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Cape Cod Cookies makes cookies of identical size from batches of cookie dough weighing 600 ounces. Cape Cod Cookies decides to modify the recipe by decreasing the weight of each cookie by 1 ounce. If Cape Cod Cookies discovers that it is now able to make 30 more cookies using the same batch of dough weighing 600 ounces, how much did each cookie originally weigh?
E.) 6Please post your answer, along with the explanation, below. Get cracking!
Edit: This challenge is now closed
Let’s use x for the number of cookies produced by the original recipe, and y for the weight of each of the cookies. Given those variables, our first equation is simply xy = 600. We also need to create an equation that represents the new recipe. Since the number of cookies produced has increased by 30, and the weight of each cookie has decreased by 1, the new equation is (x + 30)(y – 1) = 600. Remember, the total weight is still 600 ounces. Foiling this equation yields xy – x + 30y – 30 = 600.
We now have two equations with two variables. There are several different paths we can go down here, but all involve substitution of one of the variables, and all will yield a quadratic. The simplest path is to recognize that since xy = 600, we can substitute for xy in the second equation to get 600 – x + 30y – 30 = 600. Subtracting the 600 from both sides, and adding an x to each side gives us 30y – 30 = x. We can now substitute for x in the first equation.
xy = 600 → y(30y – 30) = 600. This gives us 30y2 – 30y = 600.
Next divide both sides by 30 and set the equation to zero: y2 – y – 20 = 0.
Now that the equation is set to zero, we can factor it into its roots: (y + 4)(y – 5) = 0
The weight of the original cookies was thus 5 ounces, or -4 ounces. The negative root is a nonsensical result given the constraints of the problem, so the answer is 5 ounces.
An alternate solution would be to start with each answer choice, and see which satisfies the conditions of the problem. To implement this method, it’s helpful to set up a chart.
From the chart we can see that only in answer D will we end up with 30 additional cookies after changing the recipe.The correct answer is D.
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