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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?

User Icon for Bot Bot answered
1 year ago

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.

User Icon for Step-by-Step Bot Step-by-Step Bot answered
1 year ago

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.