GMAT Word Problems Explained — Next year, if Company Q produces x units of a certain product

Next year, if Company Q produces x units of a certain product and sells y units of this product, its total profit, in dollars, from this product will be 10 y - 4 x . If x and y are related by the equation y = r x and if Company Q's average (arithmetic mean) profit per item produced next year must be at least $4.50, what is the least possible value of r?

word problemslinear equationsinequalities

Explanation	Calculations	Help
$We translate "average profit per item produced next year must be at least 4.50" into an inequality comparing profit per item to 4.50.$	$\frac{10 y - 4 x}{x} \geq 4.50$	Theory & Tactics Method Card TRANS1-F Click to view full details
$We substitute "y = rx" into the inequality to express everything in terms of r and x.$	$\frac{10 ( r x ) - 4 x}{x} \geq 4.50$
$We simplify the numerator by factoring out x and then cancel x from numerator and denominator.$	$\frac{x ( 10 r - 4 )}{x} \geq 4.50 10 r - 4 \geq 4.50$	Theory & Tactics Method Card FDPR2-A Click to view full details
$We add 4 to both sides of the inequality to isolate the term with r.$	$10 r \geq 4.50 + 4$	Theory & Tactics Method Card INEQ0-C Click to view full details
$We divide both sides by 10 to solve for r.$	$r \geq \frac{8.50}{10} = 0.85$	Theory & Tactics Method Card INEQ1-A Click to view full details

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