Difference between revisions of "1986 AJHSME Problems/Problem 16"

m (Solution)
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What we want to find is the number of hamburgers sold in the winter. Since we don't know what it is, let's call it <math>x</math>. From the graph, we know that in Spring, 4.5 million hamburgers were sold, in the Summer was 5 million and in the Fall was 4 million. We know that the number of hamburgers sold in Fall is exactly <math>\frac{1}{4}</math> of the total number of hamburgers sold, so we can say that...
 
What we want to find is the number of hamburgers sold in the winter. Since we don't know what it is, let's call it <math>x</math>. From the graph, we know that in Spring, 4.5 million hamburgers were sold, in the Summer was 5 million and in the Fall was 4 million. We know that the number of hamburgers sold in Fall is exactly <math>\frac{1}{4}</math> of the total number of hamburgers sold, so we can say that...
  
<math>4 \times Fall = Spring + Winter + Fall + Summer</math>
+
<math>4 \times \text{Fall} = \text{Spring} + \text{Winter} + \text{Fall} + \text{Summer}</math>
  
 
<math>4 \times 4 = 4.5 + 4 + x + 5</math>
 
<math>4 \times 4 = 4.5 + 4 + x + 5</math>
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<math>2.5 = x</math>
 
<math>2.5 = x</math>
  
The answer is 2.5, or A
+
The answer is 2.5, or <math>\boxed{\text{A}}</math>
  
 
==See Also==
 
==See Also==
  
 
[[1986 AJHSME Problems]]
 
[[1986 AJHSME Problems]]

Revision as of 18:47, 24 January 2009

Problem

A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $25\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter?

[asy] size(250);  void bargraph(real X, real Y, real ymin, real ymax, real ystep, real tickwidth,  string yformat, Label LX, Label LY, Label[] LLX, real[] height,pen p=nullpen) { draw((0,0)--(0,Y),EndArrow); draw((0,0)--(X,0),EndArrow); label(LX,(X,0),plain.SE,fontsize(9)); label(LY,(0,Y),plain.NW,fontsize(9)); real yscale=Y/(ymax+ystep);  for(real y=ymin; y<ymax; y+=ystep) { draw((-tickwidth,yscale*y)--(0,yscale*y)); label(format(yformat,y),(-tickwidth,yscale*y),plain.W,fontsize(9)); }  int n=LLX.length; real xscale=X/(2*n+2); for(int i=0;i<n;++i) { real x=xscale*(2*i+1); path P=(x,0)--(x,height[i]*yscale)--(x+xscale,height[i]*yscale)--(x+xscale,0)--cycle; fill(P,p); draw(P); label(LLX[i],(x+xscale/2),plain.S,fontsize(10)); } for(int i=0;i<n;++i) draw((0,height[i]*yscale)--(X,height[i]*yscale),dashed); }  string yf="%#.1f"; Label[] LX={"Spring","Summer","Fall","Winter"}; for(int i=0;i<LX.length;++i) LX[i]=rotate(90)*LX[i]; real[] H={4.5,5,4,4};  bargraph(60,50,1,5.1,0.5,2,yf,"season","hamburgers (millions)",LX,H,yellow); fill(ellipse((45,30),7,10),brown); [/asy]

$\text{(A)}\ 2.5 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 3.5 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 4.5$

Solution

What we want to find is the number of hamburgers sold in the winter. Since we don't know what it is, let's call it $x$. From the graph, we know that in Spring, 4.5 million hamburgers were sold, in the Summer was 5 million and in the Fall was 4 million. We know that the number of hamburgers sold in Fall is exactly $\frac{1}{4}$ of the total number of hamburgers sold, so we can say that...

$4 \times \text{Fall} = \text{Spring} + \text{Winter} + \text{Fall} + \text{Summer}$

$4 \times 4 = 4.5 + 4 + x + 5$

$16 = x + 13.5$

$2.5 = x$

The answer is 2.5, or $\boxed{\text{A}}$

See Also

1986 AJHSME Problems

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