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# A carpenter was contracted to repair a kitchen for \$18000, the work took him 6 hours more than he has estimated and as a consequence, he earned \$100 per hour less than he had estimated. how many hours had he estimated?

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I will try to solve it systematically so it will be easier to understand. Note: The solution involves solving a quadratic equation.

Let amount of hours estimated be x:

Expected rate: \$18000/(x hours)

Time spent extra: 6 hours.

Totals hours spent: (x + 6) hours

New rate: \$18000/(x+6 hours)

Now:

18000/(x+6) x

The new rate is less than old rate by \$100, therefore we can make the above inequality equality by adding \$100 to the new rate.

18000/(x+6) + 100 = 18000/x

Solve by cross multiplaction:

(18000 + 100(x+6) * x = 18000 * (x+6)

18000x + 100x(x+6) = 18000x + 108000

Simplify:

100x(x+6) = 108000 (cancel 18000x on both sides)

x(x+6) = 1080 (divide by 100)

x^2 + 6x - 1080 = 0

Solve the above quadratic equation by any method (completing square, factorization or quadratic formula)

x = (-b+-(b^2 - 4ac)^.5)/2a where a = 1; b = 6; c = -1080

x = 30 or -36 (hours cannot be negative so disregard this result)

Therefore x = Estimated time = 30 hours

Check:

18000/30 = \$600 per hour x+6 = 36 \$18000/36 = \$500 or \$100 per hour less than estimated

Posted on Nov 13, 2010

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