Word Problems

Question

Slowly Katie peeked around the door. the huge monster wiped its mouth, smiled, went to sleep, and started to snore. Katie had been watching while the monster ate the cakes in the bakery. The first hour it ate 1/2 of all the cakes in the bakery; the second hour it ate 1/2 of all the cakes left; the third hour it ate 1/2 of what was left; and the fourth hour it ate 1/2 the cakes again. Now there are 3 cakes left How many cakes did the monster eat?

 

Explain/Break down how you got your Answer.

Answer 1

Suppose you start out with N cakes.

 

After the first hour, N/2 are eaten, and N - N/2 = N/2 remain.

After the second hour (1/2) N/2 = N/4 are eaten, and N/2 - N/4 = N/4 remain.

After the third hour, (1/2) N/4 = N/8 are eaten, and N/4 - N/8 = N/8 remain.

After the fourth hour, (1/2) N/8 = N/16 are eaten, and N/8 - N/16 = N/16 remain.

 

But after 4 hours, there are 3 left, so N/16 = 3, hence N = 3*16 = 48

 

You started out with 48 cakes and 3 remain, so 48 - 3 = 45 cakes were eaten.

Answer 2

Hi Kranda:

 

   I set this up as an algebra problem, to determine first how many cakes there were to start with (n).  After the first hour, there are (1/2)*n cakes left.  After the second hour, half of those cakes are left, so (1/2)*[(1/2)*n], or (1/4)*n (remember that products can be evaluated in any order, so (1/2)*[(1/2)*n] = [(1/2)*(1/2)]*n = (1/4)*n.

 

   You can likewise get an expression for how many cakes are left after the fourth hour.  Then we know that expression equals 3.  Use that to solve for n, then n-3 is the number of cakes the monster ate.  Hope this helps!

Answer 3

You can follow the logic to solve for how many cakes we had at the start.

Let C = number of beginning cakes

Each hour for 4 hours the monster ate 1/2 of the cakes.

Each subsequent amount taken away is 1/2 the previous number.

 

C - C/2 - C/4 - C/8 - C/16 = 3

 

C - C(1/2 + 1/4 + 1/8 + 1/16) = 3

C - C(8/16 + 4/16 + 2/16 + 1/16) = 3

C - C(15/16) = 3

 

C/16 = 3

C = 3(16)

C = 48

 

We started with 48 cakes, 3 are left, 45 were eaten

 

Answer 4

There are (1) start-to-finish solutions and (2) finish-to-start solutions.  For projects and tasks (like homework for instance), we often ask "when is it due?" and work backwards to determine when we will work on it.

 

The start-to-finish solutions use a variable (say, N) for the initial number of cakes.  Then, either the total number eaten each hour is calculated and set equal to (N-3) or else the total number left is calculated and set equal to 3.  Either method is O.K.

 

Other tutors have used the start-to-finish approach and arrived at 48 cakes with 45 eaten:

    48 - (1/2)(48) - (1/2)(1/2)(48) - (1/2)(1/2)(1/2)(48) - (1/2)(1/2)(1/2)(1/2)(48) =  3

    48 -   24      -         12            -             6               -                     3                  =  3

     48 -                                                 45                                                         =  3

    48 to start                             -        45 eaten                                                   = 3 left

 

Let's use the finish-to-start approach (it is often easier):

   At the end, there are 3 cakes left:                 3 cakes

   Half the cakes are eaten in 4th hour:             6 cakes       3 eaten        3 left

   Half the cakes are eaten in 3rd hour:           12 cakes       6 eaten        6 left

   Half the cakes are eaten in 2nd hour:          24 cakes      12 eaten       12 left

   Half the cakes are eaten in 1st hour:           48 cakes      24 eaten       24 left

                                                                                                  ------------

                                                                                                    45 eaten

 

Note:  If you use one method to solve the problem, you may use the other method to check your answer.