Question

A school has 10 classes with the same number of students in each class. One day, the weather was bad and many students were absent. 5 classes were half full, 3 classes were 3/4 full and 2 classes were 1/8 empty. A total of 70 students were absent. How many students are in this school when no students are absent?

Answer 1

Remark
The best way to do this is to figure out the fraction of empties and equate that number to 70.

Step One
givens 

x = number of students in each class.

Get fractions to represent the numbers who were absent.
Figure out the fractions for the empties.
5 classes. were 1/2 full. Therefore 5 classes were 1/2 empty. Each class has x students.

If 5 classes where 1/2 full, then these same classes were 1/2 empty. There are  
1 class = 1/2 x away.
5 classes = 5(1/2 x) away
5 classes = (5/2) x students away.

3 classes were 3/4 full. Therefore 3 classes are 1/4 empty
3(1/4)x = (3/4)x total number of students absent in 3 classes.


2 classes were 1/8 empty. 
2*(1/8)*x = (1/4) x

Step Two
Add your absences. Equate to 70
(1/4)x + (3/4)x + (5/2)x  = 70 

Step Three
Find your common denominator
the common denominator is 4. Convert all fractions to something over 4.
1/4  has a denominator of 4
3/4 has a denominator of 4
5/2 = 5*2/2*2 = 10/4

So the equation becomes.
(1/4)x + (3/4)x + (10/4)x = 70

Step Four
Solve for x
(1 + 3 + 10)x/4 = 70      Collect like terms on the left
(14x)/4 = 70                  Multiply by sides by 4
14x = 70 * 4
14x = 280                     Divide by 14
x = 280/14
x = 20

Answer
Each class contains 20 student.
10 classes contain 20 * 10 = 200 students.

Answer 2

Use the following equation to find the total amount of students:

$5( \frac{1}{2} )x + 3 (\frac{1}{4} )x + 2 (\frac{1}{8} )x = 70$

Convert the fractions so that they have the same denominator. Find the least common multiple of 2, 4, and 8.

Multiples of 2: {2,4,6,8}
Multiples of 4: {4,8}
Multiples of 8: {8,16}

The least common multiple in this set is 8. Multiply the fractions so that they have a denominator of 8.

$\frac{1}{2} * \frac{4}{4} = \frac{4}{8}$

$\frac{1}{4} * \frac{2}{2} = \frac{2}{8}$

Your equation should now read out as this:

$5( \frac{4}{8} )x + 3 (\frac{2}{8} )x + 2 (\frac{1}{8} )x = 70$

Multiply the fractions with their coefficients.

$\frac{5}{1} * \frac{4}{8} = \frac{20}{8}$

$\frac{3}{1} * \frac{2}{8} = \frac{6}{8}$

$\frac{2}{1} * \frac{1}{8} = \frac{2}{8}$

Your equation should now look like this:

$\frac{20}{8}x + \frac{6}{8}x + \frac{2}{8}x = 70$

Multiply all sides by 8 to get rid of the fractions.

$20x + 6x + 2x = 560$

Combine like terms.

$28x = 560$

Divide both sides by 28 to get x by itself.

$x = 20$

There are 20 students in each class.

Since there are 10 classrooms with this many students in school, multiply by 20 to find the total amount of students in school.

$20 * 10 = 200$

When nobody is absent, there are 200 students in the school.