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# At his regular hourly rate, Don had estimated the labor cos

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Intern
Joined: 12 Aug 2019
Posts: 1
Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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25 Aug 2019, 13:57
converge wrote:
Let t be the hourly rate and p the price: t * p = \$336

Additional 4 hours and \$2 less per hour would yield: (t+4) * (p-2) = \$336

Since both equations are equal:

336 : p = (336 : (p-2))-4

Solving for p yields 14 (the other solution is negative, so we do not consider it)

At this point probably we are very pressed on time, so the shortcut is to find the answer the last digit of which multiplied by 4 yields 6. 14 squared is not 336 so by elimination it is 24

As the quotation above, many people handle the information from the question then end with the same quadratic equation which gives us hard time to solve. Some people solve it by mechanical work while others use the answers to plug in the equation or make guess then hit the correct choice.

I did the same way when facing this equation. However, I try to find an easier way to solve this ugly equation. After few days, I find one and want to share with you. Hope you find it useful and let me know if you find any error.

In general, the quadratic equation will be from these equations

x * y = A (1)
(x - a) * (y + b) = A (2)

(2) <=> x*y + x*b - a*y - a*b = A
<=> x*b - y*a = a*b (we have x*y=A)
<=> x/a - y/b = 1 (divide both side by ab)

Set M = x/a and N = y/b
(1) <=> M * N = A/(a*b)
(2) <=> M - N = 1

These equations are much easier to solve than the original ones. Now we come back the equations above.

t * p = \$336 (1)
(p-2) * (t+4) = \$336 (2)

Set M = p/2 and N = t/4

(1) <=> M * N = 336/(2*4)
M * N = 42
(2) M - N = 1

From these equations, we can solve M = 7 and N = 6

With N = 6 => t = 24

=========================

I want to expand a little bit. If we have quite similar equations such as

x * y = A
(x + a) * (y +b) = A

We can use the same way to break them down then solve them

x * y = A
x/a + y/b = -1

M * N = A/(a*b)
M + N = -1
with M = x/a and N = y/b
Re: At his regular hourly rate, Don had estimated the labor cos   [#permalink] 25 Aug 2019, 13:57

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