Exercises. Find the selling price of the following: Most of the previous rules may easily be inverted, and thereby the number of exercises doubled; as, for example, to find the price of one, when the price of a dozen is given, we will evidently have the following: RULE.-Call the shillings in the price of one, pence, to which add as many farthings as there are threepences in the price of a dozen, and the sum will be the price of one. Again, let there be given the price of a pound avoirdupois, to find the price of an ounce; or the price of a yard, to find the price of a nail by inverting Rule VII. p. 112, we will evidently have the following for finding the price of an ounce or a nail : RULE.-Multiply the price of 16 by 3, and call the shillings in the product, farthings, and this will give the price of one; which will be the price of an ounce if the price of a pound be given, or the price of a nail if the price of a yard be given. In the same manner may many of the other rules be inverted, and the exercises given under the rule will easily be accommodated to the inverted rule. Miscellaneous Exercises, to be solved Mentally. 1. Change to decimals *, 1, 2, §, Ans. 5, 25, 75, 625. Ans. 3, 2, 714285, 36. 3. Find equivalent fractions to 1, 3, 4, 4, and §, having a common denominator 12, Ans. 1, and t 4. If of a ship be worth £200, what is the whole ship worth? Ans. £1600. 5. Change the fractions 4, 7, 4, and, to their lowest terms, Ans.,,, and . 6. If the of a number be 34, what is the whole number? Ans. 187. 7. If the of a number be 25, what is its product by 3? Ans. 120. 8. How often does 30 contain the of 8? Ans. 5 times. 9. What is the of of 120? . Ans. 75. 10. A man gave away 3s., which was of the money he had; how much had he? Ans. 15s. 11. If from one pound there be paid of, and then of; how much will remain ? Ans. 13s. 8d. * See Decimals, p. 135. RATIO. RATIO is a word used to express the comparative magnitude of two numbers or quantities of the same kind, or how many times the one number or quantity contains the other. Thus, if one field contain 9 acres, and another 3 acres, we find that 9 acres contain 3 acres three times; the ratio of 9 acres to 3 acres is therefore three. Again, since 4 shillings is the one-half of 8 shillings, the ratio of 4 shillings to 8 shillings is one-half. The two numbers are called the terms of the ratio: the first, the antecedent; the second, the consequent. Thus, in the ratio 9 to 3, the first term, 9, is the antecedent; the second, 3, is the consequent. Ratio, being just another name for quotient, is expressed by writing the numbers in the form of a fraction; the antecedent for the numerator, and the consequent for the denominator: thus, the ratio of 9 to 3 is expressed by, the ratio of 4 to 8 by 4. "The ratio of two numbers is also sometimes expressed by placing a colon between them: thus, the ratio of 9 to 3 is expressed by 9:3. A number that does not refer to any particular kind of unit is called an abstract number: thus, 9 is an abstract number. A number that denotes so many of some particular kind of unit is called a concrete number: thus, 9 acres is a concrete number. Now, though the terms of a ratio are concrete numbers of the same denomination, the ratio itself is an abstract number, whole or fractional. When both the terms of a ratio are multiplied, or both divided, by the same number, the ratio is not altered. This is very obvious, since it is only multiplying or dividing the terms of a fraction by the same number; which does not alter its value (p. 81). Thus, the ratio = †, and the ratio # = . PROPORTION. PROPORTION, in Arithmetic, denotes the equality of two Ratios. If two Ratios are equal; that is, if the two numbers forming the one Ratio are of the same comparative magnitude with regard to each other, as the two numbers forming the other Ratio, the four numbers which compose these Ratios, taken in order, make a Proportion. The first and fourth terms of a proportion are called the extremes; the second and third, the means. Thus, since the ratio of 12s. to 6s. is 2, and the ratio of 8 yards to 4 yards also 2, the ratios are equal, and the four numbers, 12, 6, 8, 4, make a proportion, which is written thus-12:6::8:4; or thus, 12:68:4, and is read, 12 is to 6 as 8 is to 4. The numbers 12 and 4 are the extremes, 6 and 8 the means. In any proportion, if the first term is greater than the second, equal to it, or less than it, the third term is also greater than the fourth, equal to it, or less than it. = third first For, the ; now, if the first be greater than the second, the first fraction is greater than unity, therefore, the second fraction is also greater than unity: hence its numerator, which is the third term, must be greater than the denominator, which is the fourth term; and in the same way when equal, equal; and when less, less. In a Proportion, the first and second terms must express things of the same kind; thus, if the first is yards, the second must also be yards; the third and fourth must also be of the same kind. There can be no comparison between things that are not of the same kind, as, for instance, between yards and minutes. ortion, the product of the extremes rtion, as 6, 9, 8, 12; then When four numbers are in prope is equal to the product of the means. For, take any four numbers in propo Pupil. Since 6 is 12 by 9 must be equal to multiply each of these equals by 9 and by 12, which gives 6 multi plied by 12 equal to 6: 9 :: 8 : 6 = 9 8 12 .. 6 × 128X9 8 multiplied by 9; that is, the product of the extremes is equal to the product of the means. From this it follows that the product of the second and third terms divided by the first, is equal to the fourti. For, since 9 × 8 = 6 × 12, if each of these equals is divided by 6, we get 9 x 8 ÷ 6 = 12. Hence, if the first, second, and third terms of a Pro portion are given, the fourth can be found, by multiplying the second and third terms together, and dividing their product by the first. It is on these principles that the Rule termed SIMPLE PROPORTION is founded. Example 1.-If 25 cwt. of sugar cost £75, how much will 16 cwt. cost at the same rate? £ 25 16: 75 16 450 750 25 = 5)1200 5)240 £48 Ans. Pupil. In this question, it is obvious that the answer will be in money; therefore, I place £75 for the third term. Again, as 25 cwt. cost £75, 16 cwt. will cost less; I therefore place 16, the less of the two numbers, for the second term, and 25 for the first. I next multiply £75 by 16, and then divide the product, £1200, by 25, and the quotient, £48, is the answer required. Note.-The reason of the operation will also be obvious from the consideration, that since £75 is the price of 25 cwt., the value of 1 cwt. will be equal to the 25th part of £75-namely, £3; and as 16 cwt. will cost 16 times as much as 1 cwt., the price of 1 cwt., multiplied by 16, will give the price of 16 cwt. = = £48. Example 2.-If a man can perform a certain piece of work in 28 days, working 9 hours each day, in how many days would he perform a similar piece of work, working 12 hours each day? days. 12 9: 28 9 12)252 This is an example of what is called inverse proportion. By the application of the rule, the work will stand as in the margin. As in last example, the reason of the operation will be obvious from the consideration, that since the workman takes 28 days of 9 hours; that is, 28 × 9 hours, or 252 to complete the work, he must require 252 hours to finish a similar piece of work; and as he now works 12 hours a day, therefore, 252, divided by 12, will give the number of days; that is, 21 days. 21 days. Example 3.-If 4 cwt. 2 qr. 11 lb. of tobacco cost £25, 17s. 64d., how much ought to be paid for 2 cwt. 1 qr.? In this example, the question is stated as before. The third term, being a compound number, is converted to farthings, the lowest denomination it contains; the first and second terms, being also compound numbers, are converted to pounds, the lowest denomination that the |