Let a represent the first, and y the second part; then we shall have the Equations, 100; and x + y = X y From these Equations we shall find x=60, and 30. y=40. 3. Find two numbers such that their sum shall be 60, and the less number of the greater. Let x represent the greater, and y the less. x+y=60; and x y= 3; which will give x=45, and y=15. 4. At a certain election, 946 men voted for two candidates, and the successful one had a majority of 558. How many votes were given for each candidate? Let x represent the number for the successful, and y the number for the unsuccessful candidate. We shall have the Equations x+y=946; and ix-y=558; from which Equations x will be found 752, and y=194. 5. Divide the number 48 into two such parts, that the quotient of the greater part divided by 4 may be equal to 4 times the quotient of the greater part divided by the less. Let x represent the greater, and У = x+y=48; and the less part. X 4.x 4 y from which we shall find x=32, and y=16. 6. A, B, and C, make a joint contribution, which, in the whole, ; amounts to $400; B contributed twice as much as A and $20 more, and C as much as the other two together. What sum did each contribute? Let x represent A's contribution, y B's, and z C's. By the conditions of the problem, the Equations will then be x+y+z=400; z=x+y. These three Equations will give x=60, y=140, and z=200. 7. Find three numbers such that the sum of the first and second shall be 35, the sum of the first and third 40, and the sum of the second and third 45. Let x represent the first, y the second, and z the third; then the Equations will be x+y=35; x+2=40; and y+z=45; from which we shall find x=15, y=20, and z=25. 8. A sum of money was divided between A and B, so that B's share was of A's, and A's share exceeded of the whole sum by $50 What was the share of each? Let x represent A's share, and y B's share; then x+y represents the whole sum; and 5x+5y represents of the whole sum. 9 By the conditions of the problem, we shall then have 3x y= ; and 5x+5y +50. These Equations will give x=$450, and y=$270. 9. The stock of three traders amounted to $760. The shares of the 1st and 2d together exceeded the share of the 3d by $240; and the share of the 1st was $360 less than the sum of the shares of the other two. What was the share of each? x= Let x represent the share of the first, y the share of the second, and z the share of the third; then we shall have the Equations x+y+z=760; x+y=2+240; and x=y+z−360. These Equations will give a $200, y=$300, and z=$260. == "If I 10. A man being asked the age of himself and son replied, were as old as I am + 3 times the age of my son I should be 45; and if he were his present age + three times mine, he would be 111." Required their ages. Let x represent the father's age, and y the son's age. x +3y=45; and y +3x=111; 4 from which x will be found 36, and y= = 12 years. 11. A and B together have $340, B and C together $384, and A and C together $356. What sum has each ? Let x represent A's, y B's, and z C's, amount of money. x+y=340; y+z=384; and x+z=356. These three Equations will give x=$156, y=$184, and z=$200. 12. A number which is expressed by two digits is equal to 4 times the sum of its digits, and if 18 be added to the number, its digits will be interchanged with each other. What is the number? Let x represent the tens' and y the units' figure of the number. Then, since the number is equal to ten times the tens' figure the units' figure, 10x+y represents the required number; and 10y+x represents the number with its digits interchanged with each other. By the conditions of the problem, the Equations will then be 10x+y=4(x+y); and 10x+y+18=10y+x; from which we shall find x=2, the tens' figure, and y=4, the units' figure. Hence the number is 24. 13. It is required to divide the number 36 into three such parts, that of the first, of the second, and of the third, shall all be equal to each other. What are the parts? Let x, y, and z represent the first, second, and third parts, respectively. The Equations will then be x+y+z=36; 812 4.x 5 4.x 5 19133 = which will give x=$125. 2160 4 From these Equations we shall find x=8, y=12, and z=16. and 14. A and B have both the same income; A saves of his annually, but B, by spending $50 per annum more than A, at the end of 4 years, finds himself $100 in debt. What is their income? Let x represent their income; then X represents what A saves annually; and represents what A spends annually; hence +50 represents what B spends annually. Now, since B at the end of 4 years is $100 in debt, his expenses for that time, amount to 100 dollars more than his income for the same time. Then, since 4x represents B's income for 4 years, we shall have 4 (+50)=4x+100; 15 A gentleman purchased a chaise, horse, and harness, for $180; the horse cost twice as much as the harness, and the chaise twice as much as the horse and harness together. What was the price of each? Let a represent the price of the harness; then x+2x+6x=180; = which will give a $20, the price of the harness; hence 2×20=$40, is the price of the horse; and 6×20= $120, is the price of the chaise. 16. A farmer purchased 100 acres of land for $2450; for a part of the land he paid $20 an acre, and for the other part $30 an acre. How many acres were there in each part? Let x represent the number of acres in the first, and y in the second part; then 20x represents the cost of the 1st part; and 30y represents the cost of the 2d part. Hence we shall have the Equations x+y=100; and 20x+30y=2450; which will give x=55, and y=45 acres. X Let-represent the fraction. y Then, by the conditions of the problem, we shall have x+1 y X y+1 which will give x=3, and y=8. 17. What fraction is that to the numerator of which if 1 be added, the value will be ; but if 1 be added to the denominator, the value of the fraction will be ? = 1 2 ; and 1 3 Hence the fraction is 3. the number 18. A and B together possess an income of $570; if A's income were three times, and B's five times as much as each really is, they would together have $2350. What is the income of each? |