Foizlar

⭐ — imtihonda eng ko'p ishlatiladigan formulalar.

Asosiy ta'riflar

• ⭐ $1\% = \dfrac{1}{100} = 0{,}01$. $p\%$ — sonning $\dfrac{p}{100}$ qismi.
• Foizdan kasrga: $25\% = \dfrac{25}{100} = \dfrac{1}{4}$. Kasrdan foizga: $\dfrac{3}{5} = 0{,}6 = 60\%$.

Uch asosiy masala

• ⭐ Sonning foizini topish: $A$ ning $p\%$ i $= A \cdot \dfrac{p}{100}$. Masalan, $240$ ning $15\%$ i $= 36$.
• ⭐ Foiz bo'yicha sonni topish: $A$ sonning $p\%$ i $B$ bo'lsa, $A = B \cdot \dfrac{100}{p}$. Masalan, $15\%$ i $36$ bo'lgan son: $36 \cdot \dfrac{100}{15} = 240$.
• ⭐ Necha foiz ekanini topish: $B$ soni $A$ ning necha foizi: $\dfrac{B}{A} \cdot 100\%$. Masalan, $36$ soni $240$ ning $\dfrac{36}{240} \cdot 100\% = 15\%$ i.

O'zgarish

• ⭐ $p\%$ ga oshgan son: $A\left(1 + \dfrac{p}{100}\right)$. $p\%$ ga kamaygan son: $A\left(1 - \dfrac{p}{100}\right)$.
• Foizli o'zgarish: $\dfrac{\text{yangi} - \text{eski}}{\text{eski}} \cdot 100\%$.
• ⭐ Chegirma: yangi narx $= \text{narx} \cdot \left(1 - \dfrac{p}{100}\right)$.
• Ketma-ket o'zgarishlar ko'paytiriladi, qo'shilmaydi: $p\%$ ga, keyin $q\%$ ga oshsa, umumiy koeffitsient $\left(1 + \dfrac{p}{100}\right)\left(1 + \dfrac{q}{100}\right)$.
• ⚠️ $20\%$ ga oshgan son keyin $20\%$ ga kamaysa, asl songa qaytmaydi: $1{,}2 \cdot 0{,}8 = 0{,}96$, ya'ni $4\%$ ga kamayadi.

Foiz va foiz punkti

• Foiz punkti — foizlar farqi: $20\%$ dan $25\%$ ga oshish $5$ foiz punkti, lekin $\dfrac{25 - 20}{20} \cdot 100\% = 25\%$ ga oshish.

Foizli o'sish (omonat, kredit)

• ⭐ Oddiy foiz: $n$ yildan keyin $A = A_0\left(1 + \dfrac{p \cdot n}{100}\right)$ (foiz faqat boshlang'ich summadan).
• ⭐ Murakkab foiz: $A = A_0\left(1 + \dfrac{p}{100}\right)^n$ (har yili foiz oldingi yilgi summadan hisoblanadi).
• Misol: $1\,000$ so'm yiliga $10\%$ dan $2$ yilga: oddiy foizda $1\,200$, murakkab foizda $1\,000 \cdot 1{,}1^2 = 1\,210$.

Amaliy

• Qo'shimcha qiymat solig'i, chegirma, ustama: hammasi $A \cdot (1 \pm \dfrac{p}{100})$ ko'rinishida.
• Aralashmada modda ulushi: $\dfrac{\text{modda massasi}}{\text{aralashma massasi}} \cdot 100\%$.