one third of the people in a convention room are seated in three-fourths of the chairs. THe rest of the people in the room decide to stand. If there are 8 empty chairs, how many people are standing?
Please fully explain because i am sooo confused!!!
let there be p people, c chairs
There are 8 unused chairs. 3/4 were used, meaning the 8 chairs are 1/4 of all.
c/4 = 8
c = 32
Now,
p/3 = 3c/4
p/3 = 96/4 = 24
p = 72
check:
1/3 of 72 = 24
3/4 of 32 = 24
8 chairs left over
One-thirds of the people in a room are seated in three-fourths of the
chairs. The rest of the people are standing. If there are 6 empty chairs,
how many people have occupied the chair?
Let's break down the information given step-by-step to find the solution.
1. One-third of the people in the convention room are seated in three-fourths of the chairs.
This means that 1/3 of the total number of people in the room are seated in 3/4 of the total number of chairs. Let's denote the total number of people in the room as "P" and the total number of chairs as "C".
Therefore, the number of people seated = (1/3)P
And the number of chairs occupied = (3/4)C
2. The rest of the people in the room decide to stand.
This means that the remaining 2/3 of the total number of people in the room are standing.
3. There are 8 empty chairs.
We can use this information to form an equation. The number of empty chairs is the difference between the total number of chairs and the number of chairs occupied.
Therefore, the number of empty chairs = C - (3/4)C = (1/4)C = 8
Now, let's solve the equation to find the value of "C" and then calculate the number of people standing.
(1/4)C = 8 [since the number of empty chairs is (1/4) of the total chairs]
C = 8 * 4
C = 32
Now that we know the total number of chairs in the room is 32, we can calculate the number of people seated and standing.
Number of people seated = (1/3)P = (1/3)*P
Number of people standing = 2/3*P
Since the number of people seated in the chairs is one-third of the total people, it must be equal to (1/3)P.
(1/3)P = (3/4)C
(1/3)P = (3/4)*32
(1/3)P = 24
P = 24 * 3
P = 72
Therefore, the total number of people in the room is 72.
Number of people standing = 2/3 * P
Number of people standing = 2/3 * 72
Number of people standing = 48
So, there are 48 people standing in the room.
To find the number of people standing in the convention room, we need to calculate the total number of chairs and the number of people seated in those chairs.
Let:
- x be the total number of people in the convention room
- y be the total number of chairs in the convention room
According to the given information, one third (1/3) of the people in the room are seated in three-fourths (3/4) of the chairs. This means that (1/3)x people are seated in (3/4)y chairs.
We also know that the rest of the people in the room (2/3)x have decided to stand. Considering that there are 8 empty chairs, we can set up the following equation:
(1/3)x = (3/4)y - (1)
Since the number of empty chairs is 8, we have:
y - (1/3)x = 8 - (2)
We can now solve the system of equations (1) and (2) to find the value of x, which represents the total number of people standing in the convention room.
First, let's rearrange equation (1) to solve for y:
(3/4)y = (1/3)x
y = (4/3)(1/3)x
y = (4/9)x - (3)
Now we substitute equation (3) into equation (2):
(4/9)x - (1/3)x = 8
Combining like terms, we get:
(4/9 - 1/3)x = 8
(12/27 - 9/27)x = 8
(3/27)x = 8
(1/9)x = 8
To isolate x, we multiply both sides of the equation by 9:
x = 8 * 9
x = 72
Therefore, there are 72 people in the convention room.
To find the number of people standing, we subtract the number of people seated from the total number of people:
Number of people standing = Total number of people - Number of people seated
Number of people standing = 72 - (1/3)(72)
Number of people standing = 72 - 24
Number of people standing = 48
Hence, there are 48 people standing in the convention room.