7x + 6y 38,000 4x + 5y 28,000 A manufacturer wants to produce x balls and y...

Question

7x + 6y 38,000 4x + 5y 28,000 A manufacturer wants to produce x balls and y boxes. Resource constraints require that x and y satisfy the inequalities shown. What is the maximum number of balls and boxes combined that can be produced given the resource constraints?

GMAT Panda · Accepted Answer

6000 We first name the variables to match the question. x = number of balls y = number of boxes We sum the two resource constraints "7x + 6y 38000" and "4x + 5y 28000", applying Adding Two Inequalities. 7x + 6y 38000 4x + 5y 28000 11x + 11y 66000 We divide both sides of the combined inequality by 11, which is positive, so the inequality direction stays the same according to Multiplying/Dividing Inequalities by Positive. 11x + 11y/11 66000/11 x + y 6000 We verify that the bound is attainable by finding values that satisfy both original constraints. 7 2000 + 6 4000 = 38000 4 2000 + 5 4000 = 28000 x + y = 2000 + 4000 = 6000

Explanation	Calculations	Help
$We first name the variables to match the question.$	$x = number of balls y = number of boxes$
$We sum the two resource constraints "7x + 6y \leq 38000" and "4x + 5y \leq 28000", applying Adding Two Inequalities.$	$7 x + 6 y \leq 38000 4 x + 5 y \leq 28000 11 x + 11 y \leq 66000$
$We divide both sides of the combined inequality by 11, which is positive, so the inequality direction stays the same according to Multiplying/Dividing Inequalities by Positive .$	$\frac{11 x + 11 y}{11} \leq \frac{66000}{11} x + y \leq 6000$
$We verify that the bound is attainable by finding values that satisfy both original constraints.$	$7 \times 2000 + 6 \times 4000 = 38000 4 \times 2000 + 5 \times 4000 = 28000 x + y = 2000 + 4000 = 6000$

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