REASON. As 7000 = 7 × 1000, therefore if we were to multiply by 7, and that product by 1000, we should arrive at the same result as that already obtained. But multiplying by 1, is simply to write down the number; and any number whatever multiplied by 0 = 0; hence it is evident why it is only necessary to multiply by the significant figure of the Multiplier, and to annex the cipher or ciphers as indicated in the Rule. No. XIII. When the Multiplier is 10, 100, 1000, &c. Find the product of the following: CASE C. When the Multiplier consists of several figures. Multiply the Multiplicand by each figure of the Multiplier separately, taking care to place the first figure of each new product under the figure you multiply by. Draw a line under these several products, and adding them together, their sum will give the total product. NOTE. If ciphers are intermixed with the significant figures of the Multiplier, it is only necessary to keep the first figure in the column under the figure of the Multiplier producing it. EXAMPLE FOR ILLUSTRATION. Multiply 673425 by 8076 8076 4040550 4713975 5387400 Here we place the multiplier under the multiplicand and draw a line as before. Then we multiply the whole of the multiplicand by the units-figure of the multiplier, viz., 6. Next we multiply by the tens-figure of the multiplier, viz., 7; and we say 7 times 5 are 35; that is, 35 tens, because the 7 is 7 tens. We set down the 5 tens in the tens-place, and carry the 30 tens, or 3 hundreds to the hundreds-figure; and so on throughout the entire multiplicand. Now, as we cannot multiply by the cipher which occupies the hundreds-place of the multiplier, we proceed to multiply by the thousands-figure, viz., 8; and we say 8 times 5 are 40; that is, 5438580300 40 thousands, or 4 tens of thousands; we set down a cipher in the thousandscolumn, because the 8 producing it occupies that place in the multiplier; and we carry the 4 to the next higher denomination, viz., tens of thousands; and thus proceed through the entire multiplicand. Lastly, draw a line, add up the several products, and the sum thus obtained will give the total product. REASON. From what has been already stated, and from the following operations, the reason of the foregoing process will be evident. Taking the same example, 673425 × 8076 8076 is made up of 6 + 70 + 8000 Therefore 673425 × 673425 × 70 = 47139750 Now it is unnecessary to write the ciphers at the end of each line, if we keep the other figures in their places without them as directed in the rule; the operation will stand thus: 673425 : No. XIV. When the Multiplier consists of several figures. Find the product of the following: CASE D. When the Multiplier is above 12, and less than 20. Multiply the units-figure of the Multiplicand by the units-figure of the Multiplier, and set down the units-figure of their product, observing what is to be carried. Next multiply the tens-figure of the Multiplicand in the same manner; to this product add the number carried, and also the significant figure of the Multiplicand previously multiplied. Proceed thus throughout, taking care always to add the last figure multiplied to the product obtained in the usual way. Instead, however, of setting down the complete product of the last figure of the Multiplicand, as is usual, merely set down the units-figure of such product, and add the remaining figure or figures to the last of the Multiplicand, in order to form the total product. EXAMPLE FOR ILLUSTRATION. Multiply 743265 by 15 15 11148975 As before, we say here 5 times 5 are 25; set down the 5 units and carry the 2 tens: then 5 times 6 are 30, and 2 we carried are 32, and the figure previously multiplied, viz., 5, will make 37; set down the 7 tens, and carry the 30 tens or 3 hundreds to the next figure. Again, we say 5 times 2 are 10; and 3 we carried are 13, and the last figure multiplied 6, and we have 19; set down the 9 and carry the 10 hundreds or 1 thousand. Proceed thus until every figure is multiplied. Observe the product of the last figure when increased by that previously multiplied is 41; we set down the 1, and adding the 4 to the 7, the last figure of the multiplicand, we set down their sum. F REASON. By this operation we not only multiply by the units-figure 5 of the Multiplier; but, by adding the back figure, we at the same time multiply by ten also. This will be evident from the following process ;— No. XV. When the Multiplier is above 12, and less than 20. Find the product of the following numbers : CASE E. When the Multiplier is a composite number. Multiply by one of the component parts, and that product by the other component part or parts. NOTE. Any number which can be separated into factors is called a composite number, as 40, 125, &c.; and any number which cannot be so separated is called a prime number, as 3, 7, 11, 13, &c. No. XVI. When the Multiplier is a composite number. 1. What is the product of the following numbers?— 5307182964 × 21 2. 4567823456 × 22 14. 9876543210 × 72 15. 82319008452 × 81 1. There are twenty shillings in a pound, and twelve pence in & shilling: how many pence in fifty sovereigns? The value of the Shilling among the Anglo-Saxons was only 5d., and this was reduced to 4d. about a century before the Conquest. It afterwards underwent many alterations, sometimes containing 16d., and often 20d. After the Conquest the French solidus, in use among the Normans, was called a shilling; but the true English shilling was first coined in the reign of Henry VII., about 1505. It is said to have attained its present value in the reign of Edward I. "One lb. troy of silver, containing 11 oz. 2 dwts. pure, and 18 dwts. alloy, is coined into 66 shillings." -Imp. Dict. Haydn. A Pound contains 240 pence, which were in Saxon times equivalent to a pound troy of silver, and consequently was three times as large as it is at present. It is usually distinguished from the pound weight by the term sterling, which signifies, that it is of the National or Standard value. A Penny is the largest British copper coin. The word penny we derive from our Saxon forefathers, among whom it was the only coin current, and with whom it was a small silver piece weighing 22 grains, and worth about 24d. of present money. Till the time of Edward L., the penny was struck with a cross so deeply indented, that it might readily be divided into halves and quarters: hence our term halfpence, and quadrantes, farthings. In the reign of Henry I. the Sovereign was issued at 22 shillings, being the 24th part of a pound of gold. In the reign of Henry VIII. sovereigns were coined and passed for 22s. 6d., 24s., and 30s.; the first-named were current till the reign of James I. In 1816 the sovereign was directed to pass for 20s. One lb. troy of gold, 11 parts pure and one alloy, is coined into 46 sovereigns, andths of a sovereign, or £46 14s. 6d."-Imp. Dict. Haydn. |