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EXAMPLES 1. Reduce i35 to its equivalent decimal.
We here use two ciphers and therefore point off two decimal places in the quotient.
OPERATION. 125 635(5,08
625 1000 1000
2. Reduce 1 and 1127 to decimals.
Ans. ,25 and 200797+. 3. Reduce 10, 35, 100, and be to decimals. .
Ans. ,025; ,692 +; ,003; ,000183+. 4. Reduce } and
to decimals. Ans. t. 5. Reduce 314957123 to'a decimal. Ans. 1,496+. 6. Reduce &, 1936, 191, 174 to decimals.
Ans. 1,333+; 0,162+; 0,792+; 4,666+. REDUCTION OF DENOMINATE DECIMALS. § 134. We have seen that a denominate number is one in which the kind of unit is denominated or expressed (see § 45.)
A denominate decimal is a decimal fraction in which the kind of unit that has been divided is expressed. Thus, ,5 of a £, and ,6 of a shilling, are denominate decimals. The unit that was divided in the first fraction being £1, and that in the second 1 shilling.
Q. What is a denominate number? What is a denominate decimal ? In the decimal five-tenths of a £, what is the unit ? In the decimal six-tenths of a shilling, what is the unit ?
$ 135. To find the value of a denominate number in decimals of a higher denomination.
EXAMPLES. 1. Reduce 9d to the decimal of a £.
We first find that there are 240 OPERATION. pence in £1. We then divide 9d by 240d=£1 240, which gives the quotient ,0375 240)9(,0375 of a £. This is the true value of 9d Ans. £,0375. in the decimal of a £.
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. ,027343751b. 3. Reduce 26d to the decimal of a £. Ans. ,1083333+. 4. Reduce ,056 poles to the decimal of an acre.
Ans. A. 5. Reduce 14 minutes to the decimal of a day.
Ans. ,0097222da.+. 6. Reduce ,21 pints to the decimal of a peck.
Ans. pk. 7. Reduce 3 hours to the decimal of a day. Ans. ,125. 8. Reduce 375678 feet to the decimal of a mile.
Ans 71,151+. 9. Reduce 36 yards to the decimal of a rod. 10. Reduce ,5 quarts to the decimal of a barrel.
§ 136. To reduce denominate numbers of different denominations to an equivalent decimal of a given denomination.
1. Reduce £1 4s 9d to the denomination of pounds.
We first reduce 3 farthings to the decimal of a penny,
id=,75d, hence, by dividing by 4. We then
9d=9,75d, annex the quotient ,75 to the 12)9,750 9 pence. We next divide
,8125s, and by 12 giving ,8125 which is the decimal of a shilling.
20)4,8125s, This we annex to the shillings
£,240625, therefore, and then divide by 20.
£1 48 9;d=£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 3}d to the decimal of a £.
Ans. £19,863+. 3. Reduce 15s 6d to the decimal of a £. Ans. £,775. 4. Reduce 7}d to the denomination of shillings.
Ans. S. 5. Reduce 21b. 5oz. 12 pwt. 16gr., Troy, to the decimal of a 1b.
Ans. 2,4694441b.t. 6. Reduce 3 feet 9 inches to the denomination of yards.
Ans. 1,25yd. 7. Reduce 11b. 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 denominations, 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 £.
OPERATION. We first multiply the decimal by
,832296 20, which brings it to shillings, and
20 after cutting off from the right as
16,645920 many places for decimals as in the
12 given number, we have 16s and the
7,751040 decimal ,645920 over. This we reduce to pence by multiplying by 12, and then reduce to farthings by
3,004160 multiplying by 4.
| Ans. 16s 7d 3far.
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 ,00208416. Troy?
Ans. 12,00384gr. 3. What is the value of ,625 of a cwt. Ans. 2qr. 1416. 4. What is the value of ,625 of a gallon?
Ans. 2qt. 1 pt. 5. What is the value of £,3375 ?
Ans. 6. What is the value of ,3375 of a ton ? Ans. 6cwt. 3qr. 7. What is the value of ,05 of an acre ?
Ans. 8P. 8. What is the value of ,875 pipes of wine ?
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. 121b. 50. 1,92dr. 13. What is the difference between ,82 of a day and ,32 of an hour ?
Ans. 19hr. 21m. 36 sec. 14. What is the value of 1,089 miles ?
Ans. lm. 28rd. 7ft. 11,04in. 15. What is the value of ,09375 of a pound avoirdupois weight?
Ans. 16. What is the value of ,28493 of a year of 365 days?
Ans. 103da. 23hr. 59m. 12,48sec. 17. What is the value of £1,046 ? Ans. £i ildt. 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.
1 yard of cloth is the 1st term;
4th term. 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: 6 :: 2 : 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,