SOLUTION: Chairs are arranged in the hall into 18 short rows and 7 long rows. Each long row has 18 more chairs than each short row. 1/2 of the chairs are arranged into long rows. 3/7 of the

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Chairs are arranged in the hall into 18 short rows and 7 long rows. Each long row has 18 more chairs than each short row. 1/2 of the chairs are arranged into long rows. 3/7 of the       Log On


   

Question 1205847: Chairs are arranged in the hall into 18 short rows and 7 long rows. Each long row has 18 more chairs than each short row. 1/2 of the chairs are arranged into long rows. 3/7 of the chairs are arranged into short rows. The remaining chairs are stacked in a corner.
(a) What fraction of the chairs are stacked in the corner?
(b) How many chairs are arranged in long rows?
Found 4 solutions by josgarithmetic, greenestamps, ikleyn, Edwin McCravy:
Answer by josgarithmetic(39621) About Me  (Show Source):
Answer by greenestamps(13203) About Me  (Show Source):
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1/2 of the chairs are arranged in long rows; 3/7 of them are arranged in short rows. The remainder are stacked in a corner.

1/2 + 3/7 = 7/14 + 6/14 = 13/14

(a) ANSWER: 1 - 13/14 = 1/14 of the chairs are stacked in the corner.

Using the fractions 7/14 and 6/14 from the preceding calculation....

let 7x = number of chairs in the 7 long rows
let 6x = number of chairs in the 18 short rows

Then the number of chairs in each long row is 7x/7 = x
And the number of chairs in each short row is 6x/18 = x/3

The number of chairs in each long row is 18 more than the number in each short row:

x - (x/3) = 18
(2/3)x = 18
x = 18(3/2) = 27

(b) ANSWER: the number of chairs in long rows is 7x = 7*27 = 189

Answer by ikleyn(52830) About Me  (Show Source):
You can put this solution on YOUR website!
.
Chairs are arranged in the hall into 18 short rows and 7 long rows.
Each long row has 18 more chairs than each short row.
1/2 of the chairs are arranged into long rows.
3/7 of the chairs are arranged into short rows.
The remaining chairs are stacked in a corner.
(a) What fraction of the chairs are stacked in the corner?
(b) How many chairs are arranged in long rows?
~~~~~~~~~~~~~~~~~~~~~~~~


        I use different setup comparing with that of @greenestamps. 


(a)  The answer for question (a) is simple arithmetic

         1 - 1%2F2 - 3%2F7 = write with common denominator of 14 = 1 - 7%2F14 - 6%2F14 = 1 - 13%2F14 = 1%2F14.

     We will use this value 1/14 in part (b).



(b)  Part (b) is more interesting. Its setup is quite unusual.


     Let x be the number of chairs in each long row.

     Then the number of rows in short rows is (x-18), according to the problem.


     The number of rows in all 7 long rows is 7x.

     The number of chairs in all 18 short rows is 18*(x-18).


     Since 1/2 of the chairs are arranged into long rows, the total number of chairs is twice 7x, or 14x.
     Hence, the other half is 7x, and it consists of 18*(x-18) chairs in short rows AND 1/14 of the total chairs, 14x, in the corner.


     Based on it, we can write this equation for the second half

         7x = 18*(x-18) + %281%2F14%29%2A14x.


     The setup is complete. Now simplify the last equation and find x

         7x = 18x - 324 + x

         324 = 18 + x - 7x

         324 = 12x

           x = 324/12 = 27.


     Thus we get for (b) 27 chairs in each long row and 27*7 = 189 chairs in all 7 long chairs, altogether.

Solved.



Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
x = no. of chairs in a short row.
18x = no. of chairs in all 18 short rows
x+18 = no. of chairs in each long row.
7(x+18) = 7x+126 = no. of chairs in all 7 long rows.
y = no. of chairs stacked.
Total no. of chairs = 18x+7x+126+y = 25x+y+126



system%28%0D%0A25x%2By%2B126=2%287x%2B126%29%2C%0D%0A3%2825x%2By%2B126%29=7%2818x%29%29

system%28%0D%0A25x%2By%2B126=14x%2B252%2C%0D%0A3%2825x%2By%2B126%29=126x%29%29

system%28%0D%0A11x%2By=126%2C%0D%0A25x%2By%2B126=42x%29%29

system%28%0D%0A11x%2By=126%2C%0D%0A-17x%2By=-126%29%29

Subtract the two equations:

28x=252

x=9, substitute in 11x+y=126

11(9)+y=126
99+y=126
y=27

9 chairs in each short row and 27 chairs stacked.

(a) 1-1%2F2-3%2F7+=+14%2F14-7%2F14-6%2F14=1%2F14

(b) 7x+126 = 7(9)+126 = 189 in all 7 long rows.

Edwin