The total cost C of printing a school magazine is partly constant and partly varies direct… — JAMB UTME

The total cost $C$ of printing a school magazine is partly constant and partly varies directly as the number of copies $n$ printed. If it costs $\text{N}70,000$ to print $500$ copies and $\text{N}140,000$ to print $1,200$ copies, what is the cost of printing $2,000$ copies?

A
$\text{N}200,000$
$\text{N}220,000$ Answer
C
$\text{N}240,000$
D
$\text{N}280,000$

Answer

\text{N}220,000

In partial variation, the total cost

C

is represented by

C = k_1 + k_2 n

. Subtracting the two linear equations

70,000 = k_1 + 500 k_2

and

140,000 = k_1 + 1,200 k_2

gives

700 k_2 = 70,000

, so

k_2 = 100

. Substituting

k_2 = 100

yields the fixed constant

k_1 = 20,000

. Substituting

n = 2,000

into

C = 20,000 + 100n

yields

20,000 + 200,000 = \text{N}220,000

Step-by-Step Solution

Set up the partial variation equation

C = k_1 + k_2 n

, where

k_1

is the fixed cost and

k_2

is the rate per copy

Partial variation consists of a constant term and a term that varies directly with the independent variable.

Substitute given values to form simultaneous equations

Equation 1:

70,000 = k_1 + 500 k_2

Equation 2:

140,000 = k_1 + 1,200 k_2

Using the two given data points

(n=500, C=70,000)

and

(n=1,200, C=140,000)

creates a system of linear equations.

Solve for the variation constants

k_1

and

k_2

Subtracting Equation 1 from Equation 2 gives

70,000 = 700 k_2 \implies k_2 = 100

.
Substituting

k_2 = 100

into Equation 1 gives

70,000 = k_1 + 500(100) \implies k_1 = 20,000

Determines the specific values for the fixed overhead and rate per copy.

Calculate the total cost for

2,000

copies

C = 20,000 + 100(2,000) = 20,000 + 200,000 = \text{N}220,000

Evaluates the completed formula

C = 20,000 + 100n

n = 2,000

Key Concept

Partial Variation and Simultaneous Equations

Estimated Time:1m 30s

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