struck off from any number of figures, it is immediately divided by 10, and the figure (if any) so struck off is so many tenths, and consequently, is so many times 2s., (for every tenth of a £. is 2s.); hence the necessity for halving the even number of shillings the price consists of, to make its denominator 10, and for doubling the last right hand figure in the product for shillings; and instead of the above operation, the following only would have been necessary. 647 4 £258,8=£258 and 8 tenths, or, 8 times 2 or 16s. answer. If the quantity end with a fraction, the fraction and the whole number in the units' place must be doubled in the product for shillings. EXAMPLES. What will 377 yards come to at 6s. per yard? 377 Answer, £113 6s. 6d. The best method, perhaps, is to say 3 times 7 is 231; double the 34, which makes 63 shillings, and carry the 2. 1763 yards at 10s. Answer, £88 8s. 9d. 490 lb. at 12s. Answer, £294 2s. When a quantity has fractional parts, and the price is pounds, shillings, and pence, convert the fractional parts of the quantity into pounds, &c., and proceed to find the value in the usual way. The value of the ton being given to find the value of any given number of lb. Rule. For every 71b. add three farthings; multiply the quantity thus turned to money by the pounds sterling per ton, and take aliquot parts for the rest, will be the answer. EXAMPLES. What will 14lb. come to at £7 6s. 8d. per ton? Here 14lb. 7×2; and × 2=14d.; hence the operation is as follows: 6s. 8d. 11⁄2d. 10 Answer, 11d. Had it been required to have found the value of 161b., the above operation might still have been the same, only with the difference of adding of the above answer to itself, thus, 11+=1s. Od. 4, which would be the exact answer; 13d. 12d. for 14+16; thus, 1d.+. ; this multiplied 7 7 1s. Od., as before; but the other This and the following rules are founded on these principles. A ton being as many cwt. as there are shillings in a £. sterling; and this being divided and subdivided, we have a cwt. in weight, corresponding to one shilling in money; a quarter of a cwt. will then be 3d.; 7lb. will be three farthings; and, should the given number of lb. not be exactly the multiple of 7, either of the above measures may be used to obtain the answer. What will 211b. come to at £9 per ton? * Answer, 1s. 84d. At £100 per ton, what is the value of 71b.? Answer, 6s. 3d. In addition to the above, the following rule may be useful. The product of the lb. in weight and £. sterling multiplied by, will be the answer in farthings; for 2240 the lb. in a ton being put as a denominator, and 960, the farthings in a £. as a numerator, the fraction will be 18, which shows that for 1lb. at £1 per ton the answer will be of a farthing; from these considerations the above rule is made. EXAMPLES. What will 291b. come to at £21 per ton? Here 29×21 × 2=29 × 3 × 3=71. Answer, 5s. 3 d What will 191b. come to at £48 per ton? Answer, 8s. 11⁄2d. ‡ The following is the converse of this. When a rule is made similar to the above, another rule may also immediately be made from it, by inverting the multiplier, &c., of which the following is an instance. The value of any number of lb. being given to find the value of a ton. Rule. Divide by the given number of lb.; this quotient multiplied by the value in farthings will be the answer in pounds sterling per ton. EXAMPLES. If 291b. cost 5s. 54d., what is the value of a ton? First 32937 And 5s. 54d.=261 farthings.} Hence1£21 the answer. If 17lb. cost 5s. 3 d., what is the value of a ton? Answer, £35. What is the value of a ton when 35lb. cost 7s. 64d.? Answer, £24 1s. 4d. Note. If the quantity be any multiple of 7, put the same multiples of of a penny for a denominator, and the cost price in pence as a numerator; the quotient will be the £. sterling required; or, it may be as well to reduce both to farthings; and it will give the answer at once. EXAMPLES What is the value of a ton when 7lb. cost 1s. 6d. ? Hence £25, the answer. What is the value of a ton when 14lb. cost 3s. 9d.? First, 3s. 9d.=180 farthings. And 14-twice seven; therefore twice 3 or 6 is the divisor. Hence 1-£30, the answer. If 63lb. cost 15s. 64d., what will 1 ton cost? Answer, £27 11s. 10d. When the quantity consists of quarters of cwt., add a many threepences as there are quarters, because threepence is the fourth of a shilling—a quarter being the fourth of a cwt. EXAMPLES What will 3qr. 21lb. come to at £10 6s. 8d. per ton? Here 3qrs.=9d. } sum 11 d. 6s. 8d. 11 d. 10 9 4 33 Answer, 9s. 84d What will 1qr. 7lb. come to at £5 7s. 6d. per ton? What will 2qr. 21lb. come to at £25 per ton? Answer, 17s. 2 d. When there are cwt. qr. and lb. to find the value at so much per ton. Rule. Consider the cwt. as so many shillings, proceed with the qr. and lb. as before; which being multiplied by the price per ton will be the answer. EXAMPLES. What will 13cwt. 1qr. 14lb. come to at £7 6s. 8d. per ton? At £16 13s. 4d. per ton, what is the value of 19cwt. 3qr. 14lb.? Answer, £16 11s. 3d. The answer to the above example may be obtained better by the following method of reasoning. First, 19cwt. 3qr. 14lb. is less than a ton by 14lb., or of a cwt.; but of a cwt. is Too of a ton; now, if this be considered a fractional part of a £. sterling, its value will be 1d., or 1 d. to be deducted from each of the above number of £.; therefore £16 13s. 4d. -2s. 1d. (16 times 1d.) we have £16 11s. 3d., as above. What will 12cwt. 2qr. 20lb. come to at £10 10s. per ton? Answer, £6 13s. 1 d. When there are tons, cwt. qr. and lb. given, to find their value. Convert the tons into £. sterling, and proceed with the rest as before directed. What EXAMPLES. will 3tons 7cwt. 3qr. 14lb. come to at £15 What will 17tons, 3cwt. 2qr. 94lb. come to at £10 5s. per ton? Answer, £176 1s. 83d. The price of a lb. being given to find the value of a cwt. Rule. Multiply the farthings in the price by, and the answer in pounds is obtained. The reason of this rule is founded upon this principle; namely, by putting the lb. in a cwt. as a numerator, and subscribing the farthings in a £. sterling as a denominator; thus, ; but, if instead of the 960, we put the pence in a £. sterling for a denominator, the fraction will be ; consequently, the pence in the price multiplied by 7, and divided by 15, for £. sterling, will be both one and the same thing; but the former method is preferable for general use. What is the value of 1cwt. at 10 d. per lb. ? (By the former method.) Here 10 d. 43 farthings. (By the latter method.) 102d. At 1s. 14d. per lb., what is the value of 1cwt. ? Answer, £6 3s. 8d. What will lcwt. come to at 1s. 51d. per lb. ? Answer, £8 1s. To find the value of several cwt. The rule for finding the value of 1cwt. having been found to be, it follows that if this fraction be multiplied by any given number of cwt., the answer will be obtained. And if be found to be the rule, so will also its equal; 10 14 viz, 90 120 &c. Hence the raising of the terms of a fraction; so that the denominator be raised to such a number as to make it divisible by the given quantity, in order that the fraction may be increased, (which has been pointed out in multiplication) is another instance of the great utility of the fractions in their application to mental calculation; and a further proof of the excellence of a right understanding of fractions. It will often happen that the greater the quantity the less the trouble it will take to obtain the answer; for we multiply by to obtain the answer for 1cwt.; to obtain the answer for 15cwt.; and lastly, we multiply by 7, or 4, to find the answer for 60cwt., or 3 tons. |