Difference between revisions of "2012 AMC 12B Problems/Problem 6"
Humzaiqbal (talk | contribs) (Created page with "== Problem== In order to estimate the value of x-y where x and y are real numbers with x <math>\g</math> y") |
(→Solution) |
||
(8 intermediate revisions by 7 users not shown) | |||
Line 1: | Line 1: | ||
− | == Problem== | + | == Problem == |
− | In order to estimate the value of x-y where x and y are real numbers with x <math>\ | + | In order to estimate the value of <math>x-y</math> where <math>x</math> and <math>y</math> are real numbers with <math>x>y>0</math>, Xiaoli rounded <math>x</math> up by a small amount, rounded <math>y</math> down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? |
+ | |||
+ | <math>\textbf{(A) } \text{Her estimate is larger than } x - y \qquad \textbf{(B) } \text{Her estimate is smaller than } x - y \qquad \textbf{(C) } \text{Her estimate equals } x - y</math> | ||
+ | <math>\textbf{(D) } \text{Her estimate equals } y - x \qquad \textbf{(E) } \text{Her estimate is } 0</math> | ||
+ | |||
+ | == Solution == | ||
+ | The original expression <math>x-y</math> now becomes <math>(x+k) - (y-k)=(x-y)+2k>x-y</math>, where <math>k</math> is a positive constant, hence the answer is <math>\boxed{\textbf{(A)}}</math>. | ||
+ | |||
+ | == Solution 2 == | ||
+ | The problem never says what <math>x</math> and <math>y</math> are, so we can decide what they are. Let <math>x = 1.6</math> and <math>y = 1.4</math>. We round <math>x</math> to <math>2</math> and <math>y</math> to <math>1</math>. Then the new <math>x - y = 1</math>, while the original <math>x - y = 0.2</math>. Thus, the new <math>x - y</math> is greater than the original <math>x - y</math>. The answer is <math>\boxed{\textbf{(A)}}</math>. ~Extremelysupercooldude | ||
+ | |||
+ | == See Also == | ||
+ | {{AMC12 box|year=2012|ab=B|num-b=5|num-a=7}} | ||
+ | {{MAA Notice}} |
Latest revision as of 07:32, 29 June 2023
Contents
Problem
In order to estimate the value of where
and
are real numbers with
, Xiaoli rounded
up by a small amount, rounded
down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?
Solution
The original expression now becomes
, where
is a positive constant, hence the answer is
.
Solution 2
The problem never says what and
are, so we can decide what they are. Let
and
. We round
to
and
to
. Then the new
, while the original
. Thus, the new
is greater than the original
. The answer is
. ~Extremelysupercooldude
See Also
2012 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.