Hence, we have the following RULE. I. Consider how many units of the given denomination make one unit of the denomination to which you would reduce. II. Divide the given denominate number by the number so found, and the quotient will be the value in the required denomination. Q How do you find the value of a denominate number in a decimal of a higher denomination? 2. Reduce 7 drams to the decimal of a lb. avoirdupois. Ans. ,02734375lb. 3. Reduce 26d to the decimal of a £. Ans.,1083333+. 4. Reduce,056 poles to the decimal of an acre. Ans. 5. Reduce 14 minutes to the decimal of a day. A. Ans. ,0097222da.+. 6. Reduce,21 pints to the decimal of a peck. Ans. pk. Ans.,125. 7. Reduce 3 hours to the decimal of a day. Ans 71,151+. 9. Reduce 36 yards to the decimal of a rod. 10. Reduce,5 quarts to the decimal of a barrel. CASE II. § 136. To reduce denominate numbers of different denominations to an equivalent decimal of a given denomination. EXAMPLES. 1. Reduce £1 4s 93d to the denomination of pounds. We first reduce 3 farthings to the decimal of a penny, by dividing by 4. We then annex the quotient,75 to the 9 pence. We next divide by 12 giving,8125 which is the decimal of a shilling. This we annex to the shillings and then divide by 20. OPERATION. 2d=,75d, hence, 93d 9,75d, 12)9,75d ,8125s, and 20)4,8125s, £,240625, therefore, £1 48 93d £1,240625. Hence, we have the following RULE. Divide the lowest denomination named, by that number which makes one of the denomination next higher, annexing ciphers if necessary: then annex this quotient to the next higher denomination, and divide as before: proceed in the same manner through all the denominations to the last the last result will be the answer sought. 2. Reduce £19 17s 31d to the decimal of a £. : Ans. £19,863+. 3. Reduce 15s 6d to the decimal of a £. Ans. £,775. 4. Reduce 7d to the denomination of shillings. Ans. S. 5. Reduce 2lb. 5oz. 12pwt. 16gr., Troy, to the decimal of a lb. Ans. 2,4694447b.+. Reduce 3 feet 9 inches to the denomination of yards. Ans. 1,25yd. 6. 7. Reduce 1lb. 12dr., avoirdupois, to the denomination of pounds. Ans. 1,046875lb. 8. Reduce 5 leagues 2 furlongs to the denomination of leagues. Ans. +. Q. How do you reduce denominate numbers of different denomina. tions, to equivalent decimals of a given denomination? CASE III. § 137. To find the value of a denominate decimal in terms of integers of inferior denominations. EXAMPLES. 1. What is the value of ,832296 of a £. Hence, the following RULE. 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. 2. What is the value of ,002084lb. Troy? Ans. 12,00384gr. 3. What is the value of ,625 of a cut. 5. What is the value of £,3375? Ans. 2qr. 14lb. Ans. 2qt. 1pt. 6. What is the value of ,3375 of a ton? Ans. 6cwt. 3qr 7. What is the value of ,05 of an acre? 8. What is the value of ,875 pipes of wine? Ans. 8P Ans. 9. What is the value of ,125 hogsheads of beer? (see § 67.) Ans. 6gal. 3qt. 10. What is the value of ,375 of a year of 365 days? Ans. 136da. 21hr. 11. What is the value of ,085 of a £? Ans. 12. What is the value of ,86 of a cwt.? +. Ans. 3qr. 12lb. 5oz. 1,92dr. 13. What is the difference between ,82 of a day and ,32 of an hour? Ans. 19hr. 21m. 36sec. 14. What is the value of 1,089 miles? Ans. Ans. 1m. 28rd. 7ft. 11,04in 15. What is the value of ,09375 of a pound avoirdupois weight? 16. What is the value of ,28493 of a year of 365 days? Ans. 103da. 23hr. 59m. 12,48sec Ans. £1 11d+. 17. What is the value of £1,046 ? 18. What is the value of £1,88 ? Ans. +. Q. 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? RULE OF THREE. § 138. If 1 yard of cloth cost $2, how much will 6 yards cost at the same rate? It is plain that 6 yards of cloth, at the same rate will cost 6 times as much as 1 yard, and therefore the whole cost is found by multiplying $2 by 6, giving $12 for the cost. 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. Three of these terms were known or given in the question, and the other was to be found. Now the 2nd term 6 contains the first term 1, 6 times, and the 4th term 12 contains the 3rd term 2, 6 times-that is, the 2nd term is as many times greater than the 1st, as the 4th term is greater than the 3rd. This relation between four numbers is called proportion; and generally Four numbers are in proportion, when the 2nd term is as many times greater or less than the 1st, as the 4th term is greater or less than the 3rd. We express that four numbers are in proportion thus: 1: 62 : 12. That is, we write the numbers in the same line and place two dots between the 1st and 2nd terms, four between the 2nd and 3rd, and two between the 3rd and 4th terms. We read the proportion thus, as 1 is to 6, so is 2 to 12. The 1st and 2nd terms of a proportion always express quantities of the same kind, and so likewise do the 3rd and 4th terms. As in the example, Q. If 1 yard of cloth cost $2, what will 6 yards cost? How many numbers are considered in this question? What are they called? How many were known or given? Name the terms. How many times does the 2nd term contain the first? How many times does the fourth contain the third? How many times is the second term greater than the first? When are four numbers in proportion? How are they written? How are they read? What terms of a proportion express quantities of the same kind? If 1 yard of cloth cost $3, what will 2 yards cost at the same rate? What will 3 cost? 4? 5? 6? 7? 8? 9? 10? If one yard of cloth cost $4 what will 2 yards cost? What will 3 yards cost? 4? 5? 6? 7? 8? 9? 10? § 139. The numbers 2:48: 16 are in proportion since the 2nd term is two times greater than the 1st, and the 4th term two times greater than the 3rd. And when four numbers are in proportion, the quotient of the 2nd term divided by the 1st, is equal to the quotient of the 4th term divided by the 3rd. This quotient is called the ratio of the proportion. Thus 2 is the ratio of the proportion 2:4 : : 8 : 16; The ratio of two numbers, simply expresses how many times the second number contains the first. Hence, it is equal to the quotient of the second number divided by the first. 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=1. The ratio of 9 to 3 is, since 3÷9=1. In every proportion, the ratio of the 1st term to the 2nd, is equal to the ratio of the 3rd term to the 4th. EXAMPLES. 1. What is the ratio of 9 to 18? 2. What is the ratio of 6 to 24? Ans. 2. Ans. Ans. Ans. 13 2. 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 Ans. to 4? |