SOLUTION: A sum of money amounting to $5.15 consists of 10 cents and 25 cents. If there are 32 coins in all, how many 25 cents are there?

Question 10957: A sum of money amounting to $5.15 consists of 10 cents and 25 cents. If there are 32 coins in all, how many 25 cents are there?
Answer by kinupanda(9) (Show Source): You can put this solution on YOUR website!
Let us define the following variables for our coins: we have 10 cent pieces and 25 cent pieces. We have 32 coins in all, so we can write .
We also know the total amount of money we have, $5.15. To avoid decimals in our problem, let's rewrite this as 515 cents. Just as in real life, to find the amount of money we have, we multiply the number of dimes by 10 to get cents, multiply the number of quarters by 25 to get cents, and then add them. Thus: .

From here, we solve the linear system:

To save time later, we can simplify the top equation: since all three of the coefficients are divisible by 5, we can divide them all by 5 and rewrite the system as:

We multiply the bottom equation by -2 (you'll see why in a minute!):

Now, we can combine the two equations by adding both of the left sides, as well as both of the right sides.

... simplifying

By multiplying the bottom equation by -2, we were able to eliminate the d variable entirely, which is an important trick in solving linear systems. Thus, we can solve for q now: .

To solve for d, all we have to do is substitute q into one of the two equations from our original system. The easier option would be to substitute into , of course... so we get , or .

Thus, our 32 coins break down to 19 dimes and 13 quarters. Checking our problem, 19 dimes is $1.90 and 13 quarters is $3.25. Added up, this is $5.15, the amount of money we should have! :)
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