Question

How many schools had none of the three viz., laboratory, library or play-ground?

• A. 20
• B. 5
• C. 30
• D. 35

Hint:

Translate each sentence into equations or ratios, and solve step-by-step. Count total known groups, subtract from total to find the "none" group.

Answer 1

The Correct Option is B

Given: Total number of schools = 100
Schools with only playground = 30
These 30 schools have neither lab nor library. So, schools with no lab and no library = 30 Let us define: - Let $L_1$ = number of schools with lab only - Let $B_1$ = number of schools with library only - Let $LB$ = number of schools with both lab and library From the passage: - Schools with lab only = $L_1$ - Schools with library only = $B_1$ - Schools with both lab and library = $LB$ - Schools with either lab or library or both = 35 Now, - The number of schools with lab only = $L_1$ - The number of schools with library only = $B_1$ - The number of schools with both = $LB$ Step 1: From statement 1: $L_1 = 2B_1$ (lab only is twice library only) Step 2: From statement 2: $LB = \dfrac{1}{4}L_1$ Step 3: Use total of lab/library/both = 35 \[ L_1 + B_1 + LB = 35 \] Substitute: \[ L_1 + \frac{1}{2}L_1 + \frac{1}{4}L_1 = 35 \Rightarrow \left(1 + 0.5 + 0.25\right)L_1 = 35 \Rightarrow 1.75L_1 = 35 \Rightarrow L_1 = 20 \] Then: \[ B_1 = \frac{1}{2} \cdot 20 = 10,\quad LB = \frac{1}{4} \cdot 20 = 5 \] Step 4: Total schools accounted for: \[ \text{Lab only} = 20,\quad \text{Library only} = 10,\quad \text{Both} = 5,\quad \text{Playground only} = 30 \] Total = $20 + 10 + 5 + 30 = 65$ Step 5: Remaining schools (none of the three): \[ 100 - 65 = \boxed{5} \]