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# A reading room is half full it becomes 1/3 full when 10 people have left the room. Determine how many people the room holds when full

It's math smart I need help with some questions

### 2 Answers by Expert Tutors

Nathan B. | Elementary and Algebraic skilledElementary and Algebraic skilled
5.0 5.0 (18 lesson ratings) (18)
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Let p = people in the room and r = room capacity. Here's what we know:
p = 1/2 r (the original # of people filled to half capacity)
p - 10 = 1/3 r (after 10 people left, it's now at 1/3 capacity)

We have a value for a variable, so we can plug it in:
1/2 r - 10 = 1/3 r
Subtract 1/2 r from both sides:
-10 = 1/3 r - 1/2 r
Find the LCD:
-10 = 2/6 r - 3/6 r
-10 = -1/6 r
Multiply both sides by -1/6:
60 = r

Check:
p = 60 / 2 (the half capacity at the start)
p = 30

30 - 10 = 60 / 3 (the half-cap minus 10 should equal one-third of the full-cap)
20 = 20
Marissa S. | Enthusiastic teacher with a passion for sharing knowledge.Enthusiastic teacher with a passion for ...
5.0 5.0 (1 lesson ratings) (1)
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1/2 full
1/3 full=1/2 -10 people
therefore, x/3 +10 =x/2
x/2 - x/3=10

x/2 = 3x/6 and x/3 = 2x/6
so x/2 - x/3 =3x/6 - 2x/6=x/6 ; then,
x/6=10 so, x=60
Therefore, it holds 60 people.
To check: half of 60 is 30, if 10 people leave that's 20, one third of 60 is 60/3  which =20 so it's correct!

Hope this helped!