Taylor bought 13 cupcakes and muffins from the bakery. Cupcakes cost $2.00 and muffins cost $1.50. Taylor spent a total of $22.00. How many cupcakes (c) and how many muffins (m) did she buy?
[tex]\begin{array}{c}
c+m=13 \\
2 c+1.5 m=22
\end{array}[/tex]
[?] cupcakes [ ] muffins
Answer (2)
Express the total number of items as an equation: c + m = 13 .
Express the total cost as an equation: 2 c + 1.5 m = 22 .
Solve for c by substituting m = 13 − c into the cost equation and simplifying: c = 5 .
Solve for m using the number of items equation: m = 8 . The final answer is 5 cupcakes and 8 muffins .
Explanation
Analyze the problem Let's analyze the problem. We are given two equations:
c + m = 13 (The total number of cupcakes and muffins is 13)
2 c + 1.5 m = 22 (The total cost is $22.00, where cupcakes cost $2.00 each and muffins cost $1.50 each)
Our goal is to find the values of c and m that satisfy both equations.
Express m in terms of c We can solve this system of equations using substitution or elimination. Let's use substitution. From the first equation, we can express m in terms of c :
m = 13 − c
Substitute into the second equation Now, substitute this expression for m into the second equation:
2 c + 1.5 ( 13 − c ) = 22
Simplify the equation Expand and simplify the equation:
2 c + 19.5 − 1.5 c = 22
Combine the c terms:
0.5 c + 19.5 = 22
Isolate the c term Subtract 19.5 from both sides:
0.5 c = 22 − 19.5
0.5 c = 2.5
Solve for c Divide both sides by 0.5:
c = 0.5 2.5
c = 5
Solve for m Now that we have the value of c , we can find the value of m using the equation m = 13 − c :
m = 13 − 5
m = 8
Conclusion So, Taylor bought 5 cupcakes and 8 muffins.
Examples
Imagine you're planning a bake sale and need to figure out how many cookies and brownies to bake. You know you want to make a total of 50 items, and you want to spend exactly $30 on ingredients. If cookies cost $0.50 to make and brownies cost $0.75, you can set up a system of equations similar to the one in this problem to determine how many of each item you should bake. This helps you manage your resources and meet your goals efficiently.
Taylor bought 5 cupcakes and 8 muffins. We found this by solving the equations representing the total number of items and the total cost. By substituting and simplifying, we determined the quantities of both cupcakes and muffins she purchased.
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