## Tuesday, April 10, 2012

### this week's MGRE Math Beast Challenge

From here:

For Jack, income tax is between 15 and 35 percent of total income after an “exclusion” amount has been subtracted (that is, Jack does not have to pay any income tax on the exclusion amount, only on the remainder of his total income). If the exclusion amount is between \$5200 and \$9800, and Jack’s income tax was \$8700, which of the following could have been Jack’s total income?

(Choose all that are appropriate)

\$13,000

\$23,100

\$33,200

\$43,300

\$53,400

\$63,500

\$73,600

Go to it! I'll leave my response in the comments.

_

Kevin Kim said...

This is one of those GRE problems for which multiple responses are possible. I have a bad feeling that my own answer is wrong, because I ended up choosing four of the seven possibilities. Let me take you through my reasoning, though, and you can tell me whether I'm off my rocker.

I = total income (the thing we want to find out)

E = exclusion amount, which is a range:

\$5200 < E < \$9800

T = income tax = \$8700

I' = net income (i.e., I - E)

We can render the tax as an inequality:

0.15•I' < T < 0.35•I'

or

0.15•I' < \$8700 < 0.35•I'

which is

0.15(I - E) < \$8700 < 0.35(I - E)

It seems obvious to me that the upper-range figure for the exclusion amount should go on the left: by subtracting it from I, we end up with a smaller number, and thus we can define the lower end of the tax range. So let's plug in \$9800 for E on the left end of the inequality, and \$5200 for E on the right end:

0.15(I - 9800) < 8700 < 0.35(I - 5200)

This distributes out:

0.15I - 0.15(9800) < 8700 < 0.35I - 0.35(5200)

We can solve for I for the left-hand inequality:

0.15I - 1470 < 8700

0.15I < 10,170

I < 67,800

Be careful now: this gives us the upper range of Jack's salary, because it's the lower end of the tax spectrum. Let's turn, now, to the right-hand side of the inequality.

8700 < 0.35I - 1820

10520 < 0.35I

approx. 30,057 < I

So the range for I, if I've done this correctly, would appear to be:

\$30,057 < I < \$67,800

By that reasoning, I conclude that the answer to the question is:

\$33,200
\$43,300
\$53,400
\$63,500

If I've gone wrong somewhere, here's your chance to tell me.

Dave said...

Kevin, you are correct. Another methodology derives the same result:

Lower tax percentage of 15%:
.15 time X = 8700
X = 8700 / .15
X = 58,000

Upper tax percentage of 35%:
.35 times X = 8700
X = 8700 / .35
X = 24,857.14

Both amounts are higher than the exclusion range, so you add the total amount of the range to both the upper and lower tax rates:
9800 - 5200 = 4600

58,000 + 4,600 = 62,600
24,857.14 + 4,600 = 29,457,14

All choices between these amounts, inclusive, are possible income amounts that yield an \$8,700 tax liability.

That would be the ones you identified in your solution.

Dave

Charles said...

I just want to know where Jack lives that he has to deal with such an insane tax system.