There are 120 students in a test, and their average score is 70. The average score of the passed students is 78, and the average score of the failed students is 30. Find the number of failed students.

1. Problem Description

There are 120 students in a test, and their average score is

7

0. The average score of the passed students is 78, and the average score of the failed students is

3

0. Find the number of failed students.

2. Solution Steps

Let $n$ be the total number of students.

Let $n_p$ be the number of passed students, and $n_f$ be the number of failed students.

Let $a$ be the average score of all students.

Let $a_p$ be the average score of passed students, and $a_f$ be the average score of failed students.

We are given:

$n = 120$

$a = 70$

$a_p = 78$

$a_f = 30$

We know that $n = n_p + n_f$, so $n_p = n - n_f = 120 - n_f$.

The sum of the scores of all students is $n \cdot a = 120 \cdot 70 = 8400$.

The sum of scores of passed students is $n_p \cdot a_p = (120 - n_f) \cdot 78$.

The sum of scores of failed students is $n_f \cdot a_f = n_f \cdot 30$.

The sum of scores of all students equals the sum of scores of passed students and failed students.

$n \cdot a = n_p \cdot a_p + n_f \cdot a_f$

$8400 = (120 - n_f) \cdot 78 + n_f \cdot 30$

$8400 = 120 \cdot 78 - 78 n_f + 30 n_f$

$8400 = 9360 - 48 n_f$

$48 n_f = 9360 - 8400$

$48 n_f = 960$

$n_f = \frac{960}{48}$

$n_f = 20$

3. Final Answer

The number of failed students is 20.