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

Question

It's math smart I need help with some questions 

Answer 1

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

Answer 2

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!