Let x represent A's income, and У B's income. +y=570; and 3x+5y=2350. These Equations will give x=$250, and y=$320. 19. "How old are we?" said a person to his father: "6 years ago," replied the latter, "I was a third more than 3 times as old as you were; and in three years, if I multiply your age by 2, it will then be equal to mine." What were their ages? Let x represent the father's age, and У the son's age; then y-6 represents the son's age 6 years ago; Then, by the conditions of the problem, we shall have from which we shall find x=36, and y=15 years. 20. Find a number such that if we subtract it from 4980, divide the remainder by 8, and subtract 123 from the quotient, we shall find a remainder equal to the number itself. Let x represent the number; then 4980- represents the remainder after subtracting the number from 4980; and by 8. 4980-x represents the quotient after dividing the remainder Then, subtracting 123 from this quotient, we shall have the Equation 4980-x −123=x; which will give x=444. 21. A laborer engaged for 40 days upon these conditions; that for every day he worked he should receive 80 cents, but for every day he was idle he should forfeit 32 cents. At the end of the time he was entitled to $15.20. How many days did he work, and how many was he idle? Let a represent the number of days he worked, and y the number he was idle; then 80x represents the number of cents he earned; and The Equations will then be x+y=40; and 80x-32y=1520. These Equations will give x=25, and y=15 days. 22. A cistern containing 820 gallons is filled in 20 minutes by 3 pipes, the first of which conveys 10 gallons more, and the second 5 gallons less than the third, per minute. How much flows through each pipe in a minute? Let x, y, and z respectively represent the number of gallons that flow through each pipe per minute. Then, since 20x, 20y, and 20z respectively represent the number of gallons that flow through each pipe in 20 minutes, we shall have the Equation 20x+20y+20z=820. And the other Equations will evidently be x=2+10; and From these three Equations we shall find x=22, y=7, and z=12 gallons. 23. A trader maintained himself for 3 years at an expense of £50 a year, and each year augmented that part of his stock which was not thus expended by thereof. At the end of the third year his original stock was doubled. What was that stock? Let x represent his stock; then х -50 is what was left after the 1st expenditure; (x−50+ 3 ing his stock the 1st time; 4x-200 or is what he had after augment-. 3 had after augmenting his stock the 3d time. Then, by the problem, the Equation will be The first of what 24. A and B began to trade with equal sums of money. year A gained $40; and B lost $40; the second year A lost he had at the end of the first, and B gained $40 less than twice what A lost; when it appeared that B had twice as much money as A. What sum did each begin with? Let x represent the sum; then x+40 is what A had at the end of the 1st year ; -40), or 3 5–160 -40 represents what B gained the 2d year; and x+40 is what B had at the 25. What fraction is that, whose numerator being doubled, and denominator increased by 7, the value becomes; but the denominator being doubled, and the numerator increased by 2, the value becomes ? Let-represent the fraction; then the Equations will be y From these Equations we shall find x=4, and y=5. Hence the fraction is . 26. A and B together can perform a piece of work in 8 days, A and C in 9 days, and B and C in 10 days. How many days would it take each person to perform the work alone? Let x, y, and z respectively represent the number of days in which A, B, and C could do it; then And, since A and B together, in 1 day, would do of the work, A and C together of it, and B and C together of it, the Equations will be Subtracting the 2d Equation from the 1st, we shall have which will give y=1723. The values of x and z will now be found equal to 143, and 23, respectively. 27. From two places which are 154 miles apart, two persons set out at the same time to meet each other, one traveling at the rate of 3 miles in 2 hours, and the other at the rate of 5 miles in 4 hours; in how many hours will they meet? Let x represent the number of hours; then, since 3 2 5 3x 2 5x 4 is the number of miles the first goes per hour; and is the number of miles the second goes per hour; is the number of miles the first goes in x hours; and is the number of miles the second goes in x hours. Then, since both together traveled 154 miles, the Equation will be Fif 28. In a naval engagement, the number of ships captured was 7 more, and the number burned was 2 less, than the number sunk. teen escaped, and the fleet consisted of 8 times the number sunk. Of how many ships did the fleet consist? |