Difference between revisions of "2014 AMC 10B Problems/Problem 7"

Latest revision as of 11:03, 2 July 2023

Problem

Suppose $A>B>0$ and A is $x$ % greater than $B$ . What is $x$ ?

$\textbf {(A) } 100\left(\frac{A-B}{B}\right) \qquad \textbf {(B) } 100\left(\frac{A+B}{B}\right) \qquad \textbf {(C) } 100\left(\frac{A+B}{A}\right)\qquad \textbf {(D) } 100\left(\frac{A-B}{A}\right) \qquad \textbf {(E) } 100\left(\frac{A}{B}\right)$

Solution

We have that A is $x\%$ greater than B, so $A=\frac{100+x}{100}(B)$ . We solve for $x$ . We get

$\frac{A}{B}=\frac{100+x}{100}$

$100\frac{A}{B}=100+x$

$100\left(\frac{A}{B}-1\right)=x$

$100\left(\frac{A-B}{B}\right)=x$ . $\boxed{\left(\textbf{A}\right)}$

Solution 2

The question is basically asking the percentage increase from $B$ to $A$ . We know the formula for percentage increase is $\frac{\text{New-Original}}{\text{Original}}$ . We know the new is $A$ and the original is $B$ . We also must multiple by $100$ to get $x$ out of it's fractional/decimal form. Therefore, the answer is $100\left(\frac{A-B}{B}\right)$ or $\boxed{\text{A}}$ .

Solution 3 (Answer Choices)

Without loss of generality, let $A = 125$ and $B = 100,$ forcing $x$ to be $25$ . Plugging our values for $A$ and $B$ into these answer choices, we find that only $\boxed{\textbf{(A)}}$ returns $25$ .

Video Solution (CREATIVE THINKING)

https://youtu.be/NTqYTNTU130

~Education, the Study of Everything

Video Solution

https://youtu.be/jj_rRTuWL14

~savannahsolver

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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 13: / Line 13: @@
 <math>100\frac{A}{B}=100+x</math>
-<math>100(\frac{A}{B}-1)=x</math>
+<math>100\left(\frac{A}{B}-1\right)=x</math>
-<math>100(\frac{A-B}{B})=x</math>.  <math>\boxed{(\textbf{A})}</math>
+<math>100\left(\frac{A-B}{B}\right)=x</math>.  <math>\boxed{\left(\textbf{A}\right)}</math>
+== Solution 2==
+The question is basically asking the percentage increase from <math>B</math> to <math>A</math>. We know the formula for percentage increase is <math>\frac{\text{New-Original}}{\text{Original}}</math>. We know the new is <math>A</math> and the original is <math>B</math>. We also must multiple by <math>100</math> to get <math>x</math> out of it's fractional/decimal form. Therefore, the answer is <math>100\left(\frac{A-B}{B}\right)</math> or <math>\boxed{\text{A}}</math>.
+==Solution 3 (Answer Choices)==
+Without loss of generality, let <math>A = 125</math> and <math>B = 100,</math> forcing <math>x</math> to be <math>25</math>. Plugging our values for <math>A</math> and <math>B</math> into these answer choices, we find that only <math>\boxed{\textbf{(A)}}</math> returns <math>25</math>.
+==Video Solution (CREATIVE THINKING)==
+https://youtu.be/NTqYTNTU130
+~Education, the Study of Everything
+==Video Solution==
+https://youtu.be/jj_rRTuWL14
+~savannahsolver
 ==See Also==
 {{AMC10 box|year=2014|ab=B|num-b=6|num-a=8}}
 {{MAA Notice}}

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