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Difference between revisions of "1982 AHSME Problems/Problem 15"

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==Problem==
 
==Problem==
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Let <math>[z]</math> denote the greatest integer not exceeding <math>z</math>. Let <math>x</math> and <math>y</math> satisfy the simultaneous equations
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Let <math>[z]</math> denote the greatest integer not exceeding <math>z</math>. Let <math>x</math> and <math>y</math> satisfy the simultaneous equations  
  
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\begin{align*} y&=2[x]+3 \\ y&=3[x-2]+5. \end{align*}
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<cmath>\begin{align*} y&=2[x]+3 \\ y&=3[x-2]+5. \end{align*}</cmath>
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If <math>x</math> is not an integer, then <math>x+y</math> is
 
  
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<math>\text {(A) } \text{ an integer} \qquad \text {(B) } \text{ between 4 and 5} \qquad \text{(C) }\text{ between  -4 and 4}\qquad\\ \text{(D) }\text{ between 15 and 16}\qquad \text{(E) } 16.5</math>
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If <math>x</math> is not an integer, then <math>x+y</math> is
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<math>\text {(A) } \text{ an integer} \qquad  
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\text {(B) } \text{ between 4 and 5} \qquad  
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\text{(C) }\text{ between  -4 and 4}\qquad\\
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\text{(D) }\text{ between 15 and 16}\qquad
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\text{(E) } 16.5 </math>
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==Solution==
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We simply ignore the floor of <math>x</math>. Then, we have <math>y</math> = <math>2x + 3</math> = <math>3(x-2)+5</math>. Solving for <math>3x - 1 = 2x + 3</math>, we get <math>x = 4</math>. For the floor of <math>x</math>, we have <math>x</math> is between <math>4</math> and <math>5</math>. Plugging in <math>8</math> + <math>3</math> = <math>11</math> for <math>y</math>, we have <math>y = 11</math>. We have <math>11 + 4.x</math> = <math>\boxed {(D)}</math>
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~Arcticturn
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== Solution 2 (RIGID) ==
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Since <math>x</math> is not an integer, we let <math>x=a+b</math>, where <math>0<b<1</math>.
 +
 
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So <math>2[x]+3=2a+3</math>. <math>3[x-2]+5=3a-1</math>.
 +
 
 +
<math>2a+3=3a-1</math>. <math>a=4</math>. So we know that <math>x</math> is between 4 and 5. <math>y=11</math>. So <math>x+y</math> is between <math>15</math> and <math>16</math>. Select <math>\boxed{D}</math>.
 +
 
 +
~Alvie567

Latest revision as of 21:41, 16 August 2022

Problem

Let $[z]$ denote the greatest integer not exceeding $z$. Let $x$ and $y$ satisfy the simultaneous equations

\begin{align*} y&=2[x]+3 \\ y&=3[x-2]+5. \end{align*}

If $x$ is not an integer, then $x+y$ is

$\text {(A) } \text{ an integer} \qquad \text {(B) } \text{ between 4 and 5} \qquad \text{(C) }\text{ between -4 and 4}\qquad\\ \text{(D) }\text{ between 15 and 16}\qquad \text{(E) } 16.5$

Solution

We simply ignore the floor of $x$. Then, we have $y$ = $2x + 3$ = $3(x-2)+5$. Solving for $3x - 1 = 2x + 3$, we get $x = 4$. For the floor of $x$, we have $x$ is between $4$ and $5$. Plugging in $8$ + $3$ = $11$ for $y$, we have $y = 11$. We have $11 + 4.x$ = $\boxed {(D)}$

~Arcticturn

Solution 2 (RIGID)

Since $x$ is not an integer, we let $x=a+b$, where $0<b<1$.

So $2[x]+3=2a+3$. $3[x-2]+5=3a-1$.

$2a+3=3a-1$. $a=4$. So we know that $x$ is between 4 and 5. $y=11$. So $x+y$ is between $15$ and $16$. Select $\boxed{D}$.

~Alvie567

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