Write out, in the form of a bill, each of the following exercises, and find the total amount of each bill: B. (1) 15 lbs. of tea at 3s. 4d.; 17 lbs. of sugar at 61d.; 18 lbs. of sugar at 43d.; 14 lbs. of coffee at is. 8d.; 19 lbs. of rice at 3d. (2) 74 yds. of black cloth at 17s. 6d. ; 59 yds. of drab cloth at 18s. 9d.; 794 yds. of green cloth at 15s. 7d.; 117 yds. of brown cloth at 14s. 10d.; 94 yds. of olive cloth at 16s. Id. (3) 157 yds. of calico at Is. 74d.; 138 yds. of linen at 2s. 11d.; 1794 yds. of sheeting at Is. 5d.; 1923 yds. of sheeting at 2s. 9d.; 237 yds. of cambric at 25. 10d. (4) 327 lbs. of congou at 4s. 91d.; 219 lbs. of hyson at 5s. 6d. ; 273 lbs. of sugar at 7d.; 587 lbs. of soap at 81d.; 2971 lbs. of starch at 10d. (5) 387 yds. of velvet at 13s. 9d.; 579 yds. of velvet at 17s. 71d.; 6071 yds. of holland at 6s. 9d. ; 495 yds. of muslin at 1s. 5d.; 283 yds. of linen at 3s. 4 d. CHAPTER II. RATIO. 7. If one concrete quantity be divided by another concrete quantity of the same kind, and expressed in the same denomination, the quotient shows how many times, or parts of a time, the one contains the other. 5S. 18s. Thus 18s. 6s., or, as it may be written, gives the abstract number 3, that is 3 times; and (read 5s. divided by 2s. 6d.) gives the abstract number 2, or 2 times. Again, 3s. ÷ 5s., or 35 gives the fraction or part, for since is. is of 5s., there 2s. 6d. 5S. fore 3s. is of 5s. ; that is, 3s. contains 5s. (threefifths) of a time. 8. Ratio is the relation which exists between two numbers, or between two quantities of the same kind, when the comparison is made by considering how many times, or parts of a time, the one contains the other; that is when the comparison is made by division, as above. 9. The terms of a ratio are the two quantities compared; the first term is called the antecedent, and the second term the consequent. 10. The value of a ratio is found by dividing the antecedent by the consequent. 11. Whatever may be the nature of the quantities compared their ratio is always an abstract number; thus the ratio of 15s. to 5s., or, as it is written, 15s. 5s., is the abstract number 3, not 3s., for it has no reference to the particular kind of quantities compared, but only to the magnitude of the one compared with that of the other. Now the ratio of 155 is also 3, therefore the ratio of 15:5 is the same as the ratio of 15s. : 5s., and therefore when the terms of a ratio are concrete quantities expressed in the same denomination, the denominations may be omitted and the numbers treated as abstract. 12. The value of a ratio is not altered by multiplying or dividing both its terms by the same number. Thus the ratios 24: 16, 12: 8, 6: 4, 18: 12 are all equal, since the first of each of these pairs of numbers is (3 halves) of the second. EXERCISE 2. What is the simplest ratio that will express the relation between (1) 3 and 4; (2) 18 and 27; (3) 15 yds. and 20 yds.; (4) 5s. 3d. and Is. 9d. (1) 31:47: 8 by multiplying each term by 2. (2) 18:27 = 2 : 3 by dividing each term by 9. (3) 15 yds. 20 yds. = ing each term by 5. (4) 5s. 3d. : IS. 9d. = 15: 203: 4 by divid 63d. : 21d. = 63: 21 = 31, by dividing each term by 21. What is the simplest value of the following ratios? (1) 8s.: 125.; (2) 4s. 2d. : 3s. 4d.; (3) 2 yds. 3 qrs. I qr. 3 nls.; (4) £1 17s. 6d: £3 2s. 6d. (1) 8s. 125. = 8 : 12 = 2 : 3 (2) 4s. 2d. : 3s. 4d. 4 = = === 50d.: 40d. (3) 2 yds. 3 qrs. : 1 qr. 3 nls. (4) £1 17s. 6d. : = = 50: 40 = 5: £3 2s. 6d. 450d.: 750d. 450 : 750 3:5 ; or, £1 17s. 6d. : 15 half-crowns: 25 half-crowns What is the simplest ratio that will express the relation between the following? A. (1) 41 : 5; (2) 3 : 21; (3) 51 : 61; (4) 23 : 4; (5) 343; (6) 31: 41; (7) 15: 18; (8) 24: 30; (9) 28:21; (10) 12 cwt.: 20 cwt; (11) 1 yd. 2 qrs.: 3 yds. 2 qrs.; (12) 3s. 6d. : 5s. 9d. ; (13) 2s. : Is. 4d.; (14) 12s. 6d. : 15s. ; (15) £6 : 4 gui. ; (16) 10 fls. 12 half-crowns. What is the simplest value of the following ratios ? B. (1) 12: 18; (2) 95.: 15s.; (3) 1s. 8d. : 2s. 6d.; (4) 35. : is. 6d.; (5) 3s. 4d.: 10s.; (6) 2 cwt. 2 qrs.: 3 cwt. 3 qrs. ; (7) 1 qr. 7 lbs. 2 qrs. 4 lbs.; (8) 1 ft. 7 in.: 2 ft. 4 in. 151 CHAPTER III. PROPORTION. 13. Proportion is the equality of ratios. Thus, the ratios 23. 4s. and 3 yds. 6 yds., being each equal to form a proportion. 14. Proportion is generally expressed by placing a double colon between the two ratios, thus, 2s. : 4s. 3 yds. 6 yds. This is read, 2s. is to 4s. as 3 yds. is to 6 yds. : 15. Since every ratio consists of two terms, a proportion must consist of at least four terms. Of these the 1st and 4th are called the extremes, and the 2nd and 3rd the means. The two extremes or the two means are sometimes called similar terms, and one extreme and one mean dissimilar terms. 16. Taking the proportion 2 : 4:3: 6, we find that the four terms are connected as follows:= 4 × 3; Ist. 2 x 6 = or the product of the extremes is equal to the product of the means. = 2nd. 4 × 3 ÷ 6 = 2; 2 × 6 ÷ 3 = 4; 2 X 6 ÷ 4 3; 4 × 3÷ 2 6. Hence, if three terms of a proportion are given, the missing term can always be found by dividing the product of the similar terms by the other given term. EXERCISE 3. Find the missing term in each of the following: A. (1) : 12: 2: 8 (2) 3: :: 4:8 (5) 18: :: 12: 16 17. When the antecedent of a ratio is less than |