Problem Solving

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?

A. 2
B. 3
C. 4
D. 5
E. 6

The OA is D

Source: Manhattan Prep

Let the initial number of cookies be x
Total weight of the mixture = 600 ounces

Since they are identical the weight per cookie will be the same
Weight per cookie is \(\frac{600}{x}\)

Let the new cookies be y (cookies produced after reducing the weight by 1 ounce)
Given that they produced 30 cookies more than the previous batch
Hence

y=30+x

y-x=30

Total weight of the mixture = 600 ounces

New weight per cookie after reducing the weight = \(\frac{600}{y}\)

Given that the difference in weight is 1 ounce

or

\(\frac{600}{x}\) - \(\frac{600}{y}\) =1

Simplifying we get

\(\frac{600\left(y-x\right)}{xy}\) =1

We know that y-x=30, substituting we get

\(\frac{18000}{xy}\) =1

xy=18000

Returning back to our initial equation that is

y-x=30

Multiplying it by x on both sides we get

\(xy\ -\ x^2=30x\)

Substituting the value of xy and rearranging we get

\(x^2+30x\ -\ 18000\) =0

Solving the quadratic

\(x^2+150x\ -\ 120x\ -\ 18000\) = 0

\(x\left(x+150\right)\ -\ 120\left(x\ +\ 150\right)=0\)

\(\left(x+150\right)\left(x\ -\ 120\right)=0\)

x=-150 or x=120

Since the number of cookies can't be negative we get that the initial mixture had 120 cookies

Weight per cookie in the initial case = \(\frac{600}{120}=5\)