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76 + x = 156 — 3x
XC 156 3 x — 76
4 X = 15676
Ans. 20 guineas. Proof. If A won 20 guineas of B, A will have 96 and B 32. 3 times 32 are 96.
This equation is rather more difficult to solve than any of the preceding. In the first place I subtract 76 from both members, so as to remove it from the first member. Then to get 3 x out of the second member, which is there subtracted, I add 3 r to both members, then the r's are all in the first member, and the known numbers in the other.
N. B. Any term which has the sign +, either expressed or understood, may be removed from one member to the other by giving it the sign --; for this is the same as subtracting it from both sides. Thus x +3= 10; x is not so much as 10 by 3, we therefore say x = 10 -- 3. Again, 5x = 18 + 3x. Now 5 x is more than 18 by 3x, therefore we may say 5 x 18.
Any term which has the sign before it may be removed from one member to the other by giving it the sign +. This is equivalent to adding the number to both sides. Thus 5 x
-3= 17. In this it appears that 5 x is more than 17 by 3; therefore we say 5 x = 17 + 3. Again, 5 x = 32. 3 x. Here it appears that 5 x is not so much as 32 by 3x ; therefore we say 5 x + 3x = 32. This is called transposition.
Hence it appears that any term may be transposed from one member to the other, care being taken to change the sign.
In the last example, 76 was transposed from the first member to the second, and the sign changed from + to —; and 3 x was transposed from the second member to the first, and the sign changed from to +., This has been done in many of the preceding examples.
When a number, consisting of two or more terms, is to be multiplied, all the terms must be multiplied, and their signs preserved. In the last example, 52 — , multiplied by 3, gave a product 156
8. A person bought two casks of wine, one of which held exactly three times as much as the other. From each he drew
4 gallons, and then there were four times as many gallons remaining in the larger as in the smaller. How many gallons were there in each
at first ?
-4 The greater is now 4 times as large as the less ; 4 times x -4 is 4 x - 16.
16 = 3 x 4
BC = 12 = less.
3x = 36 = greatër.
Ans. Less 12 gallons, greater 36 gallons. Proof. 36 is three times 12 according to the conditions. Take 4 from each, then one contains 32 and the other 8. 32 is 4 times 8.
9. A man when he was married was three times as old as his wife ; after they had lived together 15 years, he was but twice
How old was each when they were married ? 10. A farmer has two flocks of sheep, each containing the same number. From one of these he sells 39, and from the other 93 ; and finds just twice as many remaining in the one as in the other. How many did each flock originally contain ?
11. A courier, who travels 60 miles per day, had been despatched 5 days, when a second was sent to overtake him ; in order to which, he must go 75 miles per day ; in what time will he overtake the former ?
12. A and B engaged in trade, A with £240, and B with £96.
A lost twice as much as B ; and upon settling their accounts it appeared that A had three times as much remaining as B. How much did each lose ?
Let x denote B's loss, then 96 x will denote what he had remaining. 2 x will denote A's loss, and 240 — 2 x what he had remaining, &c.
13. Two persons began to play with equal sums of money ; the first lost 14 shillings, and the other won 14 shillings, and then the second had twice as many shillings as the first. What sum had each at first ?
14. Says A to B, I have 5 times as much money as you; yes, says B, but if you will give me $17, I shall have 7 times as much as you. How much had each ?
15. Two men, A and B, commenced trade; A had $500 less than 3 times as much money as B; A lost $1500, and B gained $900, then B had twice as much as A. How much had each at first ?
16. From each of 15 coins an artist filed the value of 2 shillings, and then offered them in payment for their original value; but being detected, the whole were found to be worth no more than $145. What was their original value ?
17. A boy had 41 apples, which he wished to divide between three companions, as follows ; to the second he wished to give twice as many as to the first, and three apples more; and to the third he wished to give three times as many as to the second, and 2 apples more. How many must he give to each ?
18. A person buys 12 pieces of cloth for 149 crowns : 2 are white, 3 are black, and 7 are blue. A piece of the black costs 2 crowns more than a piece of the white, and a piece of the blue costs 3 crowns more than a piece of the black. Required the price of each kind.
See example 4th of this Art.
19. A man bought 6 barrels of flour and 4 firkins of butter; he gave $2 more for a firkin of butter, than for a barrel of flour
; and the butter and flour both cost the same sum. What did he give for each !
20. A grocer sold his brandy for 25 cents a gallon more than his wine, and 37 gallons of his wine came to as much as 32 gallons of his brandy. What was each per gallon?
21. A man bought 7 oxen and 36 cows; he gave $18 apiece more for the oxen than for the cows, and the cows came to three times as much as the oxen wanting $3. What was the price of each?
22. A man sold 20 oranges, some at 4 cents apiece, and some at 5 cents apiece, and the whole amounted to 90 cents. How many were there of each sort?
If he had sold 13 at 5 cents apiece, then the number sold at 4 cents apiece would be 20— 13, or 7.
In the same manner, if he sold x oranges at 5 cents apiece, then he sold 20
& oranges at 4 cents apiece. x oranges at 5 cents apiece would come to 5 x cents, and 20 x oranges at 4 cents apiece would come to 4 times 20 - ix cents, which is 80 L 4 x cents. These added together must make 90 cents, therefore
5x + 80 - 4x = 90 By transposing 80 and uniting terms, x = 10 at 5 cents.
Ans. 10 of each sort. 23. A man dying left an estate of $2500 to be divided between his two sons, in such a manner, that twice the elder son's share should be equal to three times the share of the second. Required the share of each.
Let w denote the younger son's share.
By the conditions, =.5000
2500 - 1000 1500
Ans. Elder son $1500, younger son $1000. 24. Two robbers, after plundering a house, found they had 35 guineas between them; and that if one of them had 4 guineas more, he would have twice as many as the other. How many had each ?
25. A man sold 45 barrels of flour for $279 ; some at $5 and some at $8 per barrel. How many barrels were there of each sort ?
26. A man sold some oxen and some cows for $330; the whole number was 15. He sold the cows for $17 apiece, and the oxen for $32 apiece. How many were there of each sort?
27. After A had lost 10 guineas to B, he wanted only 8 guineas in order to have as much money as B; and together they had 60 guineas. What money had each at first ?
Let u be the number of guineas A had.
A lost 10 to B, therefore A's is diminished by 10, and B's increased by 10, which makes A's x — 10, and B's 70 — x.
By the conditions, X - 10 +8=70-
x = 36 = what A had.
60 -- 36 = 24 = what B had. 28. Divide the number 197 into two such parts, that four times the greater may exceed five times the less by 50.
29, Two workmen were employed together for 50 days, at 5 shillings per day each. A spent 6 pence a day less than B did, and at the end of the 50 days he found he had saved twice as much as B, and the expense for two days over. What did each spend per day?
Let w denote what A spent per day (in pence).
Then 60 — * (5s. being 60d.) will be what he saved per day.
B saved 6d. less than A.
By the conditions A saved 2 x more than twice what B
50 = what A spent. 50 +6 = 56 = what B spent.
V. 1. Two persons talking of their ages, A said he was 25 years older than B, and that one half of his age was equal to three times that of B wanting 35 years. What was the age of each? Let x denote the
of B. Then the age of A will be x + 25. 1 of x + 25 is expressed *+ 25
2 Hence we have
2 Multiplying by 2,
6x - 70= x + 25 By transposing x and 70, 6 x - x=25+ 70 Uniting terms,
5x = 95 Dividing by 5,
x = 19 = B's age. x + 25 = 44 = A's age.
x + 25