thing, remove the figures three places towards the left. Add the results, and we obtain the required product. NOTE. It can easily be perceived that whether we multiply 1087 by 5050, or 5050 by 1087, the product would be the same. Now were 5050 some other number, it would, very probably, be more convenient to consider it the multiplicand; thus, were it 5435, the product would be more concisely obtained by considering 5435 the multiplicand and 1087 the multiplier. 66 3. Explain each step in the division of £70 10s. 11d. by 820; and express clearly what is the value of the remainder." £ s. d. s. d. To take the 820th part of £70 10s. 11d., place the number and money in the form of divisor and dividend, as above. Now, as 70 divided by 820, will not produce a whole number, bring the pounds to their equivalent in shillings, adding in the 10s. The number of shillings will then contain the divisor once with a remainder of 590 shillings. Bring these to pence by multiplying by 12; the quotient is contained in the pence 8 times, with a remainder of 531 pence =2124 farthings, in which 820 are contained twice, i.e.-the farthings divided into 820 parts will allow of 2 farthings for one such part, and a remainder of 484, which is the number of farthings still undivided, being insufficient to produce one in each of 820 parts: hence there are 10s. 1d. left undistributed over the 820 portions. SECTION II. 1. "Find, by practice, the value of 5 cwt. 1 qr. 19 lbs., at £3 15s. per cwt." 2 "If 11 articles cost 15s., what would 17 cost? Explain each step of the process of working the sum.” If the price of 11 articles be 15s., the price of 17, at the same rate, will be the same multiple of the price of 11 as 17 is of 11. Thus, 3. "If 5 men receive £18 15s. wages for 12 months, what will be the wages of 16 men for 20 months?" If 5 men for 12 months receive £18 15s. 1. (1) SECTION III. "Subtract of from 11, and (2) find the of value of 16s. 8d." (2.) 168 × = S. d. s. d. 16 8 × 3 = 46 8 or 5 guineas = 315 99 .. 13 4 is of 5 guineas, 40 times the 315th part; 40 315 8 63 3. "Show that dividing the numerator of a fraction by any number gives the same result as multiplying the denominator by the same number." Let it be required to divide the fraction by 3. 6 = 11 11 x 3 33 2 = = 33 11. The same may be shown by dividing a line representing unity into 11 equal parts, and each of these again into 3 others; when the third part of 6 of the elevenths would, in length, be equal to 6 of the thirty-thirds. (See paper on Algebra, Sec. ii. Ques. 2.) SECTION IV. 1. "Multiply 0017 by 450, and give the reason for the correct placing of the decimal point in the product." •0017 850 68 *765 17 10000. Hence, In performing this operation, we are finding the product of by 450: for 0017 = after finding, as above, the product of 17 by 450, we have to divide by 10000; this is done by "pointing off" four figures. Thus the reason is seen for pointing off as many figures in the quotient as there are decimal places in the multiplier and the multiplicand together. 66 2. Reduce 4s. 7d. to the decimal of a pound." reduced to a decimal, becomes .6'; therefore the 4s. 7d., reduced to the decimal of a pound is . 22916. The form generally employed and deduced from the above is as follows: 12) 7d. £22916 3. "Divide 570 by 005, and give the reason for the correct placing of the decimal point in the quotient." 5 *005)570 114000 005 TO; therefore when we have divided by 5, as seen above, we have divided by a number 1000 times greater than the true divisor: therefore the result must be multiplied by 1000, or "place as many figures after the one obtained, by dividing the units, as there are decîmal places in the divisor." 4. "Extract the cube root of '04 to 3 places of decimals. Give the reason for the operation by which you |