Giuliana has 22 quarters and dollar coins worth a total of $10.75. Which equation can be used to find $d$, the number of dollar coins Giuliana has?
A. $d-22+0.25 d=10.75$
B. $0.25 d+22-d=10.75$
C. $0.25(22-d)+d=10.75$
D. $d+0.25(d-22)=10.75
Answer (2)
Define variables: Let d be the number of dollar coins and q be the number of quarters.
Set up equations: d + q = 22 and d + 0.25 q = 10.75 .
Substitute: Express q in terms of d as q = 22 − d and substitute into the second equation.
The correct equation is: 0.25 ( 22 − d ) + d = 10.75 .
Explanation
Problem Analysis Let's analyze the problem. Giuliana has a mix of quarters and dollar coins, totaling 22 coins and worth $10.75. We need to find the equation that helps us determine the number of dollar coins she has.
Setting up Equations Let d be the number of dollar coins and q be the number of quarters. We know two things:
The total number of coins is 22: d + q = 22
The total value of the coins is $10.75: $1.00d + 0.25q = 10.75
Substitution From the first equation, we can express the number of quarters in terms of the number of dollar coins: q = 22 − d . Now, substitute this expression for q into the second equation:
d + 0.25 ( 22 − d ) = 10.75
Finding the Correct Equation Now, let's compare our equation with the given options:
d − 22 + 0.25 d = 10.75 (Incorrect)
0.25 d + 22 − d = 10.75 (Incorrect)
0.25 ( 22 − d ) + d = 10.75 (Correct)
d + 0.25 ( d − 22 ) = 10.75 (Incorrect)
Final Answer The correct equation is 0.25 ( 22 − d ) + d = 10.75 .
Examples
Imagine you're running a school fundraiser and need to quickly calculate how many of each type of coin you have based on the total amount collected. This problem is similar to balancing your cash register at a store, where you need to account for different denominations of currency to ensure the total matches your sales records. Understanding how to set up and solve these equations helps in managing finances and inventory efficiently. For example, if you know you have a total of 50 coins, consisting of dimes and quarters, and the total value is $8.60, you can use a similar equation to find out exactly how many dimes and quarters you have: 0.10 d + 0.25 q = 8.60 and d + q = 50 .
We defined variables for the number of dollar coins and quarters and created equations based on the total coins and their value. After substitution and simplification, we verified that the correct equation that can be used to find the number of dollar coins is 0.25 ( 22 − d ) + d = 10.75 . Therefore, the correct answer is option C.
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