3 x . 5 X 35 x X = 50 408 - 2 x of the less is 5 By the conditions, 408 2 x - 4x- 7 Multiplying by 5, 15 x 408 + 2 x = 4080— 20 x 7 Multiplying by 7, 2856 + 14x = 28560 - 140 x - - 15 x. Transposing and uniting, 204 x = 31416 X = 154 204Let x denote the less number, and solve the question again. Note. Observe, that after multiplying by 5 in the above example, the signs of both terms of the numerator were changed, that of 408 to —, and that of 2 x to t; this was done because it was not required to subtract so much as 408 by 2 x. The change of signs could not be made before multiplying by 5, because the sign before the fraction showed that the whole fraction was to be subtracted. If the signs of the fraction had been changed at first, it would have been necessary to put the sign + before the fraction. This requires particular attention, because it is of great importance, and there is danger of forgetting it. 7. A man bought a horse and chaise for $341. Now if of the price of the horse be subtracted from twice the price of the chaise, the remainder will be the same as if of the price of the chaise be subtracted from three times the price of the horse. Required the price of each. 8. Two men, A and B, were playing at cards ; when they began, A had only as much money as B. A won of B $23; then of B's money, subtracted from A's, would leave one half of what A had at first. How much had each when they began ? 9. A man has a horse and chaise. The horse is worth $44 less than the chaise. If of the value of the horse be subtracted from the value of the chaise, the remainder will be the same as if from the value of the horse you subtract of the ex cess of the value of the horse above 84 dollars. What is the value of the horse ? VII. The examples in this article are intended to exercise the learner in putting questions into equation. They require no operations which have not already been explained. It was remarked, that no rule could be given for putting questions into equation, but there is a precept which very useful. Take the unknown quantity, and perform the same operations on it, that it would be necessary to perform on the answer to see if it was right. When this is done the question is in equation. 1. A and B, being at play, severally cut packs of cards so as to take off more than they left. Now it happened that A cut off twice as many as B left, and B cut off seven times as many as A left. How were the cards cut ? Let x = the number B left. Then 2 x = the number A cut off. 52 x = the number B cut off. 52 2 x = the number A left. By the conditions, 7 times 52--2 x are equal to 52-X. 364 -14 x = = 52-X. Take the numbers of the answer and endeavour to prove that they are right, and you will see that take the same course as above. 2. A man, at a card party, betted 3s. to 2 on every deal. After twenty deals he had won 5 shillings. At how many deals did he win? Let x = the number of deals he won. Every time he won, he won 2 shillings; that will be 2 x shillings. Every loss was 3 shillings; that will be 3 times 20 60 - 3x. The loss must be taken from the gain, and he will have 5 shillings left. 2x 60 + 3x 3. What two numbers are to each other as 2 to 3 ; to each of which, if 4 be added, the sums will be as 5 to 7. 4' X, or = 5. 3 x +4. Let = the first number. Then = the second. 2 3 x Adding 4 to each, they become x + 4, and 2 The first is now of the second, or the second is of the first. 7x + 28_3x +4. 5 2 4. A sum of money was divided between two persons, A and B, so that the share of A was to that of B as 5 to 3. Now A's share exceeded of the whole sum by $50. What was the share of each person ? Let x = A's share. Then = B's share. 5 3 x 5. The joint stock of two partners, whose particular shares differed by 48 dollars, was to the lesser as 14 to 5. Required the shares. 6. Four men bought an ox for $43, and agreed that those, who had the hind quarters, should pay 1 cent per pound more than those, who had the fore quarters. A and B had the hind quarters, C and D the fore quarters. A's quarter weighed 158 Ib., B's 163 lb., C's 167 lb., and D's 165 lb. What was each per lb., and what did each man pay? 7. A certain person has two silver cups, and only one cover for both. The first cup weighs 12 oz. If the first сир be covered it weighs twice as much as the other cup, but if the second be covered it weighs three times as much as the first. What is the weight of the cover, and of the second cup ? Let w = weight of the cover. + = weight of the second cup, &c. 8. Some persons agreed to give 6d. each to a waterman for carrying them from London te Gravesend ; but with this condition, that for every other person taken in by the way, three pence should be abated in their joint fare. Now the waterman took in three more than a fourth part of the number of the first passengers, in consideration of which he took of them but 5d. each. How many persons were there at first ? Let x = the number of passengers at first. Then + 3 = the number taken in, &c. 4 9. Four places are situated in the order of the four letters, A, B, C, D. The distance from A to D is 134 miles, the distance from A to B is to the distance from C to D, as 3 to 2, and one fourth of the distance from A to B, added to half the distance from C to D, is three times the distance from B to C. What are the respective distances ? 10. A field of wheat and oats, which contained 20 acres, was put out to a labourer to reap for $20 ; the wheat at $1.20 and the oats $0.95 per acre. Now the labourer falling ill reaped only the wheat. How much money ought he to receive according to the bargain ? 11. Three men, A, B, and C, entered into partnership ; A paid in as much as B and one third of C; B paid as much as C and one third of A ; and C paid in $10 and one third of A. What did each pay in ? Let x = the sum A contributed. Then + 10 = 3 and + 10 + B 66 &c. 3 3 12. A gentleman gave in charity £46 ; a part of it in equal portions to 5 poor men, and the rest in equal portions to 7 poor women. Now the share of a man and a woman together amounted to £8. What was given to the men, and what to the women ? 66 Let x = the sum a man received. · Then 8 - x = the sum a woman received, &c. 13. Suppose that for every 10 sheep a farmer kept, he should plough an acre of land, and should be allowed an acre of pasture for every 4 sheep. How many sheep may that person keep who farms 700 acres ? Let x = the whole number of sheep. The number of acres ploughed will be to of the number of sheep; and the number of acres of the pasture will be 1 of the number of sheep; both these added together must be the whole number of acres, &c. 14. A, B, and make a joint stock ; A puts in $70 more than B, and $90 less than C; and the sum of the shares of A and B is of the sum of the shares of B and C. What did each put in? Let x = the sum that B put in, &c. 15. Divide the number 85 into two such parts that if the greater be increased by 7 and the less be diminished by S, they will be to each other in the proportion of 5 to 2. 16. It is required to divide the number 67 into two such parts that the difference between the greater and 75 may be to the excess of the less over 12 in the proportion of 8 to 3. 17. A man bought 12 lemons and a pound of sugar for 56 cents, afterwards he bought 18 lemons and a pound of sugar at the same rate for 74 cents. What was the price of the sugar, and of a lemon ? Let x = the price of the sugar. Then 56 -x= the price of 12 lemons. And 56. = the price of 1 lemon. 12 = the price of a lemon. 74- XC Hence &c. 12 18 18. A man bought 5 oranges and 7 lemons for 58 cents ; afterwards he bought 13 oranges and 6 lemons at the same rate for 102 cents. What was the price of an orange, and of a lemon ? |