1 A farmer bought a number of pigs for 200 However 6 of them

1. A farmer bought a number of pigs for $200. However, 6 of them died before he could sell the rest at a profit of 4 per pig. His total profit was $28. How many pigs did he originally buy?

2. A woman bought a number of items for $24. She realized that if she had bought 9 more items for the same money, she would have paid 6 less per item. How many items did she buy?

4. Austin went on a bike ride of 48 miles. He realized that if he had gone 10 mph faster, he would have arrived 20 hours sooner. How fast did he actually ride?

Can anyone help?

Solution

1.Let us suppose that the farmer had bought x number of pigs for $ 200.Then the purchase price of a pig is $ 200/x. 6 pigs died and the farmer sold x - 6 pigs at $ ( 200/x) + 4 each. Since the farmer\'s total profit is $ 28, therefore, ( x - 6)[ ( 200/x) + 4] = 228 or, [200(x -6)/x] + 4 ( x -6) = 228 or, [200(x -6) + 4x( x -6) = 228x or, 200x - 1200 + 4x² - 24x = 228x or, 4x² + ( 200 -24 -228)x -1200 = 0 or, 4x² - 52x - 1200 = 0 or, x² - 13x -300 = 0. Then, on using the quadratic formula, we have x = [-(-13) ±{(-13)²- 4(1)(-300)}]/2*1 or, x = [ 13 ± ( 169+1200)]/ 2 or, x = [13± 1369]/2 or, x = (13± 37)/2. Since x cannot be negative, we have x = ( 13 + 37)/2 or x = 50/2 or, x = 25. Thus the farmer had originally bought 25 pigs .

2. Let us suppose that the woman had purchased x items for $ 24. Then the purchase price of each item is $ 24/x. If the woman had purchased 9+x items for $ 24, then the purchase price of each item is $ 24/( x + 9). Therefore, 24/( x + 9) = (24/x) - 6 or, 24/( x + 9).= (24 - 6x)/x or, 24x = (24 -6x) (x + 9) or, 24x = 24x + 216 - 6x² -54x or, 6x² + 54x -216 = 0 or, x² + 9x - 36 = 0. Then, on using the quadratic formula, we have x =[-9±{ (9)² - 4(1)(-36)} ]/2 or, x = [ -9± (81+ 144)]/2 or, x = (-9± 225)/2 or, x = ( -9 ± 15)/2. Since x cannot be negative, we have x = ( -9 + 15)/2 or, x = 6/2 or, x = 3.

4. Let us suppose that Austin\'s average speed is x mph Since time = distance/speed, Austin takes 48/x minutes to travel 48 miles. If Austin travels at an average sopeed of x + 10 mph, he takes 48/ (x+10) minutes to travel 48 miles Therefore, 48/( x + 10) =( 48/x) - 20 or, 48/( x + 10) = (48 -20x)/x or, 48x = (48 - 20x)(x + 10) or, 48x = 48x-20x²+480-200x or, 20x²+200x- 480 = 0 or, x²+10x-24 = 0 or, x²+ 12x- 2x-24 = 0 or, x ( x + 12) -2(x + 12) = 0 or, (x + 12) (x -2) = 0 Thus, either x =-12 or x = 2. Since x cannot be negative, we have x = 2. Thus Austin rode at an average speed of 2 mph.