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REDUCTION OF INTEGERS.
Def. 1. Concrete quantities are measured, or estimated, by considering how often they contain certain fixed quantities of the same kind, which are called units of magnitude.
Def. 2. The unit being represented numerically by 1, other quantities of the same kind are represented by the number, which shews how often they contain the unit, which must be specified in every case.
Def. 3. Since all quantities do not contain the unit an exact number of times, and are not therefore expressible by whole numbers, several units of the same kind are used. The process of converting the numbers, expressing a given quantity in terms of one or more units, into others expressing it in other units, is called Reduction.
Def. 4. When a quantity is made up of several others, or expressed by two or more numbers with different units, it is called a compound quantity.
I. TO CHANGE NUMBERS FROM A HIGHER DENOMINATION TO A LOWER.
Rule. Multiply the number of the highest denomination by the number of the next lower denomination, which make up one of the higher; and if the quantity be compound, add to the product the number of the next lower denomination. In the same manner change this number to the next lower denomination, and so on, till the denomination required is arrived at.
2. Reduce 411 guis. 18s. 2d. to two-pences and farthings.
guis. $. d.
TO CHANGE NUMBERS FROM A LOWER DENOMINATION TO A HIGHER.
Rule. Divide by the number of the lower denomination which make up one of the next higher; the quotient is the number of the higher, and the remainder, if any, is of the lower denomination. In the same way change to the next bigher, and so on, till the highest required is arrived at.
EXAMPLES. 1. Change 15302 farthings to the higher denominations.
2. Change 41073 three-pences to moidores.
(9)102685. — 1 thrp. 27
3)1140 — 8s.
3. Change 2345678 drams to tons.
Rem. = 14 dr.
Rem. = 12 oz.
Rem. = 6 lbs
III. TO CHANGE NUMBERS FROM ONE DENOMINATION TO ANOTHER,
WHEN NO EXACT NUMBER OF THE ONE IS CONTAINED IN ONE OF THE OTHER. Or, TO EXPRESS ONE QUANTITY IN TERMS OF ANOTHER.
Rule. Reduce both quantities to the same denomination, the highest possible, and divide the first by the second.
EXAMPLES. 1. Change 416 moidores into guineas. 416 moidores
1 guinea = 7 three-sh.-pieces. 9
3744 — three-sh.-pieces. 3744 - 7 = 5349.
Ans. 534 guis. 18 shill.
2. In 3804 crowns, how many pieces each 78. 6d. ?
3804 crs. = 7608 — f-crs. : 78. 6d. = 3 - h-crs.
£1 = 120 — two-pences. 7284 f. = 7284 X 19 two-pences. 7284 x 19 607 X 19 11533
= 1153 -
10 Ans. £11531o. 4. How many times will a wheel 164ft. in circumference turn round in a distance of 24 mi. 3 fur. 25 po. ?
5. A grocer wishes to weigh up 8cwt. 2qrs. 201bs. of sugar into an equal number of parcels of 4 lbs. 6 lbs. 8 lbs. 10 lbs. 12 lbs. and 14 lbs. each, what number will he have of each, and how many altogether? Weight of 1 parcel of each kind = (4 +6+8+10 +12 + 14) lbs.
TO EXPRESS IN LOWER TERMS THE VALUE OF A FRACTION OF A
-- cr. =--
Rule. Multiply the fraction by the number of the next lower denomination, which make up one of the higher ; reduce the result to a whole or mixed number. Proceed in the same manner with the fractional part, if any, till the lowest denomination required is arrived at. If the fraction be a recurring decimal, it must be reduced to a vulgar fraction.
8. = 8.2-8.
7 d. = 11 d.
.: cr. = 2 s. 11 d. 12 12
12 2. Express in lower terms i of a cubic yard. 9 9 X 27 243
cub. ft. =- cub. ft. = 22— cub. ft.
Ans. £3: 11s. : 4:20.