melons, for $2.64, selling a peach at twice the price of an apple, and a melon at 7 times the price of a peach. At what price did he sell one of each kind?

11. A man bought 7 bushels of potatoes, 5 of corn, and 9 of wheat, for £7 19 s., giving twice as much a bushel for corn as for potatoes, and twice as much a bushel for wheat as for corn. How many shillings did he give a

bushel for each?

12. A and B traded in company, and gained $240. B put in twice as much stock as A. What is each man's share of the gain?

Remark. It is evident that, since B furnished twice as much stock as A, he should have twice as much of the gain.

13. A gentleman distributed 20 shillings among 3 beggars. To the first he gave twice as much as to the second, and to the second 3 times as much as to the third. How many shillings did he give to each?

14. A man bequeathed an estate of $16,000 to his two sons and three daughters, directing that the daughters should all share alike, that the younger son should have twice as much as one daughter, and that the elder son should have as much as all the daughters. Required the share of each.

15. Three men, A, B, and C, built 670 rods of fence. A built 7, B 5, and C 4 rods a day. A wrought 3 times as many days as B, and B wrought 5 times as many days as C. Required the number of days each wrought.

16. A draper bought 16 pieces of cloth; 3 were white, 4 black, and 9 blue. A piece of black cost twice, and a piece of blue 3 times, as much as a piece of white. Required the price of a piece of each, the cost of the whole being $152.

EQUATIONS

SECTION II.

OF THE FIRST DEGREE,

HAVING

UNKNOWN

TERMS IN ONE MEMBER ONLY, AND KNOWN TERMS IN BOTH.

ART. 16. 1. The sum of the ages of A and B is 50 years, and B is 10 years older than A. Required the age of each.

Let x represent A's age in years; then x +10 will represent B's age. Hence,

x+x+10=50. Reducing the first member,
2x+10=50.

Since the two members of the equation are equal, we may now subtract 10 from each member, and the remainders will be equal, (Ax. 2.) First representing this subtraction, we have

2x+10-10=50-10. Now reducing, that is,

performing the subtraction represented,

2x=40; from which

x=20 years, A's age;

and x+10=20+10=30 years, B's age; } Ans

2. A farmer had 6 more than twice as many sheep as cows, and the number of his sheep and cows together was 66. Required the number of each.

Let x represent the number of cows;

then 2x+6 must be the number of sheep. Hence, x+2x+6=66. Reduce the first member, 3x+6=66; subtract 6 from each member, 3x+6—666—6; reduce both members, 3x=60, and

Remark. In the two preceding questions, we see that, after reducing the first equation, we subtracted from both members the known term, which was in the same member with the unknown quantity. Let the learner solve the following problems in a similar manner, first representing the subtraction, and afterwards reducing.

3. A and B hired a pasture for $75, of which A paid $15 more than B. How much did each pay?

4. A man and his son could earn together, in one day, $2.50, but the man earned 5 s. more than his son. How much could each earn?

5. A laborer wrought 10 days, having the assistance of his boy 4 days, and received for the wages of both $12 but the man earned, in a day, $0.50 more than the boy. Required the daily wages of each.

6. A is 2 years older than B, and B is 3 years older than C. Required the age of each, the sum of their ages being 68 years.

7. A grocer gave $53 for 5 barrels of flour and 4 barrels of rice, paying $2 a barrel more for the rice than for the flour. Required the price of a barrel of each.

8. Divide $33 between A, B, and C, so that B shall have $7 more than A, and C $9 more than B.

9. In a certain town there are 10 more Irish than English, and 30 more French than Irish. Required the number of each class, there being 710 persons in all.

10. A man paid £3 4 s. for 4 bushels of corn and 5 bushels of wheat, giving 2 s. more a bushel for wheat than for corn. Required the price of a bushel of each.

If, in solving the equations arising from the preceding questions, the learner had, after representing the subtraction in both members, reduced only that member in which the unknown quantity was found, he would have perceived that the resulting equation might have been obtained by carrying the known term, which was on the same side as

the unknown quantity, to the other member, and changing its sign from to. Thus, in the second question, we had the equation 3x+6= 66. If we represent the subtraction of 6 from each member, the equation becomes 3x+6-6-66-6. Reducing the first member of this, we have 3x=66-6, which might have been obtained from 3x+6=66, merely by transferring 6 from the first member to the second, and changing its sign from + to -.

In like manner, if we had the equation 3x=x+20, by representing the subtraction of x from each member, and then reducing the second member of the result, we should have 3 x — x = 20, which might have been obtained from 3x=x+20, by transferring z, supposed to have the sign, from the second member to the first, and changing its sign from + to.

ART. 17. Removing a term from one member of an equation to the other, is called transposing that term, or transposition. Hence, in an equation, any term affected by the sign may be transposed, if this sign be changed to -; for this is subtracting the same quantity from each member. (Ax. 2.)

ART. 18. 1. Two men, A and B, hired a farm for $450, of which A paid $50 more than B. Required the rent paid by each.

then x

Let x be the number of dollars A paid;

50 will be the number B paid. We have, then, xx-50-450. Reducing the first member, 2x-50-450; adding 50 to each member, 2x-50+50=450+50; reducing, (Ax. 5,) 2 x 500; and

=

lars, and half-dollars. He has twice as many dollars as eagles, and 10 more half-dollars than dollars. If x represent the number of eagles, what expression will represent the whole number of coins, and consequently be equal to 30?

7. B is 10 years older than A, and 3 times A's age is equal to twice B's. If x represent A's age in years, what two expressions will be equal? Ans. 3x and 2x+20.

8. Two men are of the same age; but if one were 18 years younger, and the other 10 years older, 3 times the age of the former would be the same as twice the age of the latter. What two expressions will be equal, if x represent the age of each in years?

9. Two flocks of sheep are equal in numbers; but if 20 sheep be transferred from one to the other, one flock will then contain 3 times as many as the other. Find two expressions which shall be equal after this change.

10. Half of a man's life was spent in Europe, of it in Asia, and the remainder, which was 10 years, in America. Find two expressions for his age.

11. A woman's age is of her husband's, and the dif ference of their ages is 9 years. Find two expressions for the man's age.

12. A man, having 60 dollars, lost a certain number of them, after which he found that 3 times what he lost was the same as twice what he had remaining. Find two equal expressions.

13. A farmer has twice as many sheep as cows; he buys 60 more sheep and 5 more cows, and finds that he has 4 times as many sheep as cows. Find two expressions for the number of his sheep after this purchase.