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To reduce Integers of different denominations to a Decimal Fraction of a higher denomination, and the reverse.

Art. 132.-1. Reduce 4 Art. 133.-2. Reduce .375 pence 2 farthings to the decimal of a shilling to integers of lower of a shilling. denominations. Operation. 4/2.0

12 4.500

2 farthings is 2 of a penny; then, by the rule for reducing vulgar frac.375 tions to decimal, we have 2.5, or 5 of a penny. This, placed at the right of 4 pence, 4.5, and divided by 12, the number of pence in a shilling, or because 4 pence is of a shil

ling, gives .375 of a shilling. Hence the

As this question is the reverse of the former, and as the decimal, .375, was obtained by dividing the integers, it is plain, that the integers may be obtained by multiplying the decimal by the same numbers.

Operation.
.375

12

4.500

4

2.000

Hence the

RULE.

RULE.

Place the numbers one above

Multiply the given decimal by

another, the highest denomination that number which expresses how

at the bottom. Divide the lowest denomination by that number which expresses how many of that it takes to make 1 of the next higher denomination, writing the quotient at the right of the next higher denomination; and so proceed until the whole shall be reduced to the required decimal.

OBS.-Integers of different denominations may be reduced to a decimal of a higher, by reducing the given numbers to the lowest denomination mentioned for a numerator, and the integer, to which the given numbers are to be reduced, to the same denomination for a denominator, and dividing the numerator by the denominator.

many of the next lower denomination it takes to make one of that in which the decimal is given; observing to point off as many places in the product, for decimals, as there are figures in the given decimal; and so proceed through all the denominations; and the several numbers at the left of the decimal points will be the answer required.

OBS.--Pointing off the product is the same as dividing by the denominator of the decimal.

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Art. 134. To reduce shillings, pence, and farthings to the decimal of a pound, by inspection.

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1. Reduce 78. 8d. 2qrs. to the decimal of a pound?

5 00

One shilling is of a pound: therefore, two shillings is or. Having, therefore, any number of shillings given, we take one half the even number, they will be reduced at once to the decimal of a pound. If there is an odd shilling, it is the same as Too of a pound: 26.05. The farthing, which of a pound, is made to occupy the 1000ths place. But 980 is greater than 1000 by 24000; there will, therefore, be a loss of 24000 on every farthing; but if we add one to the number, when they exceed 12 and do not exceed 36, and two when they exceed 36, the expression will be nearly so many 1000ths of a pound.

2. Reduce 48. 6d. to the decimal of a pound.

Operation.

.2 half of the even shillings.

.024 farthings in 6d.

.001 for excess of 12.

.225 Ans.

24

If we call the farthings in 6d. 24, there will be a loss of 24.000=1000; if we add 1 to the 1000ths place, we have, in this instance, precisely the decimal required.

3. Reduce 78. 8d. to the decimal of a pound.

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4. Find by inspection the decimal expression of 188. 31d., and 178. 84d. Ans. £.914 and £.885. 5. Reduce to a decimal by inspection the following sums, and add them together, viz:-15s. 3d.; 8s. 111⁄2d.; 10s. 61d.; 18. 81d.; 23d. Ans. 1.832.

Decimals may be reduced back to shillings, pence, and farthings, by reversing the above process. Double the left-hand figure, or tenths, for the shillings; if the second figure be 5, or greater than 5, deduct 5 from it, and add 1 to the shillings. Then consider the second and third figures so many farthings; if they exceed 12, deduct 1; if they exceed 36, deduct 2.

6. Find by inspection the value of £.385.

7. Find by inspection the value of £.927.

Ans. 18s. 6d. 2qrs. 8. Find by inspection the value of £.491, and £.984. Ans. 9s. 9d. 3qrs.; 19s. 8d. 1qr.

8d.

I

COMPOUND ADDITION.

Art. 135.—1. A boy bought a slate for 4d. and a book for What did both cost?

2. If I buy a book for 2s. 4d., another for 4s. pay for both?

Ans. 1s. 8d., what do

Ans. 7s. wagon, what Ans. 9s. 8d.

3. If a boy pay 4s. 8d. for a sled, and 5s. for a does he pay for both?

4. How many shillings in 4d. 8d. 9d. 3d. 6d.?

Ans. 2s. 6d.

5. How many pounds are 8s. 7s. 4s. 3s. 9s. 5s. ?

Ans. £1 16s.

6. How many yards are 3 feet, 4 feet, 5 feet, 6 feet?

Ans. 6 yards. 7. Bought two pieces of cloth; one 10 yards, 1 foot; the other 12 yards, 2 feet. What was the length of both pieces? Ans. 23 yards.

8. What is the amount of £1 4s. 2d. 3qrs., and £10 8s. 3d. and 2qrs.? Ans. £11 12s. 6d. 1qr. 9. Add £4 58. 6d. 3qrs., and £5 17s. 7d. 2qrs.

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In adding the first column, or column of farthings, we find the amount to be 5 farthings. Now as 4 farthings are equal to 1 penny, we write the 1 of farthings, and carry the 1 penny One to 7 is 8, and 6 are 14d.

In

farthing over, in the line to the column of pence. 14d. there is 1 shilling and 2d. over, which we write in the column of pence, carrying the 1s. to the column of shillings. One added to 17 is 18, and 5 are 23. In 23s. there is £1 and 38. over, which we write in the column of shillings, and carry 1 to the column of pounds.

Had the numbers to be added in the question been simple numbers, we should have had none to carry, because 5, in the column of units, is not equal to 1 in the column of tens. Again, had 14 been in the column of tens, we should have written 4 and carried 1. Lastly, had 23 been in the column of hundreds, we should have written 3, and carried 2, because 23 in the right-hand column, is equal to 2 in the left, and 3 remain; or, 23 hundred is equal to 2 thousand, and 3 hundreds remain.

Art. 136. From the foregoing questions and illustration we derive the following definition and rules.

COMPOUND ADDITION is the adding of numbers of different denominations. By different denominations is meant a different name—as shillings, pence, farthings, etc. Were the numbers given to be added, all pence, or all farthings, there would be but one denomination.

RULE.

I. Write numbers of the same denomination directly under each other, pounds under pounds, shillings under shillings, etc.

II. Begin to add at the right-hand column, observing to carry one for as many in that column as make one in the next left-hand column.

Proof-The same as in addition of simple numbers.

EXAMPLES.

1. Bought 4 books at the following prices, viz., £1 4s. 6d. ; £2 3s. 8d.; £2 19s. 11d.; 2s. 3d. 2qrs. To what did they Ans. £6 10s. 4d. 2qrs.

amount?

2. Add the following numbers: 18s. 5d. 1gr.; £57 17s. 9d. 2qrs.;

£46 26s. 7d. 3qrs.; £49 £102 19s. 10d. 1qr. Ans. £258 28. 8d. 3qrs.

3. Add $286 12 cts. 6 m.; $347 20 cts. 4 m.; $119 18 cts. 7 m.; $542 93 cts. 9 m.; $314 89 cts. 1 m.

Ans. $1610 34 cts. 7 m.

4. Add 45 lbs. 9 oz. 15 pwt. 18 grs.; 90 lbs. 6 oz. 16 pwt. 23 grs.; 30 lbs. 10 oz. 11 pwt. 6 grs.; 85 lbs. 11 oz. 13 pwt. 4 grs.; 91 lbs. 7 oz. 7 pwt. 23 grs.

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