Difference between revisions of "1957 AHSME Problems/Problem 3"
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| − | + | == Problem 3== | |
| + | |||
| + | The simplest form of <math>1 - \frac{1}{1 + \frac{a}{1 - a}}</math> is: | ||
| + | |||
| + | <math>\textbf{(A)}\ {a}\text{ if }{a\not= 0} \qquad \textbf{(B)}\ 1\qquad | ||
| + | \textbf{(C)}\ {a}\text{ if }{a\not=-1}\qquad | ||
| + | \textbf{(D)}\ {1-a}\text{ with not restriction on }{a}\qquad | ||
| + | \textbf{(E)}\ {a}\text{ if }{a\not= 1} </math> | ||
| + | |||
| + | ==Solution== | ||
| + | We have <math>1 - \frac{1}{1 + \frac{a}{1 - a}} = 1 - \frac{1}{\frac{1}{1-a}} = 1 - \frac{1-a}{1} = a</math> for almost all <math>a</math>. However, the first step is invalid when <math>a=1</math>, and each step is valid otherwise, so the answer is (E). | ||
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| + | ==See Also== | ||
| + | |||
| + | {{AHSME 50p box|year=1957|num-b=2|num-a=4}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 08:04, 25 July 2024
Problem 3
The simplest form of 
is:
Solution
We have 
for almost all 
. However, the first step is invalid when 
, and each step is valid otherwise, so the answer is (E).
See Also
| 1957 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
