Difference between revisions of "2001 AMC 12 Problems/Problem 3"
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{{duplicate|[[2001 AMC 12 Problems|2001 AMC 12 #3]] and [[2001 AMC 10 Problems|2001 AMC 10 #9]]}} | {{duplicate|[[2001 AMC 12 Problems|2001 AMC 12 #3]] and [[2001 AMC 10 Problems|2001 AMC 10 #9]]}} | ||
− | == Problem == | + | ==Problem== |
The state income tax where Kristin lives is levied at the rate of <math>p\%</math> of the first | The state income tax where Kristin lives is levied at the rate of <math>p\%</math> of the first | ||
<math>\textdollar 28000</math> of annual income plus <math>(p + 2)\%</math> of any amount above <math>\textdollar 28000</math>. Kristin | <math>\textdollar 28000</math> of annual income plus <math>(p + 2)\%</math> of any amount above <math>\textdollar 28000</math>. Kristin | ||
noticed that the state income tax she paid amounted to <math>(p + 0.25)\%</math> of her | noticed that the state income tax she paid amounted to <math>(p + 0.25)\%</math> of her | ||
annual income. What was her annual income? | annual income. What was her annual income? | ||
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<math>\text{(A)}\,\textdollar 28000 \qquad \text{(B)}\,\textdollar 32000 \qquad \text{(C)}\,\textdollar 35000 \qquad \text{(D)}\,\textdollar 42000 \qquad \text{(E)}\,\textdollar 56000</math> | <math>\text{(A)}\,\textdollar 28000 \qquad \text{(B)}\,\textdollar 32000 \qquad \text{(C)}\,\textdollar 35000 \qquad \text{(D)}\,\textdollar 42000 \qquad \text{(E)}\,\textdollar 56000</math> | ||
− | == Solution== | + | ==Solution== |
− | + | ===Solution 1=== | |
− | === Solution 1 === | ||
− | |||
Let the income amount be denoted by <math>A</math>. | Let the income amount be denoted by <math>A</math>. | ||
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− | === Solution 2 === | + | ===Solution 2=== |
Let <math>A</math>, <math>T</math> be Kristin's annual income and the income tax total, respectively. Notice that | Let <math>A</math>, <math>T</math> be Kristin's annual income and the income tax total, respectively. Notice that |
Revision as of 20:20, 16 March 2020
- The following problem is from both the 2001 AMC 12 #3 and 2001 AMC 10 #9, so both problems redirect to this page.
Contents
[hide]Problem
The state income tax where Kristin lives is levied at the rate of of the first of annual income plus of any amount above . Kristin noticed that the state income tax she paid amounted to of her annual income. What was her annual income?
Solution
Solution 1
Let the income amount be denoted by .
We know that .
We can now try to solve for :
So the answer is
Solution 2
Let , be Kristin's annual income and the income tax total, respectively. Notice that We are also given that Thus, Solve for to obtain .
~ Nafer
See Also
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.