Students at Apollo Elementary School receive ribbons at the end of each year at a school-wide awards ceremony. This year 120 students receive gold stars for perfect attendance, 180 receive certificates for participating in the science fair, and 80 students receive blue ribbons for outstanding grades. Of these, 40 students who receive the attendance star receive no other award, 50 students who receive the science fair certificate receive no other award. In addition, 10 students receive all three awards and 65 students receive no awards. Draw a Venn Diagram for this situation, and determine how many students attend the school this year.
Ah, the infamous Venn Diagram request! Well, who doesn't love a good diagram? Here we go:
Let's start with the three overlapping circles representing the awards: perfect attendance, science fair participation, and outstanding grades.
In the center, we have the intersection of all three circles where 10 students receive all three awards.
Now, to the left of the diagram, we'll have the region for perfect attendance. Since 40 students receive only perfect attendance stars, but don't get any other awards, we write 40 in this region.
Next, to the right of the diagram, we'll have the region for science fair participation. We know that 50 students receive only science fair certificates, so we write 50 in this region.
Lastly, below the diagram, we'll have the region for outstanding grades. We know that 80 students receive only blue ribbons for outstanding grades, so we write 80 in this region.
But wait, there's more! We still have to consider those students who didn't receive any awards. It's a sad reality, but hey, not everyone can get an award. So, we write 65 outside of all the circles to represent those students.
Now, let's do some math to determine the total number of students at the school this year.
If we add up all the numbers we have so far, we get: 120 (gold stars) + 180 (certificates) + 80 (blue ribbons) + 40 (perfect attendance only) + 50 (science fair only) + 80 (outstanding grades only) + 10 (all three awards) + 65 (no awards) = 625 students.
So, there you have it! This year, Apollo Elementary School has a total of 625 students. Keep in mind that this answer assumes everyone who attends the school is part of this award system. I hope this clarifies things for you!
To draw the Venn Diagram, we need to represent the three awards: gold stars for perfect attendance, certificates for participating in the science fair, and blue ribbons for outstanding grades. Let's label the three circles as "Attendance", "Science Fair", and "Grades".
From the given information, we know the following:
- 120 students receive gold stars for perfect attendance.
- 180 students receive certificates for participating in the science fair.
- 80 students receive blue ribbons for outstanding grades.
- 40 students who receive the attendance star receive no other awards.
- 50 students who receive the science fair certificate receive no other awards.
- 10 students receive all three awards.
- 65 students receive no awards.
Now, let's fill in the diagram step-by-step.
Step 1: Start by filling in the center region where all three circles overlap.
- 10 students receive all three awards.
Step 2: Move on to the overlapping regions between two circles.
- 40 students receive the attendance star but no other awards.
- 50 students receive the science fair certificate but no other awards.
Step 3: Fill in the circle regions that are not overlapping with any other circles.
- 65 students receive no awards.
Step 4: Fill in the remaining sections of the circles.
- 120 students receive gold stars for perfect attendance.
- 180 students receive certificates for participating in the science fair.
- 80 students receive blue ribbons for outstanding grades.
The completed Venn Diagram will look like this:
```
Attendance
40 (No other award)
+--------|--------+
| |
10 | 10 | 50
| |
+--------+-------|-- Science Fair
180 (No other award) |
|
10 | 30
+----|-------+-----+
| |
120 | 20 | 60
| |
+--------+-------|-- Grades
40 (No other award)
```
From the Venn Diagram, we can see that:
- 40 students receive the attendance star but no other awards.
- 50 students receive the science fair certificate but no other awards.
- 20 students receive both the attendance star and the science fair certificate.
- 30 students receive both the attendance star and the blue ribbon for outstanding grades.
- 10 students receive both the science fair certificate and the blue ribbon for outstanding grades.
- 10 students receive all three awards.
To determine how many students attend the school this year, we need to add up all the separate sections:
Attendance star only: 40
Science fair certificate only: 50
Grades blue ribbon only: 20
Attendance star and science fair certificate (but not grades blue ribbon): 20
Attendance star and grades blue ribbon (but not science fair certificate): 30
Science fair certificate and grades blue ribbon (but not attendance star): 10
All three awards: 10
Lastly, we need to add the number of students who receive no awards: 65
Adding them up:
40 + 50 + 20 + 20 + 30 + 10 + 10 + 65 = 255
Therefore, there are 255 students that attend Apollo Elementary School this year.
To draw the Venn diagram, we need to start by labeling the three circles. Let's label them as "Perfect Attendance," "Science Fair," and "Outstanding Grades."
Next, we can add the given information to the diagram.
- 120 students receive gold stars for perfect attendance, so we write 120 inside the Perfect Attendance circle.
- 180 students receive certificates for participating in the science fair, so we write 180 inside the Science Fair circle.
- 80 students receive blue ribbons for outstanding grades, so we write 80 inside the Outstanding Grades circle.
We also know that:
- 40 students who receive the attendance star receive no other award, so these students are only in the Perfect Attendance circle but not in any other circle.
- 50 students who receive the science fair certificate receive no other award, so these students are only in the Science Fair circle but not in any other circle.
- 10 students receive all three awards, so these students are in the overlapping area of all three circles.
Based on this information, we can update the Venn diagram. Let's denote the number of students in the overlapping area of the circles as "x."
```
-------------
| |
| |
Perfect | |
Attendance | x |
(120) | |
| |
| |
-------------
------/-------
/ \
Science / \
Fair | |
(180) | |
| x |
| (10) | |
| | |
|-------/-----------
| / \
| / |
Outstanding | / |
Grades | | |
(80) \ | |
\| |
---------------------
```
Now, let's find the number of students who receive no awards. It is stated that 65 students receive no awards, so we write 65 outside all the circles.
To find out how many students attend the school this year, we need to add up the numbers in each circle, including the overlapping area:
Number of students with Perfect Attendance = 120 + x + 40 = 160 + x
Number of students in the Science Fair = 180 + x + 50 = 230 + x
Number of students with Outstanding Grades = 80 + x
Number of students who receive no awards = 65
We know that the total number of students should be the sum of all these categories. Therefore, we can set up the equation:
Total number of students = Number of students with Perfect Attendance + Number of students in the Science Fair + Number of students with Outstanding Grades + Number of students who receive no awards
Total number of students = (160 + x) + (230 + x) + (80 + x) + 65
Simplifying the equation:
Total number of students = 475 + 3x
Now, we need more information to solve for the value of x or the total number of students attending the school this year.