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Rule.--Find a quantity that will compose a mixture by case first : then say as the sum
quan tities thus found, is to the limited quantity, so is each particular quantity found, to the part of that quantity
Examples. 1. A seller of liquors would make a cask of cher ry of 80 gallons : and would compose it of rum a •50 cts. and .90 cts. per gallon, and of water: I demand the quantity that must be taken, allowing the mixture to be worth •60 cts. per gallon.
NOTE,This question admits of but one linking.
of the Mixture
30 gallons of water. 30
30 gallons of N. E. rum. 60+10 = 70 gallons of W. I. rum.
130 sum of all the quantities.
130 : 80 :: 30 : 181907 of N.E. rum
When one of the simples is limited to a certain quantity
*70 cts. per lb.: what quantity of each must be taken, that the mixture may be worth •60 cts. per
168. lbs. lbs.
lbs. Proof. As 60 : 30: ::40 : 20 120x•70=$14.00
60: 30 :: 40 : 20 20X680 16.00
60 : 30 : : 40 : 20 120x690.7 18.00:The limited quantity 40) 40.x.30– 12.00
100 lbs. $60.00 As 100'16. : $6073 1 16.; 60 cts. Ans.
ÊXTRACTION OF THE SQUARE ROOT.
DEFINITION.- Extracting the square root, is the finding of a number, which being multiplied into itself will produce the given number, or is finding the root of a square number.* Roots 3, 4, 5, 6, 7,
9. Squares 1, 4, 9, 16, 25, 36, 49, 64, 81.
RULE. ---Point the given number into periods of two figures each, beginning at the place of units, thus 134567: If the point happen to fall upon the last figure, it must be considered as a full period, thus, 12345 ; secondly, having pointed the number into periods of two figures each : begin at the left hand, and find the greatest square that can be had in that period; place the root thereof in the quotient, and its square under the first period, subtract it therefrom, and to the remainder bring down the next period, and call it the resolvend : double the
* A number is squared when it is multiplied into itself.
quotient or root, and use it for a divisor; divid the resolvend, omitting the right hand figure, att place the answer in the quotient, and also at the right of the last divisor; multiply the divisor br the figure last put on its right,(and in the quotient place the product under the resolvend, and subtract it therefrom, and to the remainder bring down the next period ; double the right hand figure of the last divisor, and use it for a new one; divide the resolvend as before, omitting its right hand figure
; thus continue, until the periods are all brought down, the quotient is the root sought.
Examples. 1. The square root of 37491129 is required.
6X636 greatest sqr.)37491129(6123 root or Ansi
2. What is the square root of 8896451041 ?
Ans. 94321. 3. I demand the square root of 10201.
Ans. 101. 4. The square root of 36481 is required.
Ans. 191. Note.--If a remainder is left after the periods are all brought down, annex periods of cyphers, and continue the operation to any exactness: the root thus found, must be expressed decimally,
Examples. 1. The square root of 234321 is required.
2. The square root of 345678 is required.
Ans. 587.947. 3. The square root of 490023 is required.
Ans. 700.016t 4. The square root of 9432410 is required.
Ans. 3071.22t. 5. The square root of 87643211 is required.
Ans. 9361:71 6. The square root of 67432345 is required.
Ans. 8211•7+. 7. The square root of 4000Q000 is required.
To extract the square root of whole numbers and decimals.
RULE.-Prepare the decimals by annexing cyphers
2. The square root of 12.123 is required.
The decimal prepared 12·1230.
Root 3.487 Ans. 3. The square root of 9•181 is required.
4. The square root of 20-3331 is required.
Ans. 4•509f. 5. The square root of 11•1111 is required.
CASE III. To extract the square root of Vulgar Fractions. ROLE._Reduce compound fractions to simple ones, mixed numbers to improper fractions, and all to a common denominator, and also to its lowest terms: then extract the root of the numerator for a new numerator, and the root of the denominator for a new denominator.
NOTE: If the fraction be a surd, that is such an one whøse root can never exactly be found, reduce it to a decimal, and ex. tract the root.
V Ans. 2. The square root of 4 is required.
3. The square root of one is required. Ans.
SQUARE ROOT APPLIED.
CASE I. To form a square from any number, and to know how
many can be upon a side, RULE. The square root of the number given, will be the number upon the side of the
Examples. 1. It is required to lay out 25600 square rods of land in a square: I demand the side of the square that will contain the land.
✓ 25600 160 rods on a side. 2. A gentleman purchased 3025 tiles, for the purpose of paving a square yard : I demand the number that can he upon a side.
55 upon a side. 3. A certain General commanded an army of 19284.men; and the better to secure his standard,