08397 and 4068; (7) 93857 and 806; (8) 23675 and 6·089; (9) 17·839 and 6708; (10) 53876 and 4'307.

Find the product of

D. (1) 37°4325 and 4°32; (2) 6·3472 and '0608; (3) 723875 and 4036; (4) 53 685 and 28060; (5) 379875 and 06048; (6) 462 372 and 0625; (7) 38 764 and 3450; (8) 630875 and 3080; (9) 783625 and 0256; (10) ·389576 and 187.5.

Find the value of—

E. (1) 364 × 2·8 × 125; (2) 032 × 64 × 025; (3) 4'75 × 36 × 30; (4) 256 × 01 X 7'5; (5) 125 X 30x032; (6) 37.5 × 07 × 160; (7) 640 × 32 X 625; (8) 128 x 80 x '005; (9) 001 x 364 X 2'5; (10) 1600 X 4 X"075.

P

DIVISION.

133. In the following examples the capital letters U, T, H, Th, are used to denote units, tens, hundreds, thousands; and the small letters, t, h, th, to denote tenths, hundredths, thousandths.

Let it be required to divide 3896 by 4.

4)3 8 9 6 A fourth of 3 Th. is o Th. and 3 Th. over

38 H. is 9 H. and 2 H.
29 T. is 7 T. and 1 T.

22

I

16 U. is 4 U. and o U.

... The quotient is o Th. 9 H. 7 T. 4 U.,

or 974.

Hence it appears that each of the figures in the quotient has the same denomination as the figure or figures from which it was obtained in the dividend. Applying this method to divide 38.96 by 4 we find that

Hence the quotient is o T. 9 U. 7 t. 4 h.,

or 9'74.

Again, if it be required to divide 31648 by 43, TU th th TU th th we find as before that

Hence the following rule is obvious—

134. To divide a decimal by a whole number. Rule. Divide as in common division, placing a decimal point in the quotient before the first figure, which is obtained from the first decimal place in the dividend.

Examples.-(1) Divide 1376 by 8.

8) 1376

0172

(2) Divide 5.238 by 20.

2,0)5 238

Cut off the cipher in the divisor, and divide by 2, moving the 2619 point one place to the left in the dividend for the division by 10.

(3) Divide 3.836 by 700.

7,00)3.836 Divide by 7, and move the point two places to the left in the 00548 dividend for the division by 100.

(4) Divide 27:59 by 32.

32

8)27'59000
4) 3'44875

8621875

Dividing 27.59 by 8, there is a remainder 7 hundredths, or 70 thousandths, and this divided by 8 gives 8 thousandths and 6 thousandths, or 60 ten-thousandths over, &c. Hence if there is a remainder after all the figures in the dividend are exhausted, we may annex ciphers, and continue the division as far as we please.