SECTION VII. Examples of the Solution of Problems producing Simple Equations involving two unknown Quantities. 1. AFTER A had won four shillings of B, he had only half as many shillings as B had left. But had B won six shillings of A, then he would have had three times as many as A would have had left. How many had each? Let x = the number of shillings A had, then y - 4 = 2x + 8, and y + 6 = 38 - 18 ; ... by subtraction, 10 = X 26, and by transposition, 36 = x, 4 = 80; .. y = 84; .. A had 36, and B 84. 2. A person bought a quantity of brandy and rum for £19. 48. and gave for the brandy 9 shillings, and rum 6 shillings per bottle. He found however that he could have bought as many bottles of rum as he now had of brandy, and as many of brandy as he now had of rum for £1. 138. less. How much was bought? Let x = the number of bottles of brandy, then 9x + 6y = 384, ... he bought 30 bottles of brandy, and 19 of rum. 3. What fraction is that, to the numerator of which if 4 be added, the value is one-half, but if 7 be added to the denominator, its value is one-fifth ? 4. Find two numbers, the greater of which shall be to the less as their sum to 42, and as their difference to 6. Let x and y = the numbers; then x : y:: x + y : 42, and x : y:: x-y: 6. But ratios which are equal to the same ratio are equal to each other; ∴ x + y : 42 :: x - y : 6, altdo. x + y : x - y :: (42:6::)7:1; .. (Alg. 182.) 2x : 2y::8: 6, and x : y::4:3; 5. What two numbers are those, whose difference, sum, and product, are as the numbers 2, 3, and 5, respectively? 6. A and B playing at bowls, says A to B, If you will give me a guinea, I will bet you half a crown to eighteen pence on each game, and will play 36 games together. B won his guinea back again, and £1. 178. besides. How many games did each win? Let x = the number of games A won, and y = the number B won; .. y + x = 36, and 3y + 3x = 108 ; but 5y - 3x = 116 ; .. by addition, 8y = 224, and y = 28; 7. A person exchanged 12 bushels of wheat for 8 bushels of barley, and £2. 168.; offering at the same time to sell a certain quantity of wheat for an equal quantity of barley, and £3. 158. in money, or for £10 in money. Required the prices of the wheat and barley per bushel. Let x = the price of wheat per bushel, in shillings, and y = the price of barley; 200 then = the number of bushels in the second offer; ... the prices of wheat and barley per bushel were 8 and 5 shil lings, respectively. |