$16 \mathrm{Oz}$ Jeans has factories in Mydney and Selbourne. At the Mydney factory, fixed costs are $\$ 28000$ per month and the cost of producing each pair of jeans is $\$ 30$ . At the Selbourne factory, fixed costs are $\$ 35200$ per month and the cost of producing each pair of jeans is $\$ 24$ . During the next month $\mathrm{Oz}$ Jeans must manufacture $6000$ pairs of jeans. Calculate the production order for each factory, if the total manufacturing costs for each factory are to be the same.

Full solution

16 \mathrm{Oz}

Jeans has factories in Mydney and Selbourne. At the Mydney factory, fixed costs are

\$ 28000

per month and the cost of producing each pair of jeans is

\$ 30

. At the Selbourne factory, fixed costs are

\$ 35200

per month and the cost of producing each pair of jeans is

\$ 24

. During the next month

\mathrm{Oz}

Jeans must manufacture

6000

pairs of jeans. Calculate the production order for each factory, if the total manufacturing costs for each factory are to be the same.

Define Variables: Let's call the number of jeans produced in Mydney $x$ and in Selbourne $y$ . We know that $x + y = 6000$ because the total number of jeans to be produced is $6000$ .
Calculate Total Costs: The total cost for Mydney is the fixed cost plus the variable cost of producing $x$ jeans, so it's $\$28000 + \$30x$ . The total cost for Selbourne is the fixed cost plus the variable cost of producing $y$ jeans, so it's $\$35200 + \$24y$ .
Set Cost Equations Equal: We want the costs to be the same, so we set the two expressions equal to each other: $\newline$ $28000 + 30x = 35200 + 24y$
Solve for y: Now we solve for one of the variables. Let's solve for y in terms of x: $\newline$ $30x - 24y = 35200 - 28000$ $\newline$ $30x - 24y = 7200$
Simplify Equation: Divide everything by $\$6$ to simplify: \$$5x - \$\(4$y = \$$1200$
Eliminate Variable y: Now we remember that $x + y = 6000$. Let's multiply this equation by \$(\$)$4$ to help us eliminate y:$\newline$\$(\$)$4$x + \$(\$)$4$y = \$(\$)$24000$
Find Value of x: Add this to the $5x - 4y = 1200$ equation:$\newline$$5x - 4y + 4x + 4y = 1200 + 24000$$\newline$$9x = 25200$
Substitute $x$ to Find $y$: Divide by $\$9$ to find $x$:\[x = \frac{\$25200}{\$9}\]\[x = 2800\]
Substitute $x$ to Find $y$: Divide by $\$9$ to find $x$:
$x = \frac{\$25200}{\$9}$
$x = 2800$Now we substitute $x$ back into $x + y = 6000$ to find $y$:
$2800 + y = 6000$
$y$$0$
$y$$1$