Difference between revisions of "2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1"

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== Solution==
 
== Solution==
 
Let <math>d_1</math> be the common difference in the first sequence and <math>d_2</math> the common difference in the second sequence. Thus, <math>b_4=x+4d_2</math> and <math>b_3=x+2d_2</math>. In addition, <math>a_2=x+2d_1</math> and <math>a_1=x+d_1</math>. Consequently <cmath>\frac{b_4-b_3}{a_2-a_1}=\frac{x+4d_2-x-2d_2}{x+2d_1-x-d_1}</cmath> or <cmath>\frac{b_4-b_3}{a_2-a_1}=\frac{2d_1}{d_2}</cmath> Since <math>d_2=\frac{y-x}{3}</math> and <math>d_1=\frac{y-x}{4}</math>, we have <cmath>\frac{2d_1}{d_2}=\frac{\frac{2y-2x}{3}}{\frac{y-x}{4}}</cmath> which simplifies to <math>\boxed{\frac83}</math>.
 
Let <math>d_1</math> be the common difference in the first sequence and <math>d_2</math> the common difference in the second sequence. Thus, <math>b_4=x+4d_2</math> and <math>b_3=x+2d_2</math>. In addition, <math>a_2=x+2d_1</math> and <math>a_1=x+d_1</math>. Consequently <cmath>\frac{b_4-b_3}{a_2-a_1}=\frac{x+4d_2-x-2d_2}{x+2d_1-x-d_1}</cmath> or <cmath>\frac{b_4-b_3}{a_2-a_1}=\frac{2d_1}{d_2}</cmath> Since <math>d_2=\frac{y-x}{3}</math> and <math>d_1=\frac{y-x}{4}</math>, we have <cmath>\frac{2d_1}{d_2}=\frac{\frac{2y-2x}{3}}{\frac{y-x}{4}}</cmath> which simplifies to <math>\boxed{\frac83}</math>.
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- Juno
  
 
== See also ==
 
== See also ==

Latest revision as of 13:56, 6 May 2021

Problem

Let $x \ne y$ be two real numbers. Let $x,a_1,a_2,a_3,y$ and $b_1,x,b_2,b_3,y,b_4$ be two arithmetic sequences.

Calculate $\frac{b_4-b_3}{a_2-a_1}$.

Solution

Let $d_1$ be the common difference in the first sequence and $d_2$ the common difference in the second sequence. Thus, $b_4=x+4d_2$ and $b_3=x+2d_2$. In addition, $a_2=x+2d_1$ and $a_1=x+d_1$. Consequently \[\frac{b_4-b_3}{a_2-a_1}=\frac{x+4d_2-x-2d_2}{x+2d_1-x-d_1}\] or \[\frac{b_4-b_3}{a_2-a_1}=\frac{2d_1}{d_2}\] Since $d_2=\frac{y-x}{3}$ and $d_1=\frac{y-x}{4}$, we have \[\frac{2d_1}{d_2}=\frac{\frac{2y-2x}{3}}{\frac{y-x}{4}}\] which simplifies to $\boxed{\frac83}$. - Juno

See also

2018 UNM-PNM Contest II (ProblemsAnswer KeyResources)
Preceded by
First question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10
All UNM-PNM Problems and Solutions