Pytagoriade

Two fifth-grader teams compete in math competitions - Mathematical Olympiad and Pytagoriade. Out of 33 students in class 5A, 22 students competed in at least one of the competitions. Students who competed only in the Pytagoriade were twice as many who just competed in the Mathematical Olympiad. Students who competed in both competitions were four times more than those who only competed in Pytagoriade.

a) How many students competed in both competitions?
b) How many students competed in just one competition?
c) How many students competed in no more than one competition?

Correct answer:

a =  16
b =  6
c =  17

Step-by-step explanation:

$$\begin{gathered} p+a+o=22 \\ p=2o \\ a=4p \\ p+a+o=22 \\ p=2\cdot o \\ a=4\cdot p \\ a+o+p=22 \\ 2o-p=0 \\ a-4p=0 \\ Row3-Row1\to Row3 \\ a+o+p=22 \\ 2o-p=0 \\ -o-5p=-22 \\ Row3-\frac{-1}{2}\cdot Row2\to Row3 \\ a+o+p=22 \\ 2o-p=0 \\ -5.5p=-22 \\ p=\frac{-22}{-5.5}=4 \\ o=\frac{0+p}{2}=\frac{0+4}{2}=2 \\ a=22-o-p=22-2-4=16 \\ a=16 \\ o=2 \\ p=4 \end{gathered}$$

$b=p+o=4+2=6$

$c=33-a=33-16=17$

Try another example