Oddiy kasrlar

⭐ — imtihonda eng ko'p ishlatiladigan formulalar.

Asosiy tushunchalar

• Kasr $\dfrac{a}{b}$: $a$ — surat, $b$ — maxraj ($b \ne 0$). Maxraj butun nechta teng bo'lakka bo'linganini, surat nechtasi olinganini bildiradi.
• To'g'ri kasr: $a < b$ (qiymati $1$ dan kichik). Noto'g'ri kasr: $a \ge b$.
• Aralash son va noto'g'ri kasr:

$a\frac{b}{c} = \frac{ac + b}{c} \qquad \frac{m}{n} = q\frac{r}{n} \;\;(m = nq + r)$

Kasrning asosiy xossasi

• ⭐ Surat va maxrajni bir xil nolga teng bo'lmagan songa ko'paytirish yoki bo'lish kasrning qiymatini o'zgartirmaydi:

$\frac{a}{b} = \frac{a \cdot k}{b \cdot k} = \frac{a : k}{b : k}$

• Qisqartirish: surat va maxrajni ularning EKUBiga bo'lamiz. Masalan, $\dfrac{36}{48} = \dfrac{3}{4}$.

Kasrlarni taqqoslash

• Maxrajlari bir xil bo'lsa, surati katta bo'lgan kasr katta.
• Suratlari bir xil bo'lsa, maxraji kichik bo'lgan kasr katta.
• Umumiy holda: $\dfrac{a}{b} < \dfrac{c}{d} \iff ad < bc$ ($b, d > 0$).

Amallar

• ⭐ Qo'shish va ayirish (bir xil maxrajda): $\dfrac{a}{c} \pm \dfrac{b}{c} = \dfrac{a \pm b}{c}$
• ⭐ Umumiy holda: $\dfrac{a}{b} \pm \dfrac{c}{d} = \dfrac{ad \pm bc}{bd}$ (yoki EKUK orqali umumiy maxrajga keltiriladi).
• ⭐ Ko'paytirish: $\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}$ (avval qisqartirish mumkin)
• ⭐ Bo'lish: $\dfrac{a}{b} : \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} = \dfrac{ad}{bc}$
• Butun songa ko'paytirish: $n \cdot \dfrac{a}{b} = \dfrac{na}{b}$. Butun songa bo'lish: $\dfrac{a}{b} : n = \dfrac{a}{bn}$.

Kasrni topish masalalari

• Sonning $\dfrac{m}{n}$ qismi: $A \cdot \dfrac{m}{n}$. Masalan, $60$ ning $\dfrac{3}{5}$ qismi $= 36$.
• Qismi ma'lum bo'lsa, butunni topish: $A : \dfrac{m}{n} = A \cdot \dfrac{n}{m}$. Masalan, qaysi sonning $\dfrac{3}{5}$ qismi $36$: $36 \cdot \dfrac{5}{3} = 60$.
• Teskari son: $\dfrac{a}{b}$ ning teskarisi $\dfrac{b}{a}$, ko'paytmalari $1$.