Word equations

Question

part of a fundraiser and local organization collected 406 returnable bottles and cans some worth $0.05 each and the rest worth $0.10 each. If the total value of the cans and bottles was $25.15 how many $0.05 bottles or cans and how many $0.10 bottles or cans were collected

Answer 1

Answer: .05 bottles or cans are 309 and .10 bottles or cans are 97.

 

Solution:

Let us say the total number of $0.05 bottles or cans is x and the total number of $0.10 bottles or cans is y.

 

That means :  x+y = 406  ( This is our equation 1)

 

The total value of the cans or bottles is $25.15.

 

x bottles or cans at $0.05 value each raised .05 * x dollars.

y bottles or cans at $0.10 value each raised .10 * y dollars.

 

That means : .05 * x  +  .10 * y  = 25.15 (This is our equation 2)

 

To simplify equation 2, multiply the whole equation by 100 which gives us 5 *x + 10 * y = 2515  (This is our equation 3)

 

Multiply equation 1 by 5 so that it also has 5 * x just like equation 3 :   5 *x + 5 * y = 2030  (This is our equation 4)

 

Subtract equation 4 from equation 3 so that we have an equation only in y. The result of this subtraction is :

 

5 * y = 485

which gives y = 485 / 5 = 97.

 

Use equation 1 to find x which is 406 - 97 = 309.

 

To verify whether our answer is correct, use equation 3. The left hand side of this equation gives 5 * 309 + 10 * 97 which equals to 1545 + 970 which equals to 2515 which the right side of this equation.