Kat has 19 coins, all quarters and dimes, that are worth a total of $4. The system of equations that can be used to find the number of quarters, q, and the number of dimes, d, she has is shown.

q + d=19 0.25q + 0.1d=4

How many quarters does she have?

The purpose of this question is to set up a system of equations, and to solve it. However, the answer below will not do that, but will still give the correct answer.

What is being done, for relatively simple problems, especially those dealing with integer answers, can frequently solved in the head, as follows.

We know that, (at least for north americans)

1 quarter = 25 cents (so four quarters make $1)

1 dime = 10 cents.

Given 19 coins made up of quarters and dimes are worth $4.00. We need the number of quarters among the 19.

Solution as follows:

If all coins were quarters, then $4.00 would have 4*4=16 coins.

We know that there are 19 of them, so we need 3 extra coins by changing some quarters for dimes. 2 quarters (50 cents) are worth the same as 5 dimes (50 cents). If we make the swap, we will get 3 more coins while keeping the same value (of $4.00), making a total of 19 coins, which is exactly what we need.

Since we changed 2 quarters for 5 dimes, we have 16-2=14 quarters, the required answer.