Directions (48-51): The scatter chart given below shows the number of boys on (Y – axis) and number of girls (X – axis) in three different class of P, Q and R.

NOTE: If there were 10 more girls in class R then the probability of selecting a girl from R would be 40%

If the number of girls in class P is 25% of the number of boys in the class P, then find the difference between the total students in class P and Q.

1. Understand and Analyze the Scatter Chart:
By reading the scatter chart from the provided image:

• Class P: Number of girls is represented on the X-axis as $a+b$, and number of boys on the Y-axis is $40$.
• Class Q: Number of girls is represented on the X-axis as $a+3b$, and number of boys on the Y-axis is $20$.
• Class R: Number of girls is represented on the X-axis as $a+5b$, and number of boys on the Y-axis is $60$.

2. Formulate Equation 1 from Class R's probability condition:
We are given that if there were $10$ more girls in class R, the probability of selecting a girl from class R would be $40\%$.
Let the initial number of girls in class R be $G_R=a+5b$ and the number of boys be $B_R=60$.
With $10$ more girls:
New number of girls = $a+5b+10$
New total number of students = $(a+5b+10)+60=a+5b+70$

Now, set up the probability equation:
$\frac{a+5b+10}{a+5b+70}=\frac{40}{100}=\frac{2}{5}$

Cross-multiply to solve:
$5(a+5b+10)=2(a+5b+70)$
$5(a+5b)+50=2(a+5b)+140$
$3(a+5b)=90$
$a+5b=30$ ---- (Equation 1)

3. Formulate Equation 2 from Class P's condition:
We are given that the number of girls in class P is $25\%$ of the number of boys in class P.
Number of boys in class P = $40$
Number of girls in class P = $25\%\text{of}40=10$
Since the number of girls in class P is represented by $a+b$:
$a+b=10$ ---- (Equation 2)

4. Solve the system of linear equations:
Subtract Equation 2 from Equation 1:
$(a+5b)-(a+b)=30-10$
$4b=20$
$b=5$

Substitute $b=5$ into Equation 2:
$a+5=10$
$a=5$

5. Find the total students in Class P and Class Q:
For Class P:

• Girls = $a+b=10$
• Boys = $40$
• Total Students in P = $10+40=50$

• Girls = $a+3b=5+3\left(5\right)=20$
• Boys = $20$ (from the chart)
• Total Students in Q = $20+20=40$

6. Calculate the difference:
Difference = $|\text{Total Students in P}-\text{Total Students in Q}|$
Difference = $50-40=10$

The difference between the total students in class P and Q is 10.