MISCELLANEOUS PROBLEMS. 1. A, B and C together have $2000. B has $100 less than twice as much as A, and C $400 less than twice as much as the other two together. What sum has each ? Let x represent A's sum of money; then 2x-100 represents B's sum; 2(3x-100)-400 represents C's sum; and 9x-700 represents the whole sum. The Equation will then be 9x-700-2000. From this Equation we shall find x=300; then 2×300-100= $500, is B's sum, and 2000-300-500-$1200, is C's sum. 2. A gentleman has three plantations. The first contains 250 acres, the second as much as the first and of the third, and the third as much as the first and second. What is the whole number of acres? Let x represent the number of acres in the third plantation; then X 250+ represents the number in the second. 6 Then, since the third contains as much as the first and second, X x=500+ ; 6 600 6 from which x 600 acres; hence 250+ =350 acres, the number in the second; and the first contains 250 acres. Then, 600+350 +250 1200 is the whole number of acres. 3. A company of workmen had been employed on a piece of work for 24 days, and had half finished it, when, by calling in the assistance of 16 more men, the remaining half was completed in 16 days. What was the original number of men? Let x represent the original number of men; then x+16 represents the number that finished the last half. from which x=32 Since the time of performing each half of the work was inversely as the number of men employed on it, x:+16:16: 24; 4. A's money was equal to of B's. A paid away $50 less than of his, and B $50 more than of his, when it was found that the latter had remaining only as much as the former. What sum had each at first? x (話+50); 2 4 from which x=$400, B's sum; hence of 400-$300, is A's sum. -50 5. A person wishing to enclose a piece of ground with palisades, found that if he set them one foot asunder, he would not have enough by 150; but if he set them one yard asunder, he would have too many by 70. What was the number of his palisades? Let x represent the number; then since, if he set them one foot apart he would need 150, x+150 represents the number of feet around the enclosure. Since, if he set them one yard, or three feet, apart, he would have too many by 70, 3x-70x3, or 3x-210, also represents the number of feet around the enclosure. We shall then have the Equation x+150-3x-210; which will give x=180 palisades. 6. From two tracts of land of equal size, were sold quantities in the proportion of 3 to 5. If 150 acres less had been sold from the one which is now the smaller of the two, only as much would have been taken from it as from the other. How many acres were sold from each? Let x represent the number of acres sold from the first; then 5x represents the number of acres sold from the second. 3 Since the second is now the smaller of the two, by the conditions of the problem, 2x from which we shall find x=150 acres ; hence of 150=250, the number sold from the second tract. 5 4x 5x 3 -150: 7. A and B had adjoining farms, which in quantity were in the ratio of 4 to 5. A sold to B 50 acres, and afterwards purchased from B onethird of his entire tract, when it was found that the original ratio of their quantities of land had been reversed. How many acres had each at first? Let x represent the number of acres B had; then 4x represents the number A had; after buying from B; and 2 1 ·50)+(x+50), or 3 –50 represents the number A had after selling 50 acres; 5 x+50 represents the number B had after buying 50 acres ; 17x 100 (12€ -50) + is the number A had 5 15 3 68. 15 (x+50) represents the number B had after selling to A. 3 Reversing the original ratio of the quantities of land, 17x 100 2 (x+50): 5:4. 15 3 3 Developing the second term of this proportion, and multiplying together extremes and means. 400 10x 500 3 + ; 3 3 which will give x=250 acres, B's tract; hence of 250-200 acres, A's tract. 8. A waterman can row down the middle of the stream, on a certain river, 5 miles in of an hour; but it takes him 11⁄2 hours to return, though he keeps along shore where the current is but half as strong as in the middle. What is the velocity of the middle of the stream? Let x represent the velocity per hour; then since 3 20 5÷÷ is the number of miles he can row down stream 4' 3' in one hour, 20 -x is the number he can row in one hour, unassisted by 3 the stream. As the current along shore is but half as strong as in the middle, 20 is the number of miles he travels per hour up 3 stream; and or -X (~20 - 2 - 3) × ×1, or 10 x 3 2 5x 9x 4 he travels up stream. Since this distance is equal to the distance he travels down stream in of an hour, we have 10 9x 5; 4 from which we shall find x-23 miles per is the whole number of miles " hour. 9. A farmer has three flocks of sheep, whose numbers are in the proportion of 2, 3, and 5. If he sell 20 from each flock, the whole number will be diminished in the proportion of 4 to 3. How many sheep has he in each flock? Let x represent the number in the first flock; then 3x represents the number in the second; represents the number in the third. 2 By the conditions of the problem, 3x 5x (x−20)+ ( -20) + (517 - 0) = 32 (x + 32€ + 52.) 13x · 2 2 5.x-60- 30x which gives x=48; hence & of 48-72, is the number in the second flock; and 1⁄2 of 48=120, is the number in the third flock. ; 10. The sum of $1170 is to be divided between three persons, A, B, and C, in proportion to their ages. Now, A's age is to B's as 1 to 13, and to C's as 1 to 2. What are the respective shares? Let x represent A's share; then 1 x+1x+2x=1170; from which we shall find x=$270, A's share; hence 270 × 1=$360, is B's share; and 270×2=$540, is C's share. 11. A hare is 50 leaps before a greyhound, and takes 4 leaps to the greyhound's 3; but 2 of the greyhound's leaps are as much as 3 of the hare's. How many leaps must the greyhound take to catch the hare? Let x represent the number of the greyhound's leaps; then 4x 3 4x represents the number the hare makes in the same time; and +50 is the whole number of the hare's leaps. 3 Since 2 of the greyhound's leaps are as much as 3 of the hare's, each one of the greyhound's leaps will be equal to 3 of one of the hare's; and the whole number of leaps will be inversely as the length of each leap; that is X: 4x €3 +50 :: 1 : 3 12. A vintner has two casks of wine, the contents of which are in the proportion of 5 to 6, and if of the quantity in the second were to be drawn off, the contents of the two casks would be equal. How many gallons are there in each ? 5.x 6 2 Let x represent the number of gallons in the second; then 5x represents the number in the first; and represents the quantity remaining in the second after drain 6 5x 6 ing off. By the conditions of the problem, we shall have This is an identical Equation; the problem is therefore indeterminate, (271). |