Matematică >> Procente >> 8
\( \definecolor{red}{RGB}{255,0,0} \definecolor{blue}{RGB}{0,0,255} \definecolor{black}{RGB}{0,0,0} \definecolor{grey}{RGB}{115,115,115} \definecolor{pink}{RGB}{249,76,177} \definecolor{violet}{RGB}{173,18,212} \)
cunoscând preţul iniţial, \( \color{blue}pi \) şi preţul final, \( \color{red}pf \),
este de a rezolva ecuaţia
\(\displaystyle \color{blue}pi \color{grey}+ \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey}= \color{red}pf \), adică:
\(\displaystyle \color{blue}pi \color{grey}+ \frac{\color{violet}p}{100} \cdot \color{blue}pi \color{grey}= \color{red}pf \)
\(\displaystyle \color{violet}100 \color{grey} \cdot \color{blue}pi \color{grey}+ \color{violet}p \cdot \color{blue}pi \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}pf \)
\(\displaystyle \color{violet}p \cdot \color{blue}pi \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}pf \color{grey}- \color{violet}100 \color{grey} \cdot \color{blue}pi \)
\(\displaystyle \color{violet}p \cdot \color{blue}pi \color{grey}= \color{violet}100 \color{grey} \cdot ( \color{red}pf \color{grey}- \color{blue}pi \color{grey} ) \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \frac{\color{violet}100 \color{grey} \cdot ( \color{red}pf \color{grey}- \color{blue}pi \color{grey} )}{\color{blue}pi} \).
O altă metodă pentru a determina procentul, \( \color{violet}p \), cu care se scumpeşte un obiect,
cunoscând preţul iniţial, \( \color{blue}pi \) şi preţul final, \( \color{red}pf \),
este de a calcula \(\displaystyle \color{red}pf \color{grey}- \color{blue}pi \), adică valoarea cu care creşte preţul,
care reprezintă \(\displaystyle \color{violet}p\%\) din preţul iniţial.
Putem scrie \(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey} = \color{red}pf \color{grey}- \color{blue}pi \), adică
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}pi \color{grey}= \color{red}pf \color{grey}- \color{blue}pi\), de unde
\(\displaystyle \color{violet}p \color{grey}= \frac{\color{violet}100 \color{grey} \cdot ( \color{red}pf \color{grey}- \color{blue}pi \color{grey} )}{\color{blue}pi} \).
Să se determine procentul cu care s-a scumpit produsul.
Folosind prima metodă, trebuie rezolvată ecuaţia:
\(\displaystyle \color{blue}pi \color{grey}+ \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey}= \color{red}pf \), adică:
\(\displaystyle \color{blue}120 \color{grey}+ \color{violet}p\%\) din \(\displaystyle \color{blue}120 \color{grey}= \color{red}150 \)
\(\displaystyle \color{blue}120 \color{grey}+ \frac{\color{violet}p}{100} \cdot \color{blue}120 \color{grey}= \color{red}150 \)
\(\displaystyle \color{violet}100 \color{grey} \cdot \color{blue}120 \color{grey}+ \color{violet}p \color{grey} \cdot \color{blue}120 \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}150 \)
\(\displaystyle 12000 \color{grey}+ \color{blue}120 \color{violet}p \color{grey}= 15000 \)
\(\displaystyle \color{blue}120 \color{violet}p \color{grey}= 15000 - 12000 \)
\(\displaystyle \color{blue}120 \color{violet}p \color{grey}= 3000 \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \color{violet}25 \).
Folosind a doua metodă, se calculează mai întâi
\(\displaystyle \color{red}pf \color{grey}- \color{blue}pi \), adică valoarea cu care creşte preţul,
care reprezintă \(\displaystyle \color{violet}p\%\) din preţul iniţial.
Putem scrie \(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey} = \color{red}pf \color{grey}- \color{blue}pi \), adică
\(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}120 \color{grey} = \color{red}150 \color{grey}- \color{blue}120 \)
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}120 \color{grey}= \color{red}150 \color{grey}- \color{blue}120\)
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}120 \color{grey}= 30 \)
\(\displaystyle \color{blue}120 \color{violet}p \color{grey}= 3000 \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \color{violet}25 \).
Preţul unui obiect este de \(3200\) lei, iar după o scumpire preţul obiectului devine \(4800\) lei.
Procentul cu care s-a scumpit obiectul este:
exercițiu nou
Folosind prima metodă, trebuie rezolvată ecuaţia:
\(\displaystyle \color{blue}pi \color{grey}+ \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey}= \color{red}pf \), adică:
\(\displaystyle \color{blue}3200 \color{grey}+ \color{violet}p\%\) din \(\displaystyle \color{blue}3200 \color{grey}= \color{red}4800 \)
\(\displaystyle \color{blue}3200 \color{grey}+ \frac{\color{violet}p}{100} \cdot \color{blue}3200 \color{grey}= \color{red}4800 \)
\(\displaystyle \color{violet}100 \color{grey} \cdot \color{blue}3200 \color{grey}+ \color{violet}p \color{grey} \cdot \color{blue}3200 \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}4800 \)
\(\displaystyle 320000 \color{grey}+ \color{blue}3200 \color{violet}p \color{grey}= 480000 \)
\(\displaystyle \color{blue}3200 \color{violet}p \color{grey}= 480000 - 320000 \)
\(\displaystyle \color{blue}3200 \color{violet}p \color{grey}= 160000 \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \color{violet}50 \).
Folosind a doua metodă, se calculează mai întâi
\(\displaystyle \color{red}pf \color{grey}- \color{blue}pi \), adică valoarea cu care creşte preţul,
\(\displaystyle \color{red}4800 \color{grey}- \color{blue}3200 \color{grey} = 1600 \),
care reprezintă \(\displaystyle \color{violet}p\%\) din preţul iniţial.
Putem scrie \(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey} = \color{red}pf \color{grey}- \color{blue}pi \), adică
\(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}3200 \color{grey} = \color{red}4800 \color{grey}- \color{blue}3200 \)
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}3200 \color{grey}= \color{red}4800 \color{grey}- \color{blue}3200 \)
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}3200 \color{grey}= 1600 \)
\(\displaystyle \color{blue}3200 \color{violet}p \color{grey}= 160000 \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \color{violet}50 \).