Difference between revisions of "2001 AMC 12 Problems/Problem 3"

Revision as of 18:07, 30 June 2019

The following problem is from both the 2001 AMC 12 #3 and 2001 AMC 10 #9, so both problems redirect to this page.

Problem

The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $\textdollar 28000$ of annual income plus $(p + 2)\%$ of any amount above $\textdollar 28000$ . Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income?

$\text{(A)}\,\textdollar 28000 \qquad \text{(B)}\,\textdollar 32000 \qquad \text{(C)}\,\textdollar 35000 \qquad \text{(D)}\,\textdollar 42000 \qquad \text{(E)}\,\textdollar 56000$

Solution

Solution 1

Let the income amount be denoted by $A$ .

We know that $\frac{A(p+.25)}{100}=\frac{28000p}{100}+\frac{(p+2)(A-28000)}{100}$ .

We can now try to solve for $A$ :

$(p+.25)A=28000p+Ap+2A-28000p-56000$

$.25A=2A-56000$

$A=32000$

So the answer is $\boxed{B}$

Solution 2

Let $A$ , $T$ be Kristin's annual income and the income tax total, respectively. Notice that $\begin{align*} T &= p\%\cdot28000 + (p + 2)\%\cdot(A - 28000) \\ &= [p\%\cdot28000 + p\%\cdot(A - 28000)] + 2\%\cdot(A - 28000) \\ &= p\%\cdot A + 2\%\cdot(A - 28000) \end{align*}$ We are also given that $T = (p + 0.25)\%\cdot A = p\%\cdot A + 0.25\%\cdot A$ Thus, $p\%\cdot A + 2\%\cdot(A - 28000) = p\%\cot A + 0.25\%\cdot A$ $2\%\cdot(A - 28000) = 0.25\%\cdot A$ Solve for $A$ to obtain $A = 32000$ . $\boxed{B}$

~ Nafer

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

2001 AMC 10 (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 35: / Line 35: @@
 &= p\%\cdot A + 2\%\cdot(A - 28000)
 \end{align*}</cmath>
-We are also given that <cmath>T = (p + 0.25)\%\cdot A = p\%\cot A + 0.25\%\cdot A</cmath>
+We are also given that <cmath>T = (p + 0.25)\%\cdot A = p\%\cdot A + 0.25\%\cdot A</cmath>
 Thus, <cmath>p\%\cdot A + 2\%\cdot(A - 28000) = p\%\cot A + 0.25\%\cdot A</cmath> <cmath>2\%\cdot(A - 28000) = 0.25\%\cdot A</cmath>
 Solve for <math>A</math> to obtain <math>A = 32000</math>. <math>\boxed{B}</math>

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