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In the preceding problem, let 30 = a, and 120 = b; then we have for an equation,

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Or if we substitute the values of a and b ;

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Let x be the greater number, and we shall have the equation,

x2 - (a-x)2 = b,

2

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and restoring the values of a and b,
30 120
+
= 15+2= 17.
60

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3. A gentleman meeting four poor persons, distributed five shillings among them in the following manner:-to the second he gave twice, to the third, three times, and to the fourth, four times, as much as to the first. What did he give to each ?

then

and

Let x = the pence he gave to the first;
2x = the pence he gave to the second,
3x = the pence he gave to the third,
4x = the pence he gave to the fourth.

But the sum of the whole is five shillings, or sixty pence;

therefore, x + 2x + 3x + 4x = 60 ;

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Wherefore he gave 6, 12, 18, and 24 pence to them respectively.

4. It is required to divide the number 99 into five such parts, that the first may exceed the second by 3, be less than the third by 10, greater than the fourth by 9, and less than the fifth by 16.

Let x = the first part;

then x-3= the second,
x + 10 = the third,
x-9 = the fourth,

x + 16 = the fifth.

The sum of all the parts being 99, we have, xx-3+x+ 10 + x – 9 + x + 16 = 99.

Reducing

5x+14 = 99;

transposing, 5x = 99 - 14 = 85 ;

whence,

x = 17.

The parts are, therefore, 17, 14, 27, 8 and 33.

5. A courier, who travels 60 miles a day, had been despatched 5 days, when a second was sent to overtake him, in order to which he must travel-75 miles a day; in what time will he overtake him?

Let x = the number of days the second courier travels; then, since the first had been despatched 5 days before, he must travel x + 5 days. But they travel the same distance; hence, since the first travels 60 miles per day, and the second 75, we shall have the equation,

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Transposing, and changing the signs, we have,

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6. A merchant sells two pieces of cloth, the lengths of which are as 3 to 4. The shorter piece he sells for 13 cents more per yard than the longer, and the sum he receives for it, is to the sum he receives for the longer, as 7 to 8. At what price per yard were they sold?

Let x = the price per yard of the shorter ;

then, x-13 is the price of the other.

Then, since their lengths are as 3 to 4, we shall have, the sum received for the first, to the sum received for the second, as, 3 × x, is to 4 × (x-13).

But these sums are in the ratio of 7 to 8; therefore,

3x : 4(x - 13) = 7 : 8.

And by multiplying means and extremes, we have (Arith. 65),

24x = 28(x - 13) = 28x

Transposing and changing the signs,

4x = 364,
x = 91.

364.

91, and 91-13 = 78, are the prices respectively.

7. A merchant commences business with a certain capital, and borrows as much more. At the end of each year he finds that his expenses have exceeded his income by $1000, and borrows as much as he has remaining. Four years having passed in this way, his whole capital is exhausted. What capital had he when he began ?

Let x = the capital with which he commenced. Then, since he borrows as much as he has, and loses $1000 the first year, he will have at the end of the year 2x 1000.

In like manner, doubling this sum and subtracting $1000, he will have, at the end of the second year, 4x—2000—1000; at the end of the third, 8x-4000-2000—1000; and at the end of the fourth, 16x-8000-4000-2000—1000; but his capital being now exhausted, this last sum is equal to nothing; we have, therefore,

16x-8000-4000—2000—1000 = 0.

Transposing and reducing, 16x = 15000,

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8. Two workmen, A and B, are employed together for 50 days, at 60 cents per day each. A spent 6 cents less per day than B, and at the end of 50 days, he found that he had saved twice as much as B, and the expense of two days over. What did each spend per day?

Let x = what A spent per day; then 60-x is what he saves per day; and since B spends 6 cents more, he saves but 60 x - 6, or 54 - x. By the conditions of the question, what A saves in 50 days, minus the expense of two days, is equal to twice what B saves in 50 days; therefore,

or,

50(60-x) - 2x = 2 × 50(54 — х),
50x - 2x = 5400 — 100x,

3000

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A therefore spent 50 cents per day, and B 56.

9. A bookseller sells two books, one containing 100 sheets for 10 shillings, and the other, containing 50 sheets for 6 shillings, each being bound at the same price. What was that price?

Let x = the price;

then, 10-x:6— x = 100 : 50.

Multiplying the means and extremes of this proportion, we have (Arith. 65),

by transposition,

500-50x = 600-100x:

50x = 100,

x = 2s, the price of binding.

10. A man wishing to enclose a piece of ground with palisadoes, finds, that if he sets them a foot asunder, he will have too few by 150, but if he sets them a yard asunder, he will have too many by 70. How many has he?

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Comparing the difference of consequents and antecedents, with

the antecedents (Arith. 69), we have,

X 70: 220 = 1 : 2.

Dividing consequents by 2 (Arith. 68),

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11. A market woman bought a certain number of eggs at two for a penny, and as many at three for a penny, and sold them out at five for two pence, and lost four pence by so doing. How many did she buy and sell ?

Let x = the number of eggs of each price;

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which, multiplied by 2x, the whole number of eggs, gives for

2

4x

5'

the amount×2x, or which is, by the conditions of

the question, four pence less than the cost; we have, therefore,

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12. There is a certain number, consisting of 2 digits, the sum of these digits is 6, and if 18 be added to the number itself, the digits will be inverted; what is the number?

Let x = the left hand digit.

Then, 6-- x = the other; and since each figure in a number is ten times greater when removed one place to the left, the number itself will be

10x + 6-x, and 10(6-x) +x is the number

with its digits inverted;, we shall then have,

10x + 6 - x + 18 = 10(6 – x) + x;

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and 6-x, the other digit = 4.

Hence the number is 24, and this result is verified by adding 18 to 24, which gives 42, the digits inverted.

13. A merchant drew every year upon the stock he had in trade, the sum of a dollars for the expense of his family. His profits each year were the nth part of what remained after this reduction; but at the end of three years, he finds that his whole stock is exhausted; how much had he at the beginning?

Let x = the stock in trade;

then

x - a = what remains after a dollars is deducted;

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a

and

n

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is the profits of the first year; adding this to

a, the sum will be the stock at the end of the first year;

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