Difference between revisions of "1989 AJHSME Problems/Problem 24"

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From here on a blue line represents a cut, the dashed line represents the fold.
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From here on, a blue line represents a cut, the dashed line represents the fold.
  
 
<asy>
 
<asy>

Revision as of 01:49, 30 November 2020

Problem

Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?

$\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}$

[asy] draw((0,0)--(0,8)--(6,8)--(6,0)--cycle); draw((0,8)--(5,9)--(5,8)); draw((3,-1.5)--(3,10.3),dashed); draw((0,5.5)..(-.75,4.75)..(0,4)); draw((0,4)--(1.5,4),EndArrow); [/asy]

Solution

From here on, a blue line represents a cut, the dashed line represents the fold.

[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle,linewidth(1)); draw((1,0)--(1,4),blue+linewidth(1)); draw((2,0)--(2,4),dashed); draw((3,0)--(3,4),blue+linewidth(1)); label("$x$",(0.5,4),N); label("$x$",(1.5,4),N); label("$x$",(2.5,4),N); label("$x$",(3.5,4),N); label("$4x$",(0,2),W); [/asy]

From the diagram, we can tell the perimeter of one of the small rectangles is $2(4x+x)=10x$ and the perimeter of the large rectangle is $2(4x+2x)=12x$. The desired ratio is \[\frac{10x}{12x}=5/6\rightarrow \boxed{\text{E}}\]

See Also

1989 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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 AJHSME/AMC 8 Problems and Solutions