Donna bought a candy bar for $1.50. She paid with nickels and dimes. She used 21 coins in all. How many nickels ($n$) and how many dimes ($d$) did she use?
[tex]\begin{array}{c}
n+d=21 \\.05 n+.10 d=1.50\end{array}[/tex]
[?] nickels [?] dimes
Answer (2)
Set up a system of equations: n + d = 21 and 0.05 n + 0.10 d = 1.50 , where n is the number of nickels and d is the number of dimes.
Simplify the second equation by multiplying by 100 and dividing by 5: n + 2 d = 30 .
Solve for n in the first equation: n = 21 − d , and substitute into the simplified second equation: ( 21 − d ) + 2 d = 30 .
Solve for d : d = 9 , and then find n : n = 12 . Thus, the final answer is 12 nickels and 9 dimes .
Explanation
Problem Analysis Let's analyze the problem. We are given two equations:
n + d = 21 (The total number of coins is 21)
0.05 n + 0.10 d = 1.50 (The total value of the coins is $$1.50)
where n is the number of nickels and d is the number of dimes. Our goal is to find the values of n and d .
Eliminating Decimals To make the second equation easier to work with, we can multiply both sides by 100 to eliminate the decimals:
100 × ( 0.05 n + 0.10 d ) = 100 × 1.50
This simplifies to:
5 n + 10 d = 150
Simplifying the Equation Now, we can simplify this equation further by dividing both sides by 5:
5 5 n + 10 d = 5 150
Which gives us:
n + 2 d = 30
Solving for n We now have a system of two equations:
n + d = 21
n + 2 d = 30
We can solve for n in the first equation:
n = 21 − d
Substitution Substitute this expression for n into the second equation:
( 21 − d ) + 2 d = 30
Solving for d Simplify and solve for d :
21 − d + 2 d = 30
21 + d = 30
d = 30 − 21
d = 9
Solving for n Now that we have the value of d , we can substitute it back into the equation n = 21 − d to find the value of n :
n = 21 − 9
n = 12
Final Answer So, Donna used 12 nickels and 9 dimes.
Examples
Understanding systems of equations can help in everyday financial transactions. For instance, if you're managing a small business and need to track inventory and sales, you might use a system of equations to determine the optimal pricing strategy for your products. By setting up equations that represent your costs and revenue, you can solve for the price points that maximize your profit. This approach ensures you're making informed decisions that balance sales volume and profitability, similar to how Donna figured out her coin usage to pay for the candy bar.
Donna used 12 nickels and 9 dimes to pay for her candy bar. This was determined by solving a system of equations based on the total number of coins and their total value. The equations were simplified and solved step-by-step to arrive at the final counts of each type of coin.
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