Difference between revisions of "1984 AHSME Problems/Problem 6"
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==Solution== | ==Solution== | ||
| − | From the given, we have <math> 3g=b </math> and <math> 9t=g </math>, or <math> t=\frac{g}{9} </math>. The sum of these, in terms of <math> g </math>, is <math> 3g+g+\frac{g}{9} </math>, or, with a common denominator, <math> \frac{37g}{9} </math>. We can see that this isn't one of the choices. So we write it in terms of <math> b </math>. We can see from the first equation that <math> g=\frac{b}{3} </math>, so substituting this into the expression yields <math> \frac{37b}{27}, \boxed{\text{B}} </math>. | + | From the given, we have <math> 3g=b </math> and <math> 9t=g </math>, or <math> t=\frac{g}{9} </math>. The sum of these, in terms of <math> g </math>, is <math> 3g+g+\frac{g}{9} </math>, or, with a common [[denominator]], <math> \frac{37g}{9} </math>. We can see that this isn't one of the choices. So we write it in terms of <math> b </math>. We can see from the first [[equation]] that <math> g=\frac{b}{3} </math>, so substituting this into the expression yields <math> \frac{37b}{27}, \boxed{\text{B}} </math>. |
==See Also== | ==See Also== | ||
{{AHSME box|year=1984|num-b=5|num-a=7}} | {{AHSME box|year=1984|num-b=5|num-a=7}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 12:49, 5 July 2013
Problem
In a certain school, there are 
times as many boys as girls and 
times as many girls as teachers. Using the letters 
to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the expression
Solution
From the given, we have 
and 
, or 
. The sum of these, in terms of 
, is 
, or, with a common denominator, 
. We can see that this isn't one of the choices. So we write it in terms of 
. We can see from the first equation that 
, so substituting this into the expression yields 
.
See Also
| 1984 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
