On an airplane that was 2/3 full, 20% of the passengers were boys, one-fourth of the passengers were women, one-eighth of the passengers were girls, and there were 51 men. How many seats are on the plane
Let x denote the total number of seats in the plane.
Translate the sentence into numbers and operations. Keep in mind that $\displaystyle 20\% = \dfrac15$
$\displaystyle \left(\dfrac15 + \dfrac14 + \dfrac18\right)\cdot x+51=\dfrac23 x$
$\displaystyle \dfrac{23}{60} \cdot x +51=\dfrac23 \cdot x$
$\displaystyle 51=\dfrac{17}{60} x~\implies~x=\dfrac{51 \cdot 60}{17}=180\ seats$
the first thing you want to do is convert everything to either a fraction/ratio or a number
so it becomes
an airplane 2/3 full, 20% passengers were boys, 25% women, 12.5% girls, 51 men.
now, since you need 100%, you figure out what percent you have now: 20+25+12.5 = 57.5%
so 57.5% of the passengers WERE NOT men. That means that 42.5% WERE men. Since 42.5% were men, figure out how many passengers there are altogether. A percentage can become a decimal, e.g. 1% = .01; 42.5% = .425. Therefore, 42.5% of x = 51 becomes .425x=51. Solve for x and get 120. That means there were 120 total PASSENGERS on the plane. Since the plane is 2/3 full, do 2/3 y = 120. solve for y and get 180.
So there are 180 seats on the plane.
hope this helped you out.
Not sure which numbers are not clear to you I'll show you where I've got the numbers from:
$\displaystyle
\overbrace{\left(\underbrace{\dfrac15}_{boys} + \underbrace{\dfrac14}_{women} + \underbrace{\dfrac18}_{girls}\right)}^{fractions\ of\ all\ seats}\cdot x+\underbrace{51}_{men}=\underbrace{\dfrac23 x}_{occupied\ seats}
$
You are completely right. (In my first post I made a very advantageous error so I got the correct final result) In short:
Let t denote the total number of seats and x the number of occupied seats. Then you have a system of simultaneous equations:
$\displaystyle \left|\begin{array}{r}\dfrac23 t = x \\ \dfrac15 x+\dfrac14 x + \dfrac18 x + 51 = x\end{array}\right.$
You'll get x = 120 and t = 180