Question

The number of students attending an eastern university at one time was 3,000 more than the number attending a western university, and 1/2 the number in the eastern university plus 1/3 the number in the western university equaled 6,500. How many were attending each university.

Please explain STEP-by-STEP everything from the creating the formula to the answer. Thanks

Answer 1

First, you need to define variables for your unknowns.

Let e = numbers of student attending eastern university
Let w = number of students attending western university

Answer 2

Next, you need to write some mathematical relationships of the variables from the problem statement. Since you have two unknowns, you will need two equations in order to solve for them.

Answer 3

Translate "The number of students attending an eastern university at one time was 3,000 more than the number attending a western university" into the language of math.

We know that the number of students at e is 3000 more than at w, so we can write

e = 3000 + w

This is our first equation.

Answer 4

"1/2 the number in the eastern university plus 1/3 the number in the western university equaled 6,500" translates to:

(1/2)e + (1/3)e = 6500

Answer 5

Now we have two equations and two unknowns.

(1) e = 3000 + w
(2) = (1/2)e + (1/3)w = 6500

Answer 6

Substitute into equation 2 with what we know e is equal to.

(1/2)(3000 + w) + (1/3)w = 6500

Distribute

1500 + (1/2)w + (1/3)w = 6500

I like to get rid of the fractions, so I multiply everything by 6.

9000 + 3w + 2w = 39000

Combine like terms and solve for w.

9000 + 5w = 39000
5w = 30000
w = 6000

We know that e = 3000 + w
so
e = 3000 + 6000 = 9000

Answer 7

We can now check these against the original problem statement to make sure they make sense.

9000 students at eastern is 3000 more than at the western (6000).

1/2 the eastern students = 4500
1/3 the western students = 2000
those added = 6500

Answer 8

Thnaks:D