ANSWERS TO ARITHMETIC. (MALES.) Exercise 1 (p. 303). (1) 25; 2,025; 3,721 ; 5,776; 8,100; 10,000; 11,881; 34,969. (2) 1; 1,331; 19,683; 59,319; 389,017; 1,331,000 ; 2,924,207; 3,796,416; 1,000,000,000. (3) 1,296; 130,321; 54,700,816; 1,600,000,000. (4) 16,384; 823,543; 4,782,969. (5) 68,719,476,736; 282,429,536,481. (6); 11; 2610; 204100 25 (5) 9'79795; 11°26942; 13.63818; 14°14213. (6) I. (7) 14142135; 6·2449979; 13.6747943. (8) ; 4; 13; 120. (9) 7,072. (10) 1'046; 73. (II) 28284; 4'1683. (12) 230°46 + 4′5 = 234′96. (13) 482 yd. I ft. (14) 4295. (15) 012. Miscellaneous Examples (p. 308). (1) 128.8659793 + yrs. (4) 35383304. (5) 196 gals. water. Price 3'012 + s. per gal. (20) £328 10s., and £328 5s.; Difference 5s. (28) £5,000; £3,750; £3,125. (29) 163 p. c. (50) 5s.; 4d. (48) Difference 6s. 1d.; 3s. 11 d. and 2s. 21d. (49) 442, or 44°1379310 + perches. (FEMALES.) DECIMAL FRACTIONS. A Decimal Fraction is expressed by means of one number. This number denotes the numerator of the fraction, the denominator being understood. A dot () is always placed before a decimal fraction. A decimal fraction is often called, shortly, a decimal. A decimal fraction is equivalent to a vulgar fraction whose denominator is 10, or some power of 10, as 100, 1000, 10000, etc. (101, 103, 103, 104, etc.). The first figure after the dot denotes so many tenths of a unit, the second figure denotes so many hundredths of a unit, the third figure denotes so many thousandths of a unit, and so on to the end of the fraction. Thus 3 denotes of a unit; 03 denotes of a unit; '003 denotes of a unit. So 33 denotes+ of a unit; 333 denotes 10 +180+1000 of a unit. Therefore any decimal fraction is equivalent to a series of vulgar fractions whose numerators are the figures in the decimal taken in the order in which they stand, and whose denominators are the successive powers of 10. Thus '42378= 10+ 180 + 1000 † 10000 + 100000• In the ordinary notation employed to express whole numbers, the first figure (reckoning from the right) denotes so many units, the second figure so many tens, the third so many hundreds, the fourth so many thousands, and so on. Thus 4326=6×1+2×10 +3×100+4×1000, or (commencing with the first figure to the left of the number) = 4 × 1000 + 3 × 100 + 2 × 10 + 6 × I. Placing these expressions together, we have 4326′42378 = 4 × 1000 + 3 x 100 + 2 × 10 +6×1+4 × 10+2 × 180 +3 × 1000 + 7 × 10000 + 8 × 100000• This example will show that the method of notation for ordinary whole numbers and for decimals is the same, the value of any figure depending on its position. Any figure decreases in value 10 times for each place it is moved to the right, and increases in value 10 times for each place it is moved to the left. The same figure becomes of 10 times less value if moved one place farther to the right, of 100 times less value if moved two places farther to the right, and so on for each place it is moved. Similarly, the same figure becomes of 10 times greater value if moved one place farther to the left, of 100 times greater value if moved two places farther to the left, and so on for each place it is moved. The pupil will now be able to point out the value of any number containing a decimal. (1) Express in words the value of each digit in the following numbers: 238; 47°357. 8 2381180 + 1000 = two tenths+three hundredths + eight thou sandths. 47 357 40+ 7 + 10 + 180 + 1000 four tens + seven units + three tenths + five hundredths + seven thousandths. (2) Express the following vulgar fractions as decimals :— 426 1000 = 4. 2 10000 100+ 180 1% + 10 = 400 = 40 20 36 + 1880 + 1000 = 10 +180 + 1000 = '426 = 10 + 100 + 1000 + 10000 = 0042 (3) Express the following mixed numbers as decimals: 28 = 46 +1200 + 1000 = 467880 10000 46 +180 + 1000 = 46 + 1 + 180 + 10‰0 = 46′028. 32714 = 327 +10% +100: - 327+10+100 = 327'14. =29+10000 + 10800 =29 +100 + 10800 =29+10 + 100 + 1000 + 10800 = 29'0108. Exercise 1. Write in words the value of each digit in the following (25) Write in figures :-Five thousands and five-thousandths; three millions and three-millionths; four hundred and six and five-tenths; two hundredths; and seven thousandths. To multiply or divide a number containing a decimal by 10 or any power of 10. Rule. To multiply a number containing a decimal by 10 (= 101), move the decimal point one place farther to the right; to multiply by 100 (= 102), move the decimal point two places farther to the right; and so generally, move the decimal point as many places farther to the right as there are ciphers in the multiplier. Similarly, to divide a number containing a decimal by 10, 100, 1000, or any power of 10, move the decimal point as many places farther to the left as there are ciphers in the divisor. = The reason of this rule will appear from the principles explained above. Moving the decimal point one place farther to the right alters the position of each figure in the number, increasing the value of each ten times, and therefore multiplying the whole number by ten. Thus 42*35 4 tens, 2 units + 3 tenths+5 hundredths. If now we move the decimal point one place farther to the right, we have 423'5. The 4 tens have become 4 hundreds, the 2 units have become 2 tens, the 3 tenths have become 3 units, and the 5 hundredths have become 5 tenths. Thus each figure has been increased in value ten times. and the whole number has been multiplied by 10. Similarly it may be shown that if the decimal point be moved two places |