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

(Solution 2)
(Solution 2)
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=== Solution 2 ===
 
=== Solution 2 ===
  
Let <math>A</math> be Kristin's annual income. Notice that <cmath>p\%\cdot28000 + (p + 2)\%\cdot(A - 28000)</cmath> <cmath>= [p\%\cdot28000 + p\%\cdot(A - 28000)] + 2\%\cdot(A - 28000)</cmath> <cmath>= p\%\cdot A + 2\%\cdot(A - 28000)</cmath>
+
Let <math>A</math>, <math>T</math> be Kristin's annual income. Notice that  
 +
<cmath>\begin{align*}
 +
p\%\cdot28000 + (p + 2)\%\cdot(A - 28000)</cmath> <cmath>= [p\%\cdot28000 + p\%\cdot(A - 28000)] + 2\%\cdot(A - 28000)</cmath> <cmath>= p\%\cdot A + 2\%\cdot(A - 28000)</cmath>
 +
 
 +
\begin{align*}
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((2x+3)^3)' &= 3(2x+3)^2 \cdot (2x+3)' \
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&= 3(2x+3)^2 \cdot 2 \
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&= 6(2x+3)^2.
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\end{align*}
  
 
== See Also ==
 
== See Also ==

Revision as of 17:55, 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. Notice that

\begin{align*}
p\%\cdot28000 + (p + 2)\%\cdot(A - 28000) (Error compiling LaTeX. Unknown error_msg)

\[= [p\%\cdot28000 + p\%\cdot(A - 28000)] + 2\%\cdot(A - 28000)\] \[= p\%\cdot A + 2\%\cdot(A - 28000)\]

((2x+3)3)=3(2x+3)2(2x+3)=3(2x+3)22=6(2x+3)2.

See Also

2001 AMC 12 (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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

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