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4. A and B lay out equal shares in trade; A gains $126, and B loses $87, then A's money is double that of B; what did each lay out? Ans. $300.

5. A and B have both the same income; A saves of his yearly, but B, by spending $50 per annum more than A, at the end of 4 years, finds himself $100 in debt; what is their income, and what do they spend per annum? Ans. $125 their inc. per ann. A spends $100 per ann. B spends $150 S

QUESTIONS.

2. Of how many kinds is it? 3. What does Single Position teach?

4. What is the rule?

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5 What does Double Position teach? 6. What is the rule?

5. Permutation of Quantities.

Permutation of Quantities is a rule which shows how many different ways the order or position of any given number of things may be varied.

Problem I.

To find the number of permutations, or changes, that can be made of any given number of things, all different from each other.

RULE.*-Multiply all the terms of the natural series of numbers from 1 up to the given number, continually together, and the last product will be the answer required.

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To find how many changes can be made out of a given number of different things, by taking any given number at a time.

RULE. Take a series of numbers, beginning at the number of things given, and decreasing by 1 till the number of terms taken be equal to the number of things to be taken at a time, and the product of all these terms will be the answer.

Examples.

1. How many changes can be rung on 3 bells out of 8?

8X7X6=336 Ans.

2. How many words can be made with 5 letters of the alphabet, supposing 24 letters in all, and that a number of consonants alone will make a word? Ans. 5100480.

QUESTIONS.

1. What is Permutation of Quanti- | changes that can be made of any ties? number of things all different from

2. How do you find the number of each other?

The reason of the rule may be shown thus: any thing, a, is capable of only one position, as a. Any two things, a and b, are capable of only two variations, as ab, ba; when a number is expressed by 1×2. If there be three things, a, b, and c, then any two of these, leaving out the third, will have 1×2 variations; and consequently, when the third is taken in, these will be 1×2×3 variations; and so on as far as you please.

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The area of any figure is the space contained within the bounds of its surface, without any regard to thickness, and is estimated by the number of squares contained in the same; the side of those squares being either an inch, a foot, a yard, a rod, &c. Hence the area is said to be so many square inches, square feet, square yards, or square rods, &c.

Problem I.

To find the area of a parallelogram, whether it be a square, a rectangle, a rhombus, or a rhomboid.

RULE.-Multiply the length by the breadth, or perpendicular height, and the product will be the area.

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2. What is the area of a rectangular piece of ground 56 rods

tangle whose length is 9, and breadth 4 feet? Ans. 36 feet.

long, and 26 wide?

5626-160-9

Problem II.

To find the area of a triangle.

acres, Ans.

RULE 1.-Multiply the base by half the perpendicular height, and the product will be the area.

RULE 2.-If the three sides only are given, add these together, and take half the sum; from the half sum subtract each side separately; multiply the half sum and the three remainders continually together, and the square root of the last product will be the area of the triangle.

Examples.

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Problem III.

To find the area of a trapezoid.

RULE.-Multiply half the sum of the two parallel sides by the perpendicular distance between them, and the product will be the

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To find the area of a trapezium, or an irregular polygon.

RULE. Divide it into triangles, and then find the area of these. triangles by Problem 11. and add them together.

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