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CASE 11.

.
When the mixture is limited to a certain quantity.

Rule.--Find a quantity that will compose a mixture by case first : then say as the sum

of the

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.

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00

The price

of the Mixture

cts.
.60

30

30 gallons of water. 30

30 gallons of N. E. rum. 60+10 = 70 gallons of W. I. rum.

90

130 sum of all the quantities.
Galls, galls. galls.
As 130 : 80 : : 30 • 18-19 of water

130 : 80 :: 30 : 181907 of N.E. rum
130 : 80 : 70: 43737 of W.I. rum.

Ans

CASE III.

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When one of the simples is limited to a certain quantity

.
RULE.~-Find the quantities that will make a mix-
ture by case first, (the simple that is limited must
be linked) then say, as the quantity that stands a-
gainst the price of the limited quantity, is to the
other quantities taken separately, so is the limited
quantity to the quantity of each.

Examples
1. A grocer has 40lbs. of tea worth •30 cts. per

Ib.
which he would mix yith other teas at :90, 80, and

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*70 cts. per lb.: what quantity of each must be taken, that the mixture may be worth •60 cts. per

1b.?

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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

Answers.

100 lbs. $60.00 As 100'16. : $6073 1 16.; 60 cts. Ans.

1, 2,

8,

Rule--Point

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

36

121)149

121

1222)2811

2444

12243)36729

36729

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.

Ans. 484.067

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.

Ans. 632457

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CASE II.

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To extract the square root of whole numbers and decimals.

RULE.-Prepare the decimals by annexing cyphers
{if occasion require) so that a dot may fall on units
place of the whole numbers, then proceed as in
case first.

Examples.
1. The square root of 1.4884 is required.
1.4884(1-22 root

Ans.

1

22) 48

44

242)484

484

2. The square root of 12.123 is required.

The decimal prepared 12·1230.

Root 3.487 Ans. 3. The square root of 9181 is required.

Ans. 3.037

4. The square root of 20-3331 is required.

Ans. 4•509f. 5. The square root of 111111 is required.

Ans. 3.33331.

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.

Examples.
1. The square root of this required.

zint=
675

V Ans. 2. The square root of 4 is required.

2

81

486 T26

:9 Ans.

TI

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

square.

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.

V 3025

55 upon a side. 3. A certain General commanded an army of 19284.men; and the better to secure his standard,

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