1. ## percents and fractions

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

2. Originally Posted by nancymcwilliams
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 $\displaystyle 20\% = \dfrac15$

$\displaystyle \left(\dfrac15 + \dfrac14 + \dfrac18\right)\cdot x+51=\dfrac23 x$

$\displaystyle \dfrac{23}{60} \cdot x +51=\dfrac23 \cdot x$

$\displaystyle 51=\dfrac{17}{60} x~\implies~x=\dfrac{51 \cdot 60}{17}=180\ seats$

3. I really do not understand where some of the numbers came from in your answer

4. Originally Posted by nancymcwilliams
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
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.

5. Originally Posted by nancymcwilliams
I really do not understand where some of the numbers came from in your answer
Not sure which numbers are not clear to you I'll show you where I've got the numbers from:

$\displaystyle \overbrace{\left(\underbrace{\dfrac15}_{boys} + \underbrace{\dfrac14}_{women} + \underbrace{\dfrac18}_{girls}\right)}^{fractions\ of\ all\ seats}\cdot x+\underbrace{51}_{men}=\underbrace{\dfrac23 x}_{occupied\ seats}$

6. Originally Posted by earboth
Not sure which numbers are not clear to you I'll show you where I've got the numbers from:

$\displaystyle \overbrace{\left(\underbrace{\dfrac15}_{boys} + \underbrace{\dfrac14}_{women} + \underbrace{\dfrac18}_{girls}\right)}^{fractions\ of\ all\ seats}\cdot x+\underbrace{51}_{men}=\underbrace{\dfrac23 x}_{occupied\ seats}$
i think the question says 20%of the passengers not 20%of the total seats
The same goes to women and girls in which case supposing x=total seats then
the no of the passengers is 2x/3 then 20% of the passengers=1/5 of 2x/3
pardon me if i am wrong though!!

7. Originally Posted by syathish
i think the question says 20%of the passengers not 20%of the total seats
The same goes to women and girls in which case supposing x=total seats then
the no of the passengers is 2x/3 then 20% of the passengers=1/5 of 2x/3
pardon me if i am wrong though!!
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

8. It is the number of seats not passengers but i really appreaciate both answers. together I was able to figure the problem out.

9. Originally Posted by earboth
Let x denote the total number of seats in the plane.
Translate the sentence into numbers and operations. Keep in mind that $\displaystyle 20\% = \dfrac15$

$\displaystyle \left(\dfrac15 + \dfrac14 + \dfrac18\right)\cdot x+51=\dfrac23 x$

$\displaystyle \dfrac{23}{60} \cdot x +51=\dfrac23 \cdot x$

$\displaystyle 51=\dfrac{17}{60} x~\implies~x=\dfrac{51 \cdot 60}{17}=180\ seats$

should be $\displaystyle \left(\dfrac15 + \dfrac14 + \dfrac18\right)\cdot \dfrac23x+51=\dfrac23 x$