Question

Some members of a social service organization in Kolkata decide to prepare 2400 laddoos to gift to children in various orphanages and slums in the city, during Durga Puja. The plan is that each of them makes the same number of laddoos. However, on laddoo-making day, ten members are absent, thus each remaining member makes 12 laddoos more than earlier decided.
How many members actually make the laddoos?

• A. 50
• B. 90
• C. 40
• D. 24
• E. 100

Answer 1

The Correct Option is C

To solve the problem, let's define the variables and break down the situation step by step:

1. Let the original number of members in the organization be \(x\).
2. Each member was supposed to make an equal number of laddoos, and the total is 2400 laddoos. Thus, originally, each member would make \(\frac{2400}{x}\) laddoos.
3. On the laddoo-making day, 10 members are absent. Thus, the number of members actually making laddoos becomes \(x - 10\).
4. Each remaining member now makes 12 more laddoos than originally planned. Hence, each member makes \(\left(\frac{2400}{x} + 12\right)\) laddoos.
5. Since the total number of laddoos remains 2400, the expression for the laddoos made by the present members is:
6. \[ (x - 10) \times \left(\frac{2400}{x} + 12\right) = 2400 \]
7. Simplify the equation:
   • Multiply out: \[ (x - 10) \times \frac{2400}{x} + (x - 10) \times 12 = 2400 \]
   • Distribute and simplify: \[ 2400 - \frac{24000}{x} + 12x - 120 = 2400 \]
   • Combine terms: \[ 12x - \frac{24000}{x} - 120 = 0 \]
   • Multiply through by \(x\) to clear the fraction: \[ 12x^2 - 24000 - 120x = 0 \]
8. Rearrange and simplify further: \[ 12x^2 - 120x - 24000 = 0 \]
9. To simplify the equation, divide the entire equation by 12: \[ x^2 - 10x - 2000 = 0 \]
10. This is a quadratic equation, which can be solved using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where \(a = 1\), \(b = -10\), and \(c = -2000\).
11. Calculate the discriminant: \[ b^2 - 4ac = (-10)^2 - 4 \times 1 \times (-2000) = 100 + 8000 = 8100 \]
12. Calculate the roots: \[ x = \frac{10 \pm \sqrt{8100}}{2} \] \[ x = \frac{10 \pm 90}{2} \]
13. This gives two possible values for \(x\): \[ x = \frac{100}{2} = 50 \quad \text{or} \quad x = \frac{-80}{2} = -40 \]

Since the number of members cannot be negative, the only plausible solution is \(x = 50\). However, since 10 members are absent, the number of members who actually make the laddoos is \(x - 10 = 40\). Hence, the correct answer is 40.

Answer 2

Let the total number of members in the organization be N.

If all N members were present, each member would have made x laddoos.

Since 2400 laddoos are made, the equation is: 
N×x=2400

When 10 members are absent, the number of members making laddoos is N-10.

Each remaining member makes 12 more laddoos, thus making x+12 laddoos.

The equation becomes:
(N-10)×(x+12)=2400

Since both expressions equal 2400, we can equate them:
Nx=(N-10)×(x+12)

Expanding the right side:
Nx-10x+12N-120

Setting the two expressions equal:
Nx=Nx-10x+12N-120

Cancel Nx from both sides:
0=N=10x+12N-120

Simplify the equation:
10x=120-12N

This implies:
x=(120-12N)/10

From the number of total laddoos:
Nx=2400

Substituting the value of x in the equation:
N×(120-12N)⁄10=2400

Simplifying gives:
120N-12N×N=24000

Dividing throughout by 12:
10N-N=2000

This simplifies to:
2N=200

Divide by 2:
N=40

Thus, 40 members actually make the laddoos.