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Examples of putting Questions into Equations.

the same manner, B next divided with A and C, and after this, C with A and B. If, then, by these means, the intended equal division is effected, how much booty did each soldier make?

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4x-4y-4x=6y-2x-2z

4x-4y-4x=7x-x-y.

11. A certain number consists of three digits, of which the digit occupying the place of tens is half the sum of the other two. If this number be divided by the sum of its digits, the quotient is 48; but if 198 be subtracted from it, then we obtain for the remainder a number consisting of the same digits, but in an inverted order. What number

is this?

Ans. If x = the digit which is in the place of units,
that in the place of tens,

y =

z = that in the place of hundreds.

The number is 100% +10 y+x,

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100% +10 y+x-198-100 x + 10 y + z.

12. A person goes to a tavern with a certain sum of money in his pocket, where he spends 2 shillings; he then borrows as much money as he had left, and going to another tavern, he there spends 2 shillings also; then borrowing again as much money as was left, he went to a third tavern,

Examples of putting Questions into Equations.

where likewise he spent 2 shillings, and borrowed as much as he had left; and again spending 2 shillings at a fourth tavern, he then had nothing remaining. What had he at first?

Ans. If x == the shillings he had at first,

the required equation is

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13. A person possessed a certain capital, which he placed out a certain interest. Another person, who possessed $10 000 more than the first, and who put out his capital 1 per cent. more advantageously than the first did, had an income greater by $ 800. A third person, who possessed $15 000 more than the first, and who put out his capital 2 per cent. more advantageously than the first, had an income greater by $1500. Required the capitals of the three persons, and the three rates of interest.

Ans. If x= the capital of the first,

=

y his rate of interest per cent.

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14. A person has three kinds of goods, which together cost $230254. The pound of each article costs as many twenty-fourths of a dollar as there are pounds of that article; but he has one third more of the second kind than he has of the first, and 3 times as much of the third as he has of the second. How many pounds has he of each article? Ans. If x = the number of pounds of the first,

the required equation is

2x2 + z2x2 + f f x2 = 230

Examples of putting Questions into Equations.

15. A person buys some pieces of cloth, at equal prices, for $60. Had he got 3 pieces more for the same sum, each piece would have cost him $1 less. How many pieces did he buy?

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16. Two drapers A and B cut, each of them, a certain number of yards from a piece of cloth; A however 3 yards less than B, and jointly receive for them $35. "At my own price," said A to B, "I should have received $24 for your cloth." "I must admit," answered the other, "that, at my low price, I should have received for your cloth no more than $124." How many yards did each sell?

Solution. Let x = the number of yards sold by A;

then

x+3= the number sold by B.

Now since A would have sold x + 3 yards for $24,

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and since B would have sold x yards for $124,

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the sum for which B sells + 3 yards = 25(x+3),

and the required equation is

24 x x+3

x

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2 x

Examples of putting Questions into Equations.

17. Two travellers, A and B, set out at the same time from two different places, C and D; A, from C to D; and B, from D to C. When they met, it appeared that A had already gone 30 miles more than B; and, according to the rate at which they are travelling, A calculates that he can reach the place D in 4 days, and that B can arrive at the place C in 9 days. What is the distance between C and D? Ans. If, when they meet,

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18. Some merchants jointly form a certain capital, in such a way that each contributes 10 times as many dollars as they are in number; they trade with this capital, and gain as many dollars per cent. as exceed their number by 8. Their profit amounts to $288. How many were there of them?

Ans. If x the number of merchants, the required equation is

To x2 (x+8)

= 288.

19. Part of the property of a merchant is invested at such a rate of compound interest, that it doubles in a number of years equal to twice the rate per cent.

f interest?

Ans.

What is the rate

If x = the rate per cent., the required equation is

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Degree of an Equation.

SECTION II.

Reduction and Classification of Equations.

104. The portions of an equation, which are separated by the sign, are called its members; the one at the left of the sign being called its first member, and the other its second member.

105. Equations are divided into classes according to the form in which the unknown quantities are contained in them. But before deciding to which class an equation belongs, it should be freed from fractions, from negative exponents, and from the radical signs which affect its unknown quantities; its members should, if possible, be reduced to a series. of monomials, and the polynomials thus obtained should be reduced to their simplest forms.

106. When the equation is thus reduced, it is said to be of the same degree as the number of dimensions of the unknown quantities in that term which contains the greater number of dimensions of the unknown quantities.

Thus, x and y being the unknown quantities, the equations

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