Kat has 19 coins, all quarters and dimes, that are worth a total of $4. The system of equations that can be used to find the number of quarters, [tex]$q$[/tex], and the number of dimes, [tex]$d$[/tex], she has is shown.
[tex]
\begin{aligned}
q+d & =19 \\
0.25 q+0.1 d & =4
\end{aligned}
[/tex]
How many quarters does she have?
A. 4
B. 5
C. 14
D. 15
Answer (1)
Express d in terms of q using the first equation: d = 19 − q .
Substitute this expression into the second equation: 0.25 q + 0.1 ( 19 − q ) = 4 .
Simplify and solve for q : 0.15 q = 2.1 , which gives q = 14 .
The number of quarters is 14 .
Explanation
Problem Analysis Let's analyze the problem. We are given a system of two equations with two variables, q and d , representing the number of quarters and dimes, respectively. Our goal is to find the value of q , which represents the number of quarters.
Express d in terms of q The given system of equations is:
q + d = 19
0.25 q + 0.1 d = 4
We can solve this system using substitution or elimination. Let's use substitution. From the first equation, we can express d in terms of q :
d = 19 − q
Substitution Now, substitute this expression for d into the second equation:
0.25 q + 0.1 ( 19 − q ) = 4
Distribute Distribute the 0.1 :
0.25 q + 1.9 − 0.1 q = 4
Combine Like Terms Combine like terms:
0.15 q + 1.9 = 4
Isolate q term Subtract 1.9 from both sides:
0.15 q = 4 − 1.9 0.15 q = 2.1
Solve for q Divide both sides by 0.15 :
q = 0.15 2.1 = 15 210 = 14
Final Answer Therefore, the number of quarters is 14.
Examples
Imagine you're running a school fundraiser and need to count the coins collected. Knowing how to solve a system of equations helps you determine the exact number of each type of coin (quarters, dimes, nickels, etc.) based on the total number of coins and their total value. This is useful for balancing the accounts and reporting the earnings accurately. This method is also applicable in inventory management, where you need to track different types of items based on their quantities and total value.
