At the Henderson Middle School awards ceremony, the principal is going to honor outstanding students with gift cards. The gift cards for excellent grades, g , are worth $40, and the gift cards for perfect attendance, a , are worth $25. The principal has $4,000 allotted for this event, and he is going to recognize 100 students.
Write the pair of linear equations that model this situation.
Let's assume that the number of students receiving gift cards for excellent grades is e and the number of students receiving gift cards for perfect attendance is p.
The first equation can be formed by considering the value of the gift cards. As each gift card for excellent grades is worth $40, the total value of those gift cards would be 40e. Similarly, the total value of the gift cards for perfect attendance would be 25p. Since the principal has a total of $4,000 allotted, we can set up the equation:
40e + 25p = 4000
The second equation represents the total number of students being recognized. As per the given information, 100 students are being recognized. Thus, the number of students receiving gift cards for excellent grades and perfect attendance must add up to 100:
e + p = 100
So, the pair of linear equations that model this situation are:
40e + 25p = 4000
e + p = 100
To model this situation with a pair of linear equations, we can consider the number of "excellent grades" gift cards given out and the number of "perfect attendance" gift cards given out.
Let's denote the number of excellent grades gift cards given out as x, and the number of perfect attendance gift cards given out as y.
From the given information, we can create the following equations:
1. The value of the excellent grades gift cards (40) multiplied by the number of cards (x) should equal the total amount spent on excellent grades gift cards:
40x = Amount spent on excellent grades gift cards
2. The value of the perfect attendance gift cards (25) multiplied by the number of cards (y) should equal the total amount spent on perfect attendance gift cards:
25y = Amount spent on perfect attendance gift cards
Additionally, we know that the total amount allotted for this event is $4,000, so we can add the constraint:
40x + 25y = 4000
Therefore, the pair of linear equations that model this situation are:
40x = Amount spent on excellent grades gift cards
25y = Amount spent on perfect attendance gift cards
40x + 25y = 4000
To write the pair of linear equations that model this situation, we need to consider the two types of gift cards: one for excellent grades and one for perfect attendance. Let's represent the number of gift cards for excellent grades as "x" and the number of gift cards for perfect attendance as "y".
Since the total number of gift cards to be given out is 100, the first equation can be written as:
x + y = 100
Now, let's consider the value of the gift cards. The gift cards for excellent grades are worth $40 each, so the total value of the excellent grade gift cards would be 40 times the number of gift cards for excellent grades (x). Similarly, the gift cards for perfect attendance are worth $25 each, so the total value of the perfect attendance gift cards would be 25 times the number of gift cards for perfect attendance (y).
Since the total allotted amount is $4,000, the second equation can be written as:
40x + 25y = 4000
Therefore, the pair of linear equations that model this situation are:
x + y = 100
40x + 25y = 4000