Case Study I

Some students planned a picnic. The total budget for hiring a bus was ₹ 1440. Later on, eight of them refused to go and instead paid their total share of money towards the fee of one economically weaker student of their class and thus, the cost for each member who went for picnic is increased by ₹ 30.

1. If x students planned for the picnic, then the share for hiring the bus per student who went for the picnic, was :

1. ₹ 30x

2. ₹ 1440x

3. ₹ $\dfrac{1440}{x}$

4. ₹ $\dfrac{1440}{x - 8}$

2. The algebraic representation of the given information in the form of a quadratic equation is:

1. x² − 8x − 384 = 0

2. x² + 8x − 384 = 0

3. x² − 8x − 184 = 0

4. x² + 8x − 184 = 0

3. How many students went for the picnic?

1. 24

2. 16

3. 32

4. 2

4. How much money was paid towards the fee?

1. ₹ 280

2. ₹ 340

3. ₹ 420

4. ₹ 480

5. What would be the share of each student if all the students had attended the picnic?

1. ₹ 90

2. ₹ 30

3. ₹ 60

4. none of these

1. Let x be the number of students planned a picnic.

Given,

Budget of hiring a bus = ₹ 1440

After 8 students refused, number of students those who went = x - 8

Share for hiring the bus per student who went for the picnic = $\dfrac{1440}{x - 8}$

Hence, option (4) is the correct option.

2. In first case:

Initially share per student for hiring the bus = $\dfrac{1440}{x}$

Share per student after 8 students refused to go to the picnic = $\dfrac{1440}{x - 8}$

According to question,

Cost per student increases by ₹ 30.

$\Rightarrow \dfrac{1440}{x - 8} - \dfrac{1440}{x} = 30 \\[1em] \Rightarrow \dfrac{1440x - 1440(x - 8)}{x(x - 8)} = 30 \\[1em] \Rightarrow \dfrac{1440x - 1440x + 1440 \times 8}{x^2 - 8x} = 30 \\[1em] \Rightarrow 1440 \times 8 = 30(x^2 - 8x) \\[1em] \Rightarrow \dfrac{1440 \times 8}{30} = x^2 - 8x \\[1em] \Rightarrow 384 = x^2 - 8x \\[1em] \Rightarrow 0 = x^2 - 8x - 384 \\[1em] \Rightarrow x^2 - 8x - 384 = 0$

Hence, option (1) is the correct option.

3. Solving,

⇒ x² - 8x - 384 = 0

⇒ x² - 24x + 16x - 384 = 0

⇒ x(x - 24) + 16(x - 24) = 0

⇒ (x + 16)(x - 24) = 0

⇒ (x + 16) = 0 or (x - 24) = 0     [Using zero-product rule]

⇒ x = -16 or x = 24

⇒ x = 24 [Number of students cannot be negative]

Number of students who went for picnic = x - 8 = 24 - 8 = 16.

Hence, option (2) is the correct option.

4. Share for hiring the bus per student who planned picnic = $\dfrac{1440}{x} = \dfrac{1440}{24}$ = ₹ 60.

Number of students who did not go to picnic = 8

Share of eight persons = 60 × 8 = ₹ 480.

Hence, option (4) is the correct option.

5. Share of each student (initially) = ₹ 60.

Hence, option (3) is the correct option.