EXAMPLES.

How much sugar at 9 cents, 11 cents, and 14 cents per lb. will be necessary to form a mixture of 20 lbs. worth 12 cents per lb.?

12

2

2

3+14

8

8

: 4 : : 20: 10 lbs. 14 cents.

2. A grocer has sugar at 24 cents per lb, and at 13 cents per lb.; and he wishes so to mix 2 cwt, of it, that he may sell it at 16 cents per lb.; how much of each kind must he take? Ans. 1621 lbs. of that at 13 cents, and 61, lbs. of that at 24 cents.

3. How many gallons of water must be mixed with wine worth 60 cents per gallon, so as to fill a vessel of 80 gallons, that may be sold at 414 cents per gallon?

Ans. 25 gallons of water, and 55 of wine.

POSITION.

Position is a rule for solving questions, by one or more supposed numbers. It is divided into two parts, namely single and double.

SINGLE POSITION.

Single position teaches to solve questions which require but one supposition.

RULE.

Suppose a number, and proceed with it as if it were the real one, setting down the result-Then, as the result of that operation, is to the number given, so is the supposed number, to the number sought.

EXAMPLES.

1. What number is that, which being multiplied by 7 and the product divided by 6, will give 14 for the quotient? Suppose 18 7

2. What number is that, of which one half exceeds

one third by 15?

Suppose 60-Then | 60 | 60

30 20

Subtract 20

10

Then, as 10 15 : : 60 90 Answer.

3. What number is that, which being increased by 4, and of itself, the sum will be 125?

Ans. 60.

4. A schoolmaster being asked how many scholars he had, answered, that if of his number were multiplied by 7, and 3 of the same number added to the product, the sum would be 292. What was his number?

Ans, 60.

5. A schoolmaster being asked what number of scholars he had, said, if I had as many, half as many, and one fourth as many more, I should have 99. What was number? Ans. 36.

6. A person, after spending and of his money, had $30 left; what had he at first? Ans. $180.

7. Seven eighths of a certain number exceed four fifths by 6. What is that number?

Ans. 80.

8. A certain sum of money is to be divided among 4 persons, in such a manner that the first shall have of it, the second, the third f, and the fourth the remainder, which is $28; what is the sum? Ans. $112.

9. What sum, at 6 per cent. per annum, will amount to £860, in 12 years? Ans. £500.

10. A person having about him a certain num er of crowns, said, if a third, a fourth and a sixth of them were added together, the sum would be 45; how many crowns had he? Ans. 60.

11. What is the age of a person who says, that if of the years he has lived be multiplied by 7, and of them be added to the product, the sum would be 292?

Ans. 60 years. 12. What number is that, which being multiplied by 7, and product divided by 6, the quotient will be 14?

Ans. 12.

DOUBLE POSITION.

Double Position teaches to resolve questions by means of two supposed numbers.

RULE.

Suppose two convenient numbers, and proceed with each according to the condition of the question, and set down the errours of the results. Multiply the errours into their supposed numbers, crosswise; that is, multiply the first supposed number by the last errour, and the last supposed number by the first errour,

If the errours be alike, that is, both too much, or both too little, divide the difference of their products by the difference of the errours-the quotient will be the answer. But if the errours be unlike, that is, one too large and the other too small, divide the sum of the products by the sum of the errours,

EXAMPLES.

1. What number is that, whose part exceeds the {} part by 16?

4

Suppose 24; and as of 24 is 8, and of it is 6, it is evident that the third part exceeds the fourth part by 2 instead of 16; and therefore the errour is 14 too small. Again, suppose 48; and of 48 being 16, and being 183 12, it is manifest that the third part exceeds the fourth by 4, instead of 16; hence the errour is 12 too small.Then, the errours being alike, proceed thus

2 dif. of er.2)384 difference of the products.

192 Answer.

2. A son asking his father

how old he was, received

this answer: Your age is now of mine; but 5 years

ago, your age was } of mine.

What are their ages?

Ans. 20 and 80.

3. Two persons, A and B, have each the same income. A saves of his; but B, by spending 50 dollars per annum more than A, finds himself at the end of 4 years one hundred dollars in debt. What was their income, and what did each spend?

Ans. Their income was $125 per annum for each A spends $100 and B spends $150 per annum.

4. What number, added to the sixty-second part of 7626, will make the sum of 200? Ans. 77.

5. A man being asked how many sheep he had in his drove, said, if I had as many more, half as many more, one fourth as many more, and 121, I should have 40.How many had he?

Ans. 10.

6. An officer had a division, of which consisted of English soldiers, of Irish, of Canadians, and 50 of } Ludians. How many were there in the whole?

Ans. 600.

7. A servant being hired for 30 days, agreed to receive 2s. 6d. for every day he laboured, and to forfeit 1s. for every day he played. At the end of the term his pay amounted to £2. 14s. How many of the days did he labour? Ans. 24. 8. What number is that, which being multiplied by 6, the product increased by adding 18 to it, and the sum divided by 9, the quotient will be 20? Ans. 27,

ARITHMETICAL PROGRESSION. Arithmetical Progression is a series of numbers increasing or decreasing by a common difference; as, 1, 2, 3, 4, 5; 1, 3, 5, 7, 9; 5, 4, 3, 2, 1; 9, 7, 5, 3, 1, &c. The numbers in a series are called terms-the first and last terms are called extremes, and the common difference is the number by which the terms in a series differ from each other, as in 2, 5, 8. 11, &c.—the common difference is 3.

In any series in Arithmetical Progression, the sum of the two extremes is equal to the sum of any two terms, equally distant from them, or equal to double the middle term when there is an uneven number of terms in

the series. Thus, in the series 2, 4, 6, 8 10, 12,-the extremes are 2 and 12, equal to 14, and if you add 10 and 4, or 8 and 6, the result will be the same; and in the series 2, 4, 6, 8, 10, the extremes are 10 and 2, and as the number of terms is uneven 6 is the middle one, which, when doubled makes 12, and the extremes when added together make the same amount.

CASE I.

The first term, common difference, and number of terms, being given, to find the last term and sum of all the||

terms.

RULE.

Multiply the common difference by one less than the number of terms, and to the product add the first term, the sum will be the last. Add the first and last terms together, multiply their sum by the number of terms, and half the product will be the sum of all terms.

EXAMPLES.

1. The first term in a certain series is 3, the common difference 2, and the number of terms 9; to find the last term, and the sum of all the terms.

One less than the number of terme is 8.

2 common difference.

8 number of terms less one.

16 product.

3+ first term.

19 last term.

3+ first term.

22

9X number of terms.

2)198

Answer 99 sum of all the terms.

2. A person sold 80 yards of cloth at 3 cents for the first yard, 6 for the second, and thus increasing 3 cents every yard; what was the whole amount? Ans. $97,20.