Final answer:
Kate has 18 dimes.
Step-by-step explanation:
To find the number of dimes that Kate has, we need to set up a system of equations. Let's let x represent the number of quarters and y represent the number of dimes. We are given that Kate has a total of 50 quarters and dimes, so we have the equation x + y = 50. We are also given that the total value of the quarters and dimes is $9.80, which can be expressed as 25x + 10y = 980. To solve these equations, we can use the method of substitution.
First, we can use the first equation to rewrite x in terms of y: x = 50 - y. Substituting this expression for x into the second equation, we get 25(50 - y) + 10y = 980. Expanding and simplifying, we have 1250 - 25y + 10y = 980. Combining like terms, we get -15y = -270. Dividing both sides by -15, we find that y = 18.
Therefore, Kate has 18 dimes.