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The greater is found by adding half the excess to half the number to be divided; or by taking half the sum of the number to be divided and the excess.

In the above example,

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Let the learner generalize this question by making x = the greater part. The same results will be obtained.

This is a general rule, and will apply to all questions like it, and should be remembered, for it is frequently useful.

Let the learner find the answers to the 2d, 3d, and 7th examples of Art. III. by this rule. That is, by putting the numbers of those examples in the place of a and b in the formulas. It is easy to see the propriety of the rule. For the formula -b 55. 13 42 shows, that if the $13 that A pays

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or

2

2

2'

more than B, be taken out, the remainder is to be paid in equal

parts by them. Also the formula a+b or

2

55 +13 68

2

=

2

shows, that if B were to pay $13 more, he would pay as much as A, and the rent would be paid în equal parts by them.

7. A father, who has three sons (Art. III. exam. 4), leaves them 16000 crowns. The will specifies, that the eldest shall have 2000 crowns more than the second, and that the second shall have 1000 crowns more than the third. What is the share of each?

Let a represent the whole number of crowns, b what the eldest son's share exceeds that of the second, and c what the share of the second son exceeds that of the third.

This question may be expressed in general terms, thus: To divide a given number a, into three such parts, that the great

est may exceed the mean by a given number b, and the mean may exceed the least by a given number c

Let

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x= the greatest.

Ꮖ -b the mean.

By the conditions,

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α 26 с

b, or b subtracted from + + thus;

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3

3 3'

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The mean is x

c, or c, subtracted from

C,

or

x — b —

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α

b

+

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=

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3

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The eldest son's share, by the first formula, is

16000 + 2 × 2000+ 1000 = 7000 crowns.

3

The other shares may be found by the other two formulas. Let the learner solve this question by making x equal to the less part, and also by making it equal to the mean.

Exam. 5th, Art. III. may be solved by this formula. Let the learner generalize the questions in Art. III. as far as to Exam. 16th.

The examples in Art. I. may be generalized still farther.

8. A man bought corn at 4s. (a) per bushel, rye at 6s. (b) per bushel, and wheat at 8s. (c) per bushel : there was an equal quantity of each sort. The whole came to 90s. (d). How many bushels were there of each ?

It will readily be perceived that it is impossible actually to perform the operations of addition, subtraction, &c. on letters; but it is easy to represent these operations. We however frequently speak of adding, subtracting, multiplying, and dividing algebraic quantities, by which we mean, representing these operations. We have seen that to express 3 times x or 3 times a we write 3x, 3 a, that is, x or a multiplied by 3. In the same manner, if we wish to express a times x, that is, a multiplied by a, we write a x; and if we wish farther to express that a x (that is, a times x) is to be multiplied by b, we write a b x. x the number of bushels of each.

*Let Then

And

ax

the price of the corn. bx= the price of the rye. cx= the price of the wheat. a x + bx + c x = d.

Here x is taken a times, and b times, and c times, that is, (a+b+c) times. This may be expressed thus, (a + b + c) x,

* Let the learner perform this example first by the numbers.

enclosing the three coefficients connected by their signs in a parenthesis.

This will be plain if we put it in numbers. 4x+6x+8x is the same as (4 + 6 +8) x, that is, 18 x.

If we had

we should divide by 18,

(a+b+c) x=d
18x d

In the same manner divide by (a+b+c),

X=

d

18

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This general formula is expressed in words as follows: Divide the price of the whole by the price of a bushel of each sort added together, and it will give the number of bushels of each sort.

9. A father dying left $25000 (or a) to be divided between his wife, son, and daughter; his son was to have 3 (or b) times as much as the daughter, and the wife 2 (or c) times as much as the son. What was the share of each?

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In this example observe that x is taken 1 time, and b times, and bc times. When a letter is written without a coefficient, it is always understood to have 1 for its coefficient; thus x is the same as 1 x.

Having found the share of the daughter, it is easy to find the shares of the other two.

The son's share is 3x=7500, or b x =

The wife's do. is 6 x 15000, or b c x=

ab
1+b+bc
a b c

1+b+b c

The learner may now generalize some of the examples in Art. I. in this manner.

10. A gentleman, distributing some money among some beggars, found, that in order to give them 8 (or a) cents apiece, he should want 5 (or b) cents; he therefore gave them 7 (or c) cents, and he had 4 (or d) cents left. How many beggars were there ?

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11. There is a cistern which is supplied by two pipes; the first will fill it alone in 7 (or a) hours, the second will fill it alone in 5 (or b) hours. In what time will it be filled if both run together?

Let the number of hours in which both together will fill it.

1

The first will fill 4 or ! of it in one hour, and the second will

1

α

fill or of it in one hour; both together will fill ÷ + f or b

1

+ of it in one hour. In & hours they will fill x times as much, that is,

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