Matematică >> Procente >> 9
\( \definecolor{red}{RGB}{255,0,0} \definecolor{blue}{RGB}{0,0,255} \definecolor{black}{RGB}{0,0,0} \definecolor{grey}{RGB}{115,115,115} \definecolor{pink}{RGB}{249,76,177} \definecolor{violet}{RGB}{173,18,212} \)
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:
\(\displaystyle \color{blue}pi \color{grey}- \frac{\color{violet}p}{100} \cdot \color{blue}pi \color{grey}= \color{red}pf \)
\(\displaystyle \color{violet}100 \color{grey} \cdot \color{blue}pi \color{grey}- \color{violet}p \cdot \color{blue}pi \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}pf \)
\(\displaystyle - \color{violet}p \cdot \color{blue}pi \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}pf \color{grey}- \color{violet}100 \color{grey} \cdot \color{blue}pi \)
\(\displaystyle \color{violet}p \cdot \color{blue}pi \color{grey}= \color{violet}100 \color{grey} \cdot \color{blue}pi \color{grey}- \color{violet}100 \color{grey} \cdot \color{red}pf \)
\(\displaystyle \color{violet}p \cdot \color{blue}pi \color{grey}= \color{violet}100 \color{grey} \cdot ( \color{blue}pi \color{grey}- \color{red}pf \color{grey} ) \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \frac{\color{violet}100 \color{grey} \cdot ( \color{blue}pi \color{grey}- \color{red}pf \color{grey} )}{\color{blue}pi} \).
O altă metodă pentru a determina procentul, \( \color{violet}p \), cu care se ieftineşte un obiect,
cunoscând preţul iniţial, \( \color{blue}pi \) şi preţul final, \( \color{red}pf \),
este de a calcula \(\displaystyle \color{blue}pi \color{grey}- \color{red}pf \), adică valoarea cu care scade preţul,
care reprezintă \(\displaystyle \color{violet}p\%\) din preţul iniţial.
Putem scrie \(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey} = \color{blue}pi \color{grey}- \color{red}pf \), adică
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}pi \color{grey}= \color{blue}pi \color{grey}- \color{red}pf \), de unde
\(\displaystyle \color{violet}p \color{grey}= \frac{\color{violet}100 \color{grey} \cdot ( \color{blue}pi \color{grey}- \color{red}pf \color{grey} )}{\color{blue}pi} \).
Să se determine procentul cu care s-a ieftinit 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:
\(\displaystyle \color{blue}340 \color{grey}- \color{violet}p\%\) din \(\displaystyle \color{blue}340 \color{grey}= \color{red}289 \)
\(\displaystyle \color{blue}340 \color{grey}- \frac{\color{violet}p}{100} \cdot \color{blue}340 \color{grey}= \color{red}289 \)
\(\displaystyle \color{violet}100 \color{grey} \cdot \color{blue}340 \color{grey}- \color{violet}p \color{grey} \cdot \color{blue}340 \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}289 \)
\(\displaystyle - \color{violet}p \color{grey} \cdot \color{blue}340 \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}289 \color{grey} - \color{violet}100 \color{grey} \cdot \color{blue}340\)
\(\displaystyle \color{violet}p \color{grey} \cdot \color{blue}340 \color{grey}= \color{violet}100 \color{grey} \cdot \color{blue}340 \color{grey} - \color{violet}100 \color{grey} \cdot \color{red}289 \)
\(\displaystyle \color{blue}340 \color{violet}p \color{grey}= 34000 - 28900 \)
\(\displaystyle \color{blue}340 \color{violet}p \color{grey}= 5100 \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \color{violet}15 \).
Folosind a doua metodă, se calculează mai întâi
\(\displaystyle \color{blue}pi \color{grey}- \color{red}pf \), adică valoarea cu care scade preţul,
care reprezintă \(\displaystyle \color{violet}p\%\) din preţul iniţial.
Putem scrie \(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey} = \color{blue}pi \color{grey}- \color{red}pf \), adică
\(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}340 \color{grey} = \color{blue}340 \color{grey}- \color{red}289 \)
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}340 \color{grey}= \color{blue}340 \color{grey}- \color{red}289 \)
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}340 \color{grey}= 51 \)
\(\displaystyle \color{blue}340 \color{violet}p \color{grey}= 5100 \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \color{violet}15 \).
Preţul unui obiect este de \(16400\) lei, iar după o ieftinire preţul obiectului devine \(15416\) lei.
Procentul cu care s-a ieftinit 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:
\(\displaystyle \color{blue}16400 \color{grey}- \color{violet}p\%\) din \(\displaystyle \color{blue}16400 \color{grey}= \color{red}15416 \)
\(\displaystyle \color{blue}16400 \color{grey}- \frac{\color{violet}p}{100} \cdot \color{blue}16400 \color{grey}= \color{red}15416 \)
\(\displaystyle \color{violet}100 \color{grey} \cdot \color{blue}16400 \color{grey}- \color{violet}p \color{grey} \cdot \color{blue}16400 \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}15416 \)
\(\displaystyle - \color{violet}p \color{grey} \cdot \color{blue}16400 \color{grey}= \color{violet}100 \color{grey} \cdot \color{red}15416 \color{grey} - \color{violet}100 \color{grey} \cdot \color{blue}16400 \)
\(\displaystyle \color{violet}p \color{grey} \cdot \color{blue}16400 \color{grey}= \color{violet}100 \color{grey} \cdot \color{blue}16400 \color{grey} - \color{violet}100 \color{grey} \cdot \color{red}15416 \)
\(\displaystyle \color{blue}16400 \color{violet}p \color{grey}= 1640000 - 1541600 \)
\(\displaystyle \color{blue}16400 \color{violet}p \color{grey}= 98400 \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \color{violet}6 \).
Folosind a doua metodă, se calculează mai întâi
\(\displaystyle \color{blue}pi \color{grey}- \color{red}pf \), adică valoarea cu care scade preţul,
\(\displaystyle \color{blue}16400 \color{grey}- \color{red}15416 \color{grey} = 984 \),
care reprezintă \(\displaystyle \color{violet}p\%\) din preţul iniţial.
Putem scrie \(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}pi \color{grey} = \color{blue}pi \color{grey}- \color{red}pf \), adică
\(\displaystyle \color{violet}p\%\) din \(\displaystyle \color{blue}16400 \color{grey} = \color{blue}16400 \color{grey}- \color{red}15416 \)
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}16400 \color{grey}= \color{blue}16400 \color{grey}- \color{red}15416 \)
\(\displaystyle \frac{\color{violet}p}{100} \cdot \color{blue}16400 \color{grey}= 984 \)
\(\displaystyle \color{blue}16400 \color{violet}p \color{grey}= 98400 \)
deci procentul este:
\(\displaystyle \color{violet}p \color{grey}= \color{violet}6 \).