Practice Problems

Problem

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Solutions

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To solve this problem, we need to place the information given into the Venn diagram, which will show how many students have which apps. Here's how we can do it:

1. Start with the center (All three apps):

• 8 students have all three apps ($WhatsApp, Instagram,$ and $Snapchat$). This number goes right in the center where all three circles overlap.

2. WhatsApp and Instagram (but not Snapchat):

• 14 students have both $WhatsApp$ and Instagram. Since 8 of these students already use all three apps, we subtract the 8 from 14 to find out how many students only have $WhatsApp$ and $Instagram$.
• $14 - 8 = :success[6]$
• So, 6 students only have $WhatsApp$ and $Instagram$ (but not $Snapchat$), and this number goes in the overlap between the $WhatsApp$ and $Instagram$ circles, but outside the $Snapchat$ circle.

3. Instagram and Snapchat (but not WhatsApp):

• 24 students have both $Instagram$ and $Snapchat$. Since 8 of these students already use all three apps, we subtract the 8 from 24 to find out how many students only have $Instagram$ and $Snapchat$.
• $24 - 8 = :success[16]$
• So, 16 students only have $Instagram$ and $Snapchat$ (but not $WhatsApp$), and this number goes in the overlap between the $Instagram$ and $Snapchat$ circles, but outside the $WhatsApp$ circle.

4. WhatsApp and Snapchat (but not Instagram):

• We are told that x students have $WhatsApp$ and $Snapchat$, but not $Instagram$. This number x goes in the overlap between the $WhatsApp$ and $Snapchat$ circles, but outside the $Instagram$ circle.

5. Total number of students using each app:

• $WhatsApp$: 36 students have WhatsApp in total.

• $Instagram$: 40 students have Instagram in total.

• $Snapchat$: 54 students have Snapchat in total. These totals include all the students who use WhatsApp, Instagram, and Snapchat in any combination (including those who use two or three apps). To find out how many students only use one app, we will need to consider the other overlaps we've already identified.

• Only WhatsApp: The number of students who only have WhatsApp is calculated by subtracting the number of students who use WhatsApp in combination with Instagram, Snapchat, or both from the total number of WhatsApp users. $\text{Only WhatsApp} = 36 - (6 + 8 + x) = :success[22 - x]$

• Only Instagram: The number of students who only have Instagram is calculated by subtracting those who use Instagram in combination with the other apps. $\text{Only Instagram} = 40 - (6 + 8 + 16) = :success[10]$

• Only Snapchat: The number of students who only have Snapchat is found similarly: $\text{Only Snapchat} = 54 - (16 + 8 + x) = :success[30 - x]$

In summary, the Venn diagram will be filled as follows:

In Part (ii), we are told there are 80 students in total. This means that all the numbers in the Venn diagram should add up to 80. We use this information to find the value of x.

6. Write the equation for the total number of students: All the regions in the Venn diagram add up to the total number of students: $(22 - x) + 10 + (30 - x) + 6 + 16 + x + 8 + x = 80$

7. Simplify the equation: Start by combining like terms: $92 - x = 80$

8. Solve for $x$: To find $x$, subtract 92 from both sides of the equation: $-x = 80 - 92$ $-x = -12$

Multiply both sides by -1 to get $x$: $x = :success[12]$

So, the value of $x$ is 12.

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