A coin collector has $25 in nickels and dimes. There are three times as many nickels as there are dimes. How many of each coin are there?
(A) 125 dimes and 300 nickels
(B) 100 dimes and 300 nickels
(C) 300 dimes and 100 nickels
(D) 150 dimes and 150 nickels
Answer (2)
Define n as the number of nickels and d as the number of dimes.
Set up the equations: 0.05 n + 0.10 d = 25 and n = 3 d .
Substitute n = 3 d into the first equation and solve for d : d = 100 .
Calculate the number of nickels: n = 3 ( 100 ) = 300 . The answer is 100 dimes and 300 nickels .
Explanation
Problem Analysis Let's analyze the problem. We are given that a coin collector has $25 in nickels and dimes. We also know that there are three times as many nickels as there are dimes. Our goal is to find out how many of each coin the collector has.
Setting up Equations Let's use variables to represent the unknowns. Let n be the number of nickels and d be the number of dimes. We can set up two equations based on the given information. The total value of the coins is 25 , w hi c h c anb e w r i tt e na s : 0.05 n + 0.10 d = 25 W e a l so kn o wt ha tt h ere a re t h ree t im es a s man y ni c k e l s a s d im es , so : n = 3 d $
Substitution and Simplification Now, we can substitute the second equation into the first equation to solve for d :
0.05 ( 3 d ) + 0.10 d = 25 Simplify the equation: 0.15 d + 0.10 d = 25 0.25 d = 25
Solving for Dimes Now, solve for d :
d = 0.25 25 = 100 So, there are 100 dimes.
Solving for Nickels Now that we know the number of dimes, we can find the number of nickels: n = 3 d = 3 ( 100 ) = 300 So, there are 300 nickels.
Final Answer Therefore, the coin collector has 100 dimes and 300 nickels.
Examples
Imagine you're managing a school fundraiser. You need to count the coins collected, which include nickels and dimes. Knowing the total amount and the relationship between the number of nickels and dimes, you can use a system of equations to quickly determine the exact number of each coin type. This helps in accurate accounting and efficient handling of the funds raised. For example, if you know you have $25 and three times as many nickels as dimes, you can calculate that you have 100 dimes and 300 nickels using the equations 0.05 n + 0.10 d = 25 and n = 3 d .
The coin collector has 100 dimes and 300 nickels, calculated by setting up equations based on their total value and the ratio of the coins. This solves the problem using algebraic substitution. The answer is option (B) 100 dimes and 300 nickels.
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