4. A merchant sold 10.5 Cwt. of sugar for 108.30 dols. for which he paid 84 dols. 39.12 cents, what did he gain per Cwt. by the sale? Ans. $2.27 7m. + 5. How many pieces of cloth at 20.8 dols per piece are equal in value to 240 pieces at 12.6 dols. per piece? Ans. 145.38+pieces.

6. If when the price of wheat is 74.6 cents per bushel, the penny roll weighs 5.2 oz. what should it be per bushel, when the penny roll weighs 3.5 oz.? Ans. $1.10 8m+

POSITION.

By this rule we are able to discover true numbers by working with supposed ones as though they were real.

Position is of two kinds Single and Double.

Single Position is when it is necessary to make use of only one supposed number, Double Position is when it is necessary to make use of two supposed numbers.

SINGLE POSITION.

Rule.

1. Suppose a number and work with it as though it was the real one, and observe the result.-Then, 2. As the result of that operation,

Is to the supposed number,
So is the number given,

To the number required.

DOUBLE POSITION.

Rule.

1. Suppose a number and work with it as directed in the question, as though it were a real number unyou mention the result which will be the error.

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2. Suppose some other number and proceed in the same way to find a second result or error.

3. Multiply the first result or error by the second supposed number, and the second result or error by the first supposed number.

4. Observe whether the errors are both of the same kind, i. e. both too great or both too little.

5. If the errors are alike divide the difference of the products by the difference of the errors, and the product will be the true number or answer.

But if the errors are one too great and the other too little divide the sum of the products by the sum of the errors and the product will be the true number

How many kinds of Position are there?
When is Single Position used ?

When is Double Position used?

What is first to be done when you commence an operation by Single Position?

After having ascertained the result of the operation how do you proceed?

How do you first proceed when commencing an operation in Double Position?

After having obtained the first error how do you proceed?

When you have obtained the second error what is then to done?

What have you to consider after you have multiplied the second supposition by the first error and the first supposition by the second error?

When you have observed whether the errors are both of the same kind how do you proceed if they are both of the same kind?

But if they are not both of the same kind how do you proceed?

SINGLE POSITION.

Examples.

1. A gentleman having received a number of coins,

POSITION.

says,,, and of the number is 87, what number

of coins was there?

Suppose he had 180

90

36

1260

14859

174

1556

87

0

2. A certain box contains a number of dollars,, I of which is 690, how many was in the box?

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3. The ages of A. B. is older than C. is their separate ages?

Ans. 1200.

B. and C. amount to 133 years, and A is older than B., what Ans. A. 56, B. 42, and C. 35 yr. 4. A person bought 3 pipes of wine for 350 dol. lars, No. 1, cost double the sum that No. 2 did and No. 2, three times the price that No. 3 did, what was the price of each?

Ans. No. 1, $210, No. 2, $105, No. 3, $35. 5. A gentleman being asked his age, replied, if the years of my life were doubled and three-fifths of the product divided by 3, the result would be 14, what was his age? Ans. 35 years.

6. A person lent a sum of money at 53 per cent. compound interest, and at the expiration of 4 years and 8 months he received 4731. 18s. what was the sum lent? Ans. 7501.

7. A cistern has two cocks to supply it with water, by the first it may be filled in 45 minutes and by the second in 55 minutes it has likewise a discharging cock by which it may when full be emptied in 30 minutes, If these three cocks be left open in what time will the cistern be filled? Ans. 2 h. 21 m. 25 sec.

DOUBLE POSITION.

1. Bought cloth for a cloak at 6 dollars per yard,

and baize to line it at 1 dollar, the number of yards was 12 and the cost 42 dollars, how many yards was there of each? Ans. 6 yards of each.

First suppose there were

8 yards of cloth at $6=48
4 yards of baize at

1== 4

2. A. and B. receive the same salary, A. saves one third of his every year, but B. by spending 250 dollars per annum more than A. finds at the expiration of 7 years that he is 350 dollars in debt, what is their income and what does each spend per annum?

Ans.

Their income $600.
A. spends
B. spends

400.

650.

3. A labourer engaged himself for 50 days on condition that for every day he worked he should receive I dollar, but for every day that he was idle he should før

feit 50 cents, at the expiration of the time he received 27 dols. 50 cents how many days did he work, and how many was be idle? Ans. worked 35, idle 15 days.

4. A Farmer having driven his cattle to market received for them all 320 dollars, being paid at the rate of 24 dollars per ox, 16 dollars per cow, and 6 dollars per calf, there were as many oxen as cows, and four times as many calves as cows, how many were there of each? Ans. 5 oxen, 5 cows, 20 calves.

5. There is a pole divided into 3 parts, No. 1, is 15 feet long, No. 3, is as long as N. 1, and half of No. 2, No. 2, is as long as No. 1 and 3 together, what is the length of the pole, and what the length of each part? Ans. Pole, 120 feet, No. 1, 15 feet, No. 2,

60 feet, and N. 3, 45 feet.

6. A father left his property to his three sons, A. B. and C., dividing in the following manner, to A. he gave half wanting 751., to B. one quarter and 1007., to C. the remainder which was 1257. less than B's. share, what was the amount of property, and what was each share separate? Ans. The whole amount 750l., A's. share 300l., B's, share 2871, 10s. C's. share 1621. 10s.

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7. The sum of 1721. 198. 44d. is to be divided among 7 men, 11 women, and 19 boys in such a manner that each woman will have but one third as much as a man and twice as much as a boy, what is the sum of each? Ans. Each man 121. 10s. 1d.+; each woman 4l. 3s. 44d.+, each boy 27. 18. 8d. +

INVOLUTION, OR THE RAISING OF POWERS.

The product arising from any number multiplied by itself any number of times, is called its power, as follows,

2 X 24 the square, or 2d power of 2.
2 X 2 X 2= 8 3d power or cube of 2.
2×2×2×2=16 4th power of 2.