Difference between revisions of "2008 AMC 8 Problems/Problem 20"
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| − | ==Problem | + | ==Problem== |
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and <math>\frac{3}{4}</math> of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? | The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and <math>\frac{3}{4}</math> of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? | ||
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\textbf{(D)}\ 27\qquad | \textbf{(D)}\ 27\qquad | ||
\textbf{(E)}\ 36</math> | \textbf{(E)}\ 36</math> | ||
| + | |||
| + | ==Solution== | ||
| + | Let <math>b</math> be the number of boys and <math>g</math> be the number of girls. | ||
| + | |||
| + | <cmath>\frac23 b = \frac34 g \Rightarrow b = \frac98 g</cmath> | ||
| + | |||
| + | For <math>g</math> and <math>b</math> to be integers, <math>g</math> must cancel out with the denominator, and the smallest possible value is <math>8</math>. This yields <math>9</math> boys. The minimum number of students is <math>8+9=\boxed{\textbf{(B)}\ 17}</math>. | ||
| + | |||
| + | ==Solution 2== | ||
| + | We know that <math>\frac23 B = \frac34 G</math> or <math>\frac69 B = \frac68 G</math>. So, the ratio of the number of boys to girls is 9:8. The smallest total number of students is <math>9 + 8 = \boxed{\textbf{(B)}\ 17}</math>. ~DY | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/HseEFQDuh_c | ||
| + | |||
| + | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2008|num-b=19|num-a=21}} | {{AMC8 box|year=2008|num-b=19|num-a=21}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 15:56, 11 January 2023
Problem
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and 
of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
Solution
Let 
be the number of boys and 
be the number of girls.
![\[\frac23 b = \frac34 g \Rightarrow b = \frac98 g\]](http://latex.artofproblemsolving.com/6/c/2/6c23a00f6e01044ecb7518140f37df28977b236d.png)
For and to be integers, must cancel out with the denominator, and the smallest possible value is 
. This yields 
boys. The minimum number of students is 
.
Solution 2
We know that 
or 
. So, the ratio of the number of boys to girls is 9:8. The smallest total number of students is 
. ~DY
Video Solution by WhyMath
~savannahsolver
See Also
| 2008 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
