Question

The cost of hiring a conference facility for one day consists of two parties,one which is fixed and the other varies aa the number of participants n attending a conference.If kshs45000 is charged for hiring the facility for 100 participants and kshs4000for 60 participants find the number of participants if n kshs 63000 is used to hire the facility

Answer 1

To solve this problem, we need to find a relationship between the fixed cost of hiring the conference facility and the variable cost that depends on the number of participants.

Let's assume:
- The fixed cost (constant) is \( C_f \).
- The variable cost per participant is \( C_v \).
- The total cost \( C(n) \) for \( n \) participants is given by the equation:

\[
C(n) = C_f + C_v \times n
\]

We have two scenarios given in the problem:

1. For 100 participants, the cost is Kshs 45000:
\[
C(100) = C_f + C_v \times 100 = 45000 \tag{1}
\]

2. For 60 participants, the cost is Kshs 40000:
\[
C(60) = C_f + C_v \times 60 = 40000 \tag{2}
\]

We can create a system of equations from these two scenarios. First, let's subtract equation (2) from equation (1):

\[
(C_f + C_v \times 100) - (C_f + C_v \times 60) = 45000 - 40000
\]

This simplifies to:

\[
C_v \times 40 = 5000
\]

From this, we can find \( C_v \):

\[
C_v = \frac{5000}{40} = 125
\]

Now we can substitute \( C_v \) back into either equation to solve for \( C_f \). Using equation (2):

\[
C_f + 125 \times 60 = 40000
\]

Calculating \( 125 \times 60 \):

\[
C_f + 7500 = 40000
\]

Now, solving for \( C_f \):

\[
C_f = 40000 - 7500 = 32500
\]

Now we have both costs:
- Fixed cost \( C_f = 32500 \)
- Variable cost \( C_v = 125 \)

Now, we can use these to find the number of participants \( n \) if Kshs 63000 is used:

\[
C(n) = C_f + C_v \times n
\]

Setting \( C(n) = 63000 \):

\[
63000 = 32500 + 125 \times n
\]

Solving for \( n \):

\[
63000 - 32500 = 125 \times n
\]

This simplifies to:

\[
30500 = 125 \times n
\]

Now divide both sides by 125:

\[
n = \frac{30500}{125} = 244
\]

Therefore, the number of participants \( n \) if Kshs 63000 is used to hire the facility is **244 participants**.