In Linguistics 101, the ratio of the number of juniors to the number of seniors is 3:2. When six more juniors join the class, and one senior drops the class, the ratio of the number of juniors to the number of seniors becomes 4:1. How many students are in the class after these changes?
Let's call the current number of juniors "3x" and the current number of seniors "2x" (since the ratio of juniors to seniors is 3:2).
After six more juniors join, we will have "3x + 6" juniors. And if one senior drops, we will have "2x - 1" seniors.
Now we're told that the new ratio of juniors to seniors is 4:1. This means that the number of juniors is four times the number of seniors. So:
3x + 6 = 4(2x - 1)
Expand the right side:
3x + 6 = 8x - 4
Subtract 3x from both sides:
6 = 5x - 4
Add 4 to both sides:
10 = 5x
Divide both sides by 5:
x = 2
So currently there are 3x = 6 juniors and 2x = 4 seniors.
After the changes, there will be 3x + 6 = 3(2) + 6 = 12 juniors and 2x - 1 = 4 - 1 = 3 seniors.
Altogether, there will be 12 + 3 = 15 students in the class after these changes.
Let's solve this problem step by step.
Step 1: Assign variables
Let's say the initial number of juniors is 3x, and the initial number of seniors is 2x.
Step 2: Calculate the new ratio
After six juniors join the class and one senior drops the class, the new ratio becomes 4:1. This means the number of juniors after the changes is 4x, and the number of seniors is x.
Step 3: Set up the equation
Based on the given information, we can set up the equation: 4x/x = 4/1.
Step 4: Solve the equation
To solve the equation, we cross-multiply:
4x = x * 4
4x = 4x
Step 5: Calculate the value of x
By canceling out the common term "4x" on both sides of the equation, we get x = 4.
Step 6: Calculate the number of students in the class
Now that we know the value of x, we can determine the initial number of juniors and seniors:
Initial number of juniors = 3x = 3(4) = 12
Initial number of seniors = 2x = 2(4) = 8
After the changes, the number of juniors is 4x = 4(4) = 16, and the number of seniors is x = 4.
Therefore, the total number of students in the class after these changes is 16 + 4 = 20.
So, there are 20 students in the class after these changes.