Question

Please solve this 4 the grade algebra problem? A group of students calculated their average score at a spelling bee. They realized that if one of them scored 9 more points , their average score will be 81 points. if one of them scored 3 points less, their average score will be 78 points. How many students were in the group?

Answer 1

Let x = original average score
let y = be the number of students

We need to write two equation using the information given in the scenario in order to work them simultaneously and obtain the results.

let $\frac{yx + 9}{y} = 81$ .... (1)
     so x is the original average, so we multiply that average by the amount of     students [y × x] in order to obtain their cumulative score then you add the       nine to that score (because a student got 9 more points) [yx + 9]. Then you    divide that sum by the amount of student in  order to get the new average      which the question says would be 81

$\frac{yx - 3}{y} = 78$ ...... (2)
   so x is the original average, so we multiply that average by the amount of     students [y × x] in order to obtain their cumulative score then you subtract       three from that score since one student got three less points [yx - 3].  Then     you divide by the number of students (y) and you should 78 like the                 question says.

$\frac{yx + 9}{y} = 81$ .... (1)
$\frac{yx - 3}{y} = 78$ ...... (2)

Now simplify each equation by separating the LHS
$\frac{yx}{y} + \frac{9}{y} = 81$..... (1a)
$\frac{yx}{y} - \frac{3}{y} = 78$...... (2a)

By subtracting eq (2a) from eq (1a) in order to eliminate x
$\frac{yx}{y} - \frac{yx}{y} + \frac{9}{y} - (- \frac{3}{y}) = 81 - 78$

$\frac{12}{y} = 3$

$\frac{12}{3} = y$

⇒ y  =  4

Since y = the number of students
then  the number of students = 4