CASE III. 179. To find the value of a denominate decimal in terms of integers of inferior denominations. What is the value of .832296 of a £.? We first multiply the decimal by 20, which brings it to shillings, and after cutting off from the right as many places for decimals as in the given number, we have 16s. and the decimal .645920 over. This we reduce to pence by multiplying by 12, and then reduce to farthings by multiplying by 4. Hence, to make the reduction, OPERATION. .832296 20 16.645920 12 7.751040 4 3.004160 Ans. 16s. 7d. 3far. I. Consider how many in the next less denomination make one of the given denomination, and multiply the decimal by this number; then cut off from the right hand as many places as there are in the given decimal. II. Multiply the figures so cut off by the number which it takes in the next less denomination to make one of a higher, and cut off as before. Proceed in the same way to the lowest denomination: the figures to the left will form the answer sought. EXAMPLES. 1. What is the value of .002084lb. Troy? Ans. 12.00384gr. 2. What is the value of .625 of a cwt.? Ans. 2qr. 12lb. 8oz. 3. What is the value of .625 of a gallon? 4. What is the value of £.3375 ? Ans. 2qt. 1pt. Ans. 6cwt. 3qr. 6. What is the value of .05 of an acre? Ans. 8P. 179. How do you find the value of a denominate decimal in integers of inferior denominations? What is the value in shillings of one-half of a £? In pence of one-half of a shilling? 7. What is the value of .875 pipes of wine? 8. What is the value of .125 hogsheads of beer? Ans. 6gal. 3qt. 9. What is the value of .375 of a year of 365 days? Ans. 136da. 21hr. 10. What is the value of .085 of a £? 11. What is the value of .86 of a cut. ? Ans. Ans. 3qr. 11lb. 12. What is the difference between .82 of a day and .32 of an hour? Ans. 19hr. 21m. 36sec. -13. What is the value of 1.089 miles? Ans. 1mi. 28rd. 7ft. 11.04in. 14. What is the value of .09375 of a pound, avoirdupois weight? Ans. 15. What is the value of .28493 of a year of 365 days? Ans. 103da. 23hr. 59m. 12.48sec. 16. What is the value of £1.046? Ans. £1 11d.+. 17. What is the value of £1.88? Ans. PROMISCUOUS QUESTIONS. 1. What will 11 tons of hay cost at $17,37 a ton? Ans. $201,92625. Ans. 2. What will 12gal. 3qi. 1pt. of wine cost at $0,28 a quart? 3. Bought a load of potash for $9,17, paying at the rate of $16 a ton: what was the weight of the potash? Ans. 11cwt. 1qr. 21lb. 4. What will 57yd. 2qr. 3na. of cloth cost at $6,78 a yard? 5. What will 7A. 2R. 38P. of land cost at $64,50 per acre? Ans. $499,06875. Ans. 6. Suppose a farmer had 4 granaries of rye: the first contained 4.67 bushels; the second 9.87; the third 10.01; and the fourth 11.0012; after using 18.0679 bushels he sold the remainder for $1,03 per bushel, and divided the money among nine persons: what did each receive? 7. What is the cost of 693 yards of cloth at $3,4775 per yard? Ans. $2409.9075. 8. What is the cost of 917 bushels of wheat at $1,125 per bushel? Ans. OF THE RATIO AND PROPORTION OF NUMBERS. 180. Two numbers having the same unit may be compared together in two ways. 1st. By considering how much one is greater or less than the other, which is shown by their difference; and 2d. By considering how many times one is greater or less than the other, which is shown by their quotient. Thus, in comparing the numbers 3 and 12 together with respect to their difference, we find that 12 exceeds 3 by 9; and in comparing them together with respect to their quotient, we find that 12 contains 3 four times, or that 12 is four times greater than 3. The quotient which arises from dividing the second number by the first, is called the ratio of the numbers, and shows how many times the second number is greater than the first, or how many times it is less. Thus, the ratio of 3 to 9 is 3, since 9÷3=3. The ratio of 2 to 4 is 2, since 4÷2=2. We may also compare a larger number with a less. For example, the ratio of 4 to 2 is; for, 2÷4=. The ratio of 9 to 3 is, since 3÷9=1. EXAMPLES. 244 Ans. 2. Ans. 4. Ans. 4. Ans. 1. What is the ratio of 9 to 18? 2. What is the ratio of 6 to 24? 3. What is the ratio of 12 to 48? 4. What is the ratio of 11 to 13? 5. What part of 20 is 4? Or what is the ratio of 20 to 4.? Ans. 180. In how many ways may two numbers having the same unit be compared? How do you determine how much one number is greater than another? How do you determine how many times it is greater or less? How much does 12 exceed 3? How many times is 12 greater than 3? What is the quotient called which arises from dividing the second number by the first? What does it show? When the second number is the least, what does it show? 6. What part of 100 is 30? Or what is the ratio of 100 to 30? 7. What part of 6 is 3? 8. What part of 9 is 3? 9. What part of 12 is 4? 10. What part of 50 is 5? 11. What part of 75 is 3? Ans. 3 10 Ans. Ans. Ans. Ans. Ans. 181. In determining what part one number is of another, it is plain that the number which makes the part must be written in the numerator, and the number of which it is a part, in the denominator, and that this fraction reduced to its lowest terms will express the part. 182. If one yard of cloth cost $2, how many dollars will 6 yards of cloth cost at the same rate? It is plain that 6 yards of cloth will cost 6 times as much as one yard; that is, the cost will contain $2 as many times as 6 contains 1. Hence the cost will be $12. In this example there are four numbers considered, viz., 1 yard of cloth, 6 yards of cloth, $2, and $12: these numbers are called terms. Now the ratio of the first term to the second is the same as the ratio of the third to the fourth. This relation between four numbers is called proportion; and generally, Four numbers are said to be in proportion when the ratio of the first to the second is the same as that of the third to the fourth. Hence, 181. How do you determine what part one number is of another? 182. If one yard of cloth cost $2, what will 6 yards cost? How many numbers are here considered? What are they called? What is the ratio of the first to the second equal to? What is this relation between numbers called? When are four numbers said to be in proportion? How do you define proportion? PROPORTION is an equality of ratios between numbers compared together two and two. 183. We express that four numbers are in proportion thus: 1 : 6 : : 2 : 12. That is, we write the numbers in the same line and place two dots between the 1st and 2d terms, four between the 2d and 3d, and two between the 3d and 4th terms. We read the proportion thus, as 1 is to 6, so is 2 to 12. The 1st and 2d terms of a proportion always express quantities of the same kind, and so likewise do the 3d and 4th terms. As in the example, yd. yd. 1 : 6 : :. 2 : 12. This is implied by the definition of a ratio; for, it is only quantities of the same kind which can be divided the one by the other. The ratio of the first term to the second, or of the third to the fourth, is called the ratio of the proportion. What are the ratios of the proportions 184. When two numbers are compared together, the first is called the antecedent, and the second the consequent; and when four numbers are compared, the first antecedent and consequent are called the first couplet, and the second antecedent and consequent the second couplet. Thus, in the last proportion, 16 and 48 are the 183. How do you indicate that four numbers are in proportion? How is the proportion read? What do you remark of the first and second terms? Also of the third and fourth? 184. When two numbers are compared together, what is the first called? What the second? When four numbers are compared, what are the two first called? What the two second? |