Dictation

A total of 30 students wrote the dictation. One third of them got a one or a four. Four times more twos than threes. How many students have an insufficient grade, if we know that 7 students got a first grade, which is also the same number as the sum of those who got a third and a fourth? Write down the numbers of students according to their grades, all from 1 to 5...

Correct answer:

a =  7
b =  16
c =  4
d =  3
f =  0

Step-by-step explanation:


a+b+c+d+f=30
a+d=30/3
b = 4c
a=7
c+d=7

a+b+c+d+f=30
a+d=30/3
b = 4·c
a=7
c+d=7

a+b+c+d+f = 30
3a+3d = 30
b-4c = 0
a = 7
c+d = 7

Pivot: Row 1 ↔ Row 2
3a+3d = 30
a+b+c+d+f = 30
b-4c = 0
a = 7
c+d = 7

Row 2 - 1/3 · Row 1 → Row 2
3a+3d = 30
b+c+f = 20
b-4c = 0
a = 7
c+d = 7

Row 4 - 1/3 · Row 1 → Row 4
3a+3d = 30
b+c+f = 20
b-4c = 0
-d = -3
c+d = 7

Row 3 - Row 2 → Row 3
3a+3d = 30
b+c+f = 20
-5c-f = -20
-d = -3
c+d = 7

Row 5 - 1/-5 · Row 3 → Row 5
3a+3d = 30
b+c+f = 20
-5c-f = -20
-d = -3
d-0.2f = 3

Row 5 + Row 4 → Row 5
3a+3d = 30
b+c+f = 20
-5c-f = -20
-d = -3
-0.2f = 0


f = 0/-0.2 = -0
d = -3/-1 = 3
c = -20+f/-5 = -20/-5 = 4
b = 20-c-f = 20-4 = 16
a = 30-3d/3 = 30-3 · 3/3 = 7

a = 7
b = 16
c = 4
d = 3
f = -0

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