Question

The cost for sending a message on telegram is fixed for the first 120 words and increases in proportion to the number of words exceeding 120. If the cost for sending a 150 word message is Rs 450 and for a 250 word message is Rs 750, how much would it cost to send a 400 word message?

• A. Rs 960
• B. Rs 1,200
• C. Rs 1,050
• D. Rs 1,350

Answer 1

Answer: (B)

Rs 1,200

Answer 2

Let $C_0$ be the fixed cost for the first 120 words. Let $k$ be the additional cost per word for words exceeding 120.

For a message with $w$ words, where $w > 120$, the total cost $C(w)$ is given by:

$$C(w) = C_0 + k \times (w - 120)$$

We are given the cost for two different message lengths:

1. For a 150-word message, the cost is Rs 450. Here $w = 150$, which is greater than 120.

   $$C(150) = C_0 + k \times (150 - 120) = C_0 + 30k = 450 \quad \text{(Equation 1)}$$

2. For a 250-word message, the cost is Rs 750. Here $w = 250$, which is greater than 120.

   $$C(250) = C_0 + k \times (250 - 120) = C_0 + 130k = 750 \quad \text{(Equation 2)}$$

We now have a system of two linear equations with two variables, $C_0$ and $k$:

1. $C_0 + 30k = 450$
2. $C_0 + 130k = 750$

Subtract Equation 1 from Equation 2:

$$(C_0 + 130k) - (C_0 + 30k) = 750 - 450$$ $$100k = 300$$ $$k = \frac{300}{100} = 3$$

Substitute the value of $k=3$ into Equation 1:

$$C_0 + 30(3) = 450$$ $$C_0 + 90 = 450$$ $$C_0 = 450 - 90 = 360$$

So, the fixed cost for the first 120 words is Rs 360, and the additional cost per word exceeding 120 is Rs 3.

The cost function for $w > 120$ is $C(w) = 360 + 3(w - 120)$.

We need to find the cost to send a 400-word message. Here $w = 400$, which is greater than 120.

$$C(400) = C_0 + k \times (400 - 120)$$ $$C(400) = 360 + 3 \times (400 - 120)$$ $$C(400) = 360 + 3 \times 280$$ $$C(400) = 360 + 840$$ $$C(400) = 1200$$

The cost to send a 400-word message is Rs 1200.