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Alligation Alternate is tho reversc of Alligation Medial, and may be proved by it Questions under this rule admit of as many different answen is there are different ways of linking.

211. When the whole composition is limited to a certain quantity. RULE.–Find the differences by linking' as before; then say, As the sum of the quantities or differences, thus determined : is to the given quantity :: so is each of the differences : to the required quantity of that rate.

100 as 25


QUESTIONS FOR PRACTICE. 6. How much water at 0 cts. 8. A grocer would mix teas per gallon, must be mixed with at 3s., 4s., and 4s. 6d. per 16., brandy at $1.25 per gallon, so and would have 30 lb. of the as to fill a vessel of 80 gallons, mixture worth 3s. 6d. per Ib: and that a gallon of the mix- how much of each must he ture may be worth $1?


lb. 1.25–100

18 at 3s.

Ans. 6 at 4s. 1.25

6 at 4s. 6d. gal. gal. gal. gal. 25 : 16 water.

9. How many

gallons of water worth Os. per gallon,

must be mixed with wine worth Given quantity 80

3s. per gallon, so as to fill a 7. How much silver of 15, cask of 100 gallons, and that of 17, of 18, and 22 carats fine, a gallon of the mixture may must be melted together to be afforded at 28. 6d. ? form a composition of 40 oz.

gall. 20 carats fine ?

16% water. oz.

5 of 15 Ans.

5 of 17
5 of 18

car. fine.
25 of 22

125 : 80 :: { 100 : 64 brandy.

Ans. 831 wine.

212. When one of the simples is limited to a certain quantitij RULE.Find the differences as before; then, As the difference standing against the given quantity : is to the given quantity 1: so are the other differences, severally, : to the several quantities required.




10. A grocer would mix teas 11. How much wine at 58.g. at 123., 10s., and 6s., with 20 at 5s. 6d., and 6s. per gallon, Ib. at 43. per lb.; how much of must be mixed with 8 gallons each sort must he take to at 4s. per gallon, so that the make the composition worth mixture may be wor 58. 4d 8s. per lb.?

per gallon? 4 4 against the given

gala quantity.

2 at 58

2 2 2

Ans. 4 at 5s. 6d. per gall. 4 h,

16 at 6s. 2:10 at 6s. 4:20 :: 2:10 at 10s. Ans.

4:20 at 12s.


6 10 12


cwt. cwt. gr.

MISCELLANEOUS 1. A has 350 yards of cloth 5. A has coffee, which he 2 ts. 4d. per yard, which he barters with B at 10d. per lb. would exchange with B. for more than it cost him, against sagar at 253. 6d. per cwt. ; | tea, which stands B in 10s. the how much sugar will the cloth lb., but puts it at 12s. 6d. : I come to?

would know how much the 350 yards at Is. 4d.-466s. coffee cost at first. 8d.-5600d. and 25s.6d.-306d.

Ans. 35. 4da d. Then 306 : 1 :: 5600

6. A and B. barter; A has. Ib.

150 gallons of brandy, at $1.20 Ans. 18 1 5nearly. per gał. ready money, but in

barter, woul

have $1.40; B 2. A has 71 cwt. of sugar, at has linen at 60 cents per yard, 8d. per lb., for which В gave ready money; how ought the him12cwt. of flour; what linen to be rated in barter, was the four per lb. ?

and how many yards are equal Ans. 44d.

to A's brandy? 3. How much tea, at 9s. 6d. Ans. barter price, 70 cents, per lb,, must be given in bar- and B. must give A 300 yards. ter for 156 gallons of wine, at

7. C has tea at 78 cents per 12s. 31d. per gallon ? Ans. 2011b. 131 oz.

Ib., ready money, but in barter,

would have 93 cents ; D has 4. B delivered 3 hhds, of shoes at 7s. 6d. per pair, ready brandy, at 6s. 8d. per gallon, to money; how ought they to be C for 126 yards of cloth ; what | rated in barter, in exchange was the cloth per yard ? for tea?

Ans. $1.49 Ans. 105.


8. C. has candles at 6s. per ture for $48; A puts in 80 dozen, ready money; but in sheep for 4 months, B 60 sheep barter he will have 6s. 6d. per for 2 months, and C 72 sheep dozen ;

D has cotton at 9d. for 5 months; what share of per lb. ready money; what the rent ought each to pay ? price must the cotton be at in

A $19.20 barter, and how much cotton

B 7.20 Ans. must be bartered for 100 dozen

C 21.60 of candles ?

12. If I have a mass of pure Ans. the cotton 9 d. per in barter, and 7cwt. Ogrs. 1616. gold, a mass of pure copper, of cotton must be given for of gold and copper, each weigh

and a mass, which is a mixture 100 doz, candles.

ing 10 lb., and by immersing Note.—The exchange of one commodity for another, is called them in water, find the quantiBurter.

ties displaced by each to be 8 9. If 6 mien build a wall 20 by the copper, 7 by the mixfeet long, 6 feet high, and 4 feet ture, and 5 by the gold; what thick, in 32 days; in what time part of the mixture is gold, and will 12 men build a wall 100 what part copper? feet long, 4 feet high, and 3

82 7

And feet thick? Ans. 40 days. 10. If a family of 8 persons

52 : 68 copper

3 : 10 :: in 24 months spend $480; how

1:33 gold. much would they spend in 8 T\is is the celebrated problem of months, if their number were Archimedes, hy which he detected doubled? Ans. $320.

the fraud of the artist employed by

Hiero, king of Syracuse, to make 11. Three men hire a pas- him a crown of pure gold (211).



1. Supposing the Legislature 2. A certain school, consistelould grant a tax of $35000 | ing of 60 scholars, is supportto be assessed on the inventory ed on the polls of the scholars, of all the rateable property in and the quarterly expense of the State, which amounts to the whole school is $75; what $3000000, what part of it is that on the scholar, and must a town pay, the inventory what does A pay per quarter, of which is $24600?

who has 3 scholars? $ inv. * tax, .

Ans. $1.25 on the scholar, 3000000 : 35000 :: 24600 : 287 and A pays $3.75 per quarter.


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3. If a town, the inventory that on the dollar, and what is of which is $24600, pay $287, C's tax, whose property invenwhat will A's tax be, the in- | tories at $76.44? ventory of whose estate is $4325 : 86.50 ::1:.02 cta. $525.75

Ans. 24600.00 : 287 ::: 525.75 : & 76.44X.02—$1.528, C's tax.

$6.133 Ans. 5. If a town, the inventory 4. The inventory of a cer- of which is $16136, pay a tax tain school district is $1325, of $493.08, what is that on the and the sum to be raised on dollar? this inventory for the support $16436 : $493.08 :: 1:.03 cts, of schools, is $86.50; what is

Ans. 213. In assessing taxes, it is generally þest, first to find what each dollar pays, and the product of each man's inventory, multiplied by this sum, will be the amount of his tax. In this case, the sum on the dollar, which is to be employed as a multiplier, must be expressed as a proper decimal of a dollar, and the product must be „pointed according to the rule for the multiplication of decimals (122); thus 2 cents must be written ..02, 3 cents, .03, 4 cents, .04, &c. It is sometimes the practice to make a table by multiplying the value on the dollar by 1, 2, 3, 4, &c. as follows:

TAL E. $1 pays .03


$100 pays 3.00




400 12.00


500 15.00


600 18.00

70 41

700 21.00
80 2.41

800 24.00
90 2,70

900 27.00

1000 30.00 This table is constructed on the supposition that the tax amounts to thiee cents on the dollar, as in example 5th. „Use.- What is B's lax, wbose rateable property is $276? By the table, it appears that $200 pay $6, that $70 pay $2.10, and that $6 pay 18 cents. Thus $200 is 6.00

Proceed in the same way to find each indi70 is 2.10

vidual's tax, then add all ihe taxes together, 6 is 0.18

and if their amount agree with the whole sum

proposed to be raised, the work is right. It is 276 88,28

sometimes best to assess the tax a trifle larger B's tax.

than the amount to be raised, to compensate for the loss of fractions.

$10 pays

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REVIEW, 1. What is meant hy ratio ? How four terms of a proportion 1 How is ratio expressed? What is the first is this truth shown term called ! the second term?

3. Does changing the place a 2. What is proportion? What the two middle terms affect the progeneral truth is stated respecting the portion ? Why not?

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ternate ?

What is meant by inverse pro- 1. What is Fellowship

What is portion ?

meant ? What 5. What is meant by the Single ny dividend ? What is the rule Rule of Three ? Whai is the geir- , when the times are equal ? What, eral rule for stating questions in the when they are unequal? What is swer then found ? If the first and

8. What is Alligation? What is third terins be of different denomi- | Alligation Medial?.-Alligation Alnations, what is to be done? What,

What is the rule for if there are different denominations finding the proportional quantities in the second term? Of what de- to form a inixture of a given rate ? nomination will the quotient be? | Explain by analysis of an example. What, if the quotient be not of the When the whole composition is same denomination of the required limited to a certain quantity, how answer? What is the method of

would you proceed ? How, when proof in this rule ?

one of the simples is limited to a 6. What is compound proportion ? certain quantity? How is Alligation By what other name is it called ? proved ? What is the rule for stating questions

9. What is Barter? What is in compound proportion ?--for per- nieant by a tax? What is the comforming the operation ?

mon method of making out laxes ?


214. 1. Fractions are parts of a unit, or of a whole of any kind.

If any number, or particular thing, be divided into two equal parts, thoso parts are called halvés ; if into 3 equal parts, they are called thirds ; if inte 4 equal parts, they are called fourths, or quarters (11); and, generally, tho parts are named from the number of parts into which the thing, or whole, is divided. If any thing be divided into 5 equal parts, the parts are called

fifths ; if into 6, they are called sixths ; if into 7, they are called sevenths; and so on. These broken,'or divided quantities are called fructions. Now if an apple be divided into five equal parts, ihe value of one of those parts would be one fifth of the apple, and the value of two parts two fifths of the opple, and so on. Thus we see that the name of the fraction shows, at the same time, the number of parts into which the thing, or whole, is divided, and how many of those parts are taken, or siguitied by the fraction. Sup. pose I wished to give a person two fifths of a dollar; I must first divide the dollar into five equal parts, and then give the person two of these parts. A dollar is 100 cents-100 cents divided in:10 5 equal parts, each of those parts would be 20 cents. Hence, one fifth of 100 cents, or of a dollar, is 20 cents, and two fifths, twice 20, or 40 cents.

The tediousness and inconvenience of writing fractions in worils has led to the invention of an abridged method of expressing them by figures. One huif is written 1, one third, }, two thirds, 3, &c. The figure below the kuo shows the number of parts into which the thing, or whole, is divided, and the figure above the line shows how many of those parts are signified by thc fraction. The number below the line gives name to the fraction, and is therefore called the denominator ; thus, if the number below the line be 3, the parts signified are thirds, if 4, fourths, if 5, fifths, and so on. The waarbor wiitten above the line is called the numerator, because it enuino

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