Jake bought 9 drinks for everyone in his family. He bought larges and smalls. The larges cost $1.50 and the smalls cost $0.90. He spent $10.50 in all. How many larges (L) and how many smalls (s) did he buy?
[tex]
\begin{array}{c}
L+s=9 \\
1.5 L+.9 s=10.5
\end{array}
[/tex]
[?] larges and [ ] smalls
Answer (2)
Set up a system of equations: L + s = 9 and 1.5 L + 0.9 s = 10.5 .
Solve the first equation for L: L = 9 − s .
Substitute this expression into the second equation and solve for s: s = 5 .
Substitute the value of s back into the equation for L: L = 4 .
The final answer is 4 larges and 5 smalls.
Explanation
Problem Analysis Let's analyze the problem. We are given a system of two equations with two variables, L and s, representing the number of large and small drinks, respectively. The equations are:
Equation 1: L + s = 9 (Total number of drinks) Equation 2: 1.5 L + 0.9 s = 10.5 (Total cost of the drinks)
Our goal is to find the values of L and s that satisfy both equations.
Express L in terms of s We can solve this system of equations using substitution or elimination. Let's use substitution. From Equation 1, we can express L in terms of s:
L = 9 − s
Substitute into Equation 2 Now, substitute this expression for L into Equation 2:
1.5 ( 9 − s ) + 0.9 s = 10.5
Simplify the equation Expand and simplify the equation:
13.5 − 1.5 s + 0.9 s = 10.5
Combine the s terms:
13.5 − 0.6 s = 10.5
Isolate the s term Isolate the s term:
0.6 s = 13.5 − 10.5
0.6 s = 3
Solve for s Solve for s:
s = 0.6 3
s = 5
Solve for L Now that we have the value of s, we can find the value of L using the expression L = 9 − s :
L = 9 − 5
L = 4
Final Answer So, Jake bought 4 large drinks and 5 small drinks.
Examples
Imagine you're planning a party and need to buy drinks. You have a budget and know the prices of different drink sizes. This problem helps you determine how many of each size you can buy to stay within your budget while getting the total number of drinks you need. This kind of problem is also useful in business when you are trying to optimize costs and quantities of goods.
Jake bought 4 large drinks and 5 small drinks. This was determined by setting up a system of equations based on the total number of drinks and total cost. By solving these equations, we found that L (larges) equals 4 and s (smalls) equals 5.
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