3. Divide $1000 between A, B, and C, so that A shall have $72 more than B, and C $100 more than A. 4. Out of a cask of wine which had leaked away a third part, 21 gallons were afterwards drawn, and the cask being then gauged, appeared to be half full: how much did it hold? Suppose the cask to have held a gallons. or +21= by the question. -x 2 2x+126=3x. x=- -126. x= 126, by changing the signs of both members, which does not destroy their equality. +21=42+21=63=- 5. A fish was caught whose tail weighed 97b.; his head weighed as much as his tail and half his body, and his body weighed as much as his head and tail together; what was the weight of the fish? Let Then 2x= the weight of the body. 9+ weight of the head. And since the body weighed as much as both head and tail 6. A person engaged a workman for 48 days. For each day that he laboured he received 24 cents, and for each day that he was idle, he paid 12 cents for his board. At the end of the 48 days, the account was settled, when the labourer received 504 cents. Required the number of working days, and the number of days he was idle. If these two numbers were known, by multiplying them respectively by 24 and 12, then subtracting the last product from the first, the result would be 504. Let us indicate these operations by means of algebraic signs. 48-x= Then 24XX Then or or the number of working days. the amount earned, and 12(48-x)= the amount paid for his board. 24x-12 (48-x) 504 what he received. 24x-576+12x=504. 36x=504+576=1080 1080 30 the working days. 36 whence, 48-30-18 the idle days. And 720-216-504, the amount received. This question may be made general, by denoting the whole number of working and idle days, The amount received, for each day he worked, The amount paid for his board, for each idle day, And the balance due the laborer, or the result of the account, As before, let the number of working days be represented by n. by a. The number of idle days will be expressed 7. A fox, pursued by a greyhound, has a start of 60 leaps. He makes 9 leaps while the greyhound makes but 6; but three leaps of the greyhound are equivalent to 7 of the fox. How many leaps must the greyhound make to overtake the fox? From the enunciation, it is evident that the distance to be passed over by the greyhound is composed of the 60 leaps which the fox is in advance, plus the distance that the fox passes over from the moment when the greyhound starts in pursuit of him. Hence, if we can find the expression for these two distances, it will be easy to form the equation of the problem. Let x= the number of leaps made by the greyhound before he overtakes the fox. Now, since the fox makes 9 leaps while the greyhound makes makes 1; and, therefore, while the greyhound makes a leaps, the Hence, the distance which the greyhound must pass over, will be It might be supposed, that in order to obtain the equation, it would 3 be sufficient to place a equal to 60+; but in doing so, a 2 manifest error would be committed; for the leaps of the greyhound are greater than those of the fox, and we would then equate heterogeneous numbers, that is, numbers referred to different units. Hence it is necessary to express the leaps of the fox by means of those of the greyhound, or reciprocally. Now, according to the enunciation, 3 leaps of the greyhound are equivalent to 7 leaps of the fox, then 1 leap of the greyhound is equivalent to 7 3 leaps of the fox, and consequently a leaps of the greyhound are equivalent making the denominators disappear 14x=360+9x, Therefore, the greyhound will make 72 leaps to overtake the fox, 3 and during this time the fox will make 72 X or 108. 2 And 60+108-168, the leaps which the fox made from the beginning. 8. A and B play together at cards. A sets down with $84 and B with $48. Each loses and wins in turn, when it appears that A has five times as much as B. How much did A win? 9. A can do a piece of work alone in 10 days, B in 13 days: in what time can they do it if they work together? Denote the time by x, and the work to be done by 1. Then in |