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6 both, they will still be equal ; hence,
and x = 30 Ans. 30 feet. Proof. One half of 30 is 15, and one third of thirty is 10. Now 30 = 15 + 10 + 5.
There is another mode of reducing the above equation which in most cases is to be preferred. It is the same in principle.
If both members of an equation be multiplied by the same number, they evidently will still be equal.
In the equation,
2 3 First multiply both members by 2, the denominator of one of the fractions, and it becomes,
2x = x +
3 Next multiply both members by 3, the denominator of the other fraction, and it becomes,
6x = 3x + 2x + 30
or 6 x = 5x + 30. Subtracting 5 x from both members,
3 = 30 as before. 5. In an orchard of fruit trees į of them bear apples, of them pears, į of them plums, 7 bear peaches, and 3 bear cherries; these are all the trees in the orchard. How many are there?
6. A farmer, being asked how many sheep he had, answered, he had them in four pastures; in the first he had į of them, in the second , in the third , and in the fourth he had 24 sheep. How many had he in the whole ?
7. A person having spent and į of his money, had $263 left. How much money had he at first?
8. A man driving his geese to market, was met by another, who said good morrow, master, with your hundred geese ; said he, I have not a hundred, but if I had as many more, and half
as many more, and two geese and a half, I should have a hundred. How many had he?
9. A and B having found a bag of money, disputed about the division of it. A said that and į and of the money made $130, and if B could tell how much money there was, he should have it all, otherwise none of it. How much money was there in the bag?
10. Upon measuring the corn produced in a field, being 96 bushels, it appeared that it had yielded only one third part more than was sown.
How much was sown 11. A man sold 96 loads of hay to two persons ; to the first į, and to the second of whąt his stack contained. How many loads did the stack contain at first ?
12, A and B talking of their ages, A says to B if , š, and of my age be added to my age, and 2 years more, the sum will be twice my age. What was his age ?
13. What sum of money is that whose }, à, and } part added together amount to £9 ?
14. The account of a certain school is as follows : of the boys learn geometry, learn grammar, i learn arithmetic, learn spelling, and 9 learn to read. What is the number of scholars in the school ?
15. There is a fish whose head weighs 9 lb. his tail weighs as much as his head and half his body, and his body weighs as much as his head and tail both. What is the weight of the fish?
Represent the weight of the body by x.
16. There is a fish whose head is 4 inches long, the tail is twice the length of the head, added to of the length of the body, and the body is as long as the head and tail both. What is the whole length of the fish?
17. A and B talking of their ages, A says to B, your age is twice and three fifths of my age, and the sum of our ages is 54. What is the age of each?
18. A man divided $40 between two persons ; to the first he gave a certain sum, and to the second only as much. How much did he give to each ?
3 #will denote the share 2
Let « denote the share of the first, a oni
of the second. These added together must make $40.
Adding together, 8 x = 200
x = 25 = share of the first.
5 19. Three persons are to share $290 in the following manner : the second is to have two thirds, and the third three fourths as much as the first. What is the share of each ?
20. A farmer wishes to mix 100 bushels of provender, consisting of rye, barley, and oats, so that it may contain as much barley as oats, and į as much rye as barley. How much of each must there be in the mixture ?
21. Divide 40 apples between two boys in the proportion of 3 to 2.
The proportion 3 to 2 signifies that the second will have as many as the first.
22. A gentleman gave to 3 persons £98. The second received five-eighths of the sum given to the first, and the third one-fifth of what the second had. What did each receive ?
23. A prize of $1280 was divided between two persons, in the proportion of 9 to 7. What was the share of each ?
24. Three men trading in company, put in money in the following proportion ; the first 3 dollars as often as the second 7, and the third 5. They gain $960. What is each man's share of the gain?
Observe, the second put in of what the first put in, and the third put in ģ.
25. Three men traded together ; the first put in $700, the second $450, and the third $950. They gained $420. What was the share of each ?
Observe, the second put in 185 = 44= of what the first put in, &c.
III. 1. Two men, A and B, hired a pasture together for $55, and A was to pay $13 more than B. What did each
Suppose B paid x dollars ; A was to pay 13 dollars more ; therefore he paid x + 13. These put together must make the whole 55 dollars.
x++13=55 Putting the x's together,
2x +13=55 It appears that 2 x is not so much as 55 by 13, therefore taking 13 from 55,
2x = 55 13
2 x = 42 Dividing by 2,
H = 21 = B’s share. B's share is $21, and A's, being 13 more, is $34,
i + 13= 21 + 13 = 34 = A's share. Proof. 34+ 21 = 55 the whole sum.
2. A man bought a horse and chaise for $300; the horse cost $28 more than the chaise. What was the price of each?
3. A man bequeathed his estate of $12000 to his son and daughter; the son was to have $2350 more than the daughter. What was the share of each ?
4. A father who has three sons, leaves them 16000 crowns. The will specifies that the eldest shal ave 2000 crowns more than the second, and that the second shall have 1000 more than the youngest. What is the share of each?
Let x denote the number of crowns in the share of the youngest, then x + 1000 will denote the share of the second, and x + 1000 + 2000 will denote the share of the eldest. These added together must make the whole sum.
8 + x + 1000 +*+ 1000 + 2000 = 16000 Putting together the w's and the numbers,
3x +4000 = 16000 It appears that 3 x is not so much as 16000 by 4000, therefore subtracting 4000 from 16000,
3x = 16000 4000
3x = 12000 Dividing by 3, x = 4000 = share of the youngest.
The share of the youngest is 1000 crowns; add to this 1000, it makes 5000, the share of the second,
x + 1000 = 5000 = share of the second.
Add 2000 more, it makes 7000, the share of the eldest,
* + 1000+ 2000 = 7000 = share of the eldest. Proof. The several shares added make 16000 crowns which is the whole estate.
5. A draper bought three pieces of cloth, which together measured 159 yards ; the second piece was 15 yards longer than the first, and the third was 24 yards longer than the second. What was the length of each ?
6. A gentleman bequeathed an estate of $65000 to his wife, two sons, and three daughters. The wife was to have $2000 less than the elder son, and $3000 more than the younger son; and the portion of each of the daughters was $3500 less than that of the younger son. Required the share of each.
The 1st example may be performed differently. Let x denote the number of dollars paid by A; B paid $13 less, therefore x - 13 will represent the number of dollars paid by B. These added together must make the whole.
x + 13 = 55 Putting the x's together,
13 = 55 It appears that 2 x is more than 55 by 13, therefore add 13 to 55 to make 2 x,
2 x = 55+ 13
2x = 68 Dividing by 2, x = 34 = A's share. This gives A's share $34, from which subtract $13, and it gives B's share $21, as before,
i-13 = 21 = B's share. In the same manner perform the 2d and 3d. be solved in a similar manner.
Let the elder son's share be represented by x. The second son's share, being $2000 less, will be x-2000. The younger son's share, being $1000 less still, will be x — 2000 - 1000. These added together must make the whole sum.
-2000 - 1000 = 16000 Putting the x's together and the numbers together,
3 x-5000 =16000. It
appears that 3 x is more than 16000 by 5000, therefore add 5000 to 16000,
3x = 16000 + 5000
3x = 21000 Dividing by 3,
x = 7000
The 4th may