Hot dogs are 1.25, pizzas are 3.50 and ice cream is 1.75. Come up with two different orders that would cost 10.50.

Question

Problem solving

Answer 1

Often, the key to problem solving is to simplify the problem, then to simplify the simplification, then to reduce the simplification to cases that are easily solved; then, to reverse that whole process until the answer is back in the form of the original problem statement (this is what computer analysts and computer programmers do).
 
If we observe that prices are all multiples of $0.25, we can reduce the complexity of the problem by saying that H=5, P=14, and I=7; also, T=42.

 

Let X=number of hot dogs (H must be less than or equal to 10.50/1.25 = 8.4)

      Y=number of pizzas (P must be less than or equal to 10.50/3.50 = 3)

      Z=number of ice creams (I must be less than or equal to 10.50/1.75 = 6)

 

The problem asks us to find some values of X,Y,Z that make the following true:

   H*X + P*Y + I*Z = 42

 

or, with numbers:
   5X + 14Y + 7Z = 42

 

Now, we choose ONE number from each row to produce a sum of 42:

      Options for cost for hot dogs:  0  5  10  15  20   25   30  35  40

      Options for cost for pizza:  0  14  28   42

      Options for cost of ice cream:  0  7   14   21   28   35   42

 

(Note: a computer could very quickly calculate all possible sums of three numbers – there are only 9*4*7=252 of them.)

 

Here are the (costs in quarters) of two possible orders:

         0 + 28 + 14 = 42

         35 + 0 + 7 = 42

       

O.K., now translate our solutions back into the problem’s dollar amounts:

 

Order 1:  0 hot dogs   +   2 pizzas  +  2 ice creams

                     0         +   7.00        +   3.50                  = $10.50

 

Order 2:  7 hot dogs  +    0 pizzas   +  1 ice cream

                 8.75         +   0             +  1.75                  = $10.50

Answer 2

Let H = Hot dogs = $1.25 each

      P = Pizza = $.3.50 each

      I = Ice cream = $1.75 each

 

2I + 2P = 2(1.75) + 2(3.50) = 3.50 + 7.00 = $10.50

2 ice cream and 2 pizzas

 

3P = 3(3.50) = $10.50

3 pizzas

 

7H + I = 7(1.25) + 1.75 = $10.50

7 hot dogs and 1 ice cream