1. Muammo: 3 va 2,1 sonlarining o’rta arifmetigi 2,3 ga teng. $a$ ni toping.
Formulasi: O’rta arifmetik = $\frac{a + b}{2}$
Hisoblash:
$$\frac{a + 2.1}{2} = 2.3$$
$$a + 2.1 = 4.6$$
$$a = 4.6 - 2.1 = 2.5$$
Javob: $a = 2.5$ (variantlar orasida yo'q, ehtimol 2,6 yaqin javob)
2. Muammo: Bir son ikkinchi sondan 6 ta ortiq, o’rta arifmetigi 20. Kattasini toping.
Formulasi: $\frac{x + (x-6)}{2} = 20$
Hisoblash:
$$2x - 6 = 40$$
$$2x = 46$$
$$x = 23$$
Katta son: $23$
3. Muammo: 8 ta yog’ochning o’rtacha uzunligi 10 dm, bitta yog’och qo’shildi, o’rtacha 12 dm bo’ldi. Qo’shilgan yog’och uzunligi?
Formulasi: $$\frac{8 \times 10 + x}{9} = 12$$
Hisoblash:
$$80 + x = 108$$
$$x = 28$$
4. Muammo: Uchta sonning o’rta arifmetigi 17,4, ikkitasi 17,5 va 21,6. Uchinchi sonni toping.
Formulasi: $$\frac{17.5 + 21.6 + x}{3} = 17.4$$
Hisoblash:
$$39.1 + x = 52.2$$
$$x = 13.1$$
5. Muammo: Bir son ikkinchisidan 15 ga kichik, o’rta arifmetigi 11,5. Kichigini toping.
Formulasi: $$\frac{x + (x+15)}{2} = 11.5$$
Hisoblash:
$$2x + 15 = 23$$
$$2x = 8$$
$$x = 4$$
6. Muammo: Uchta sonning o’rta geometrigi 6, ikkitasi 8 va 9. Uchinchi son?
Formulasi: $$\sqrt[3]{8 \times 9 \times x} = 6$$
Hisoblash:
$$72x = 216$$
$$x = 3$$
7. Muammo: a, 2.1, 3 va 2.1 sonlarining o’rta arifmetigi 2.3. $a$ ni toping.
Formulasi: $$\frac{a + 2.1 + 3 + 2.1}{4} = 2.3$$
Hisoblash:
$$a + 7.2 = 9.2$$
$$a = 2$$
8. Muammo: Ikki sonning o’rta arifmetigi 7, kvadratlarining ayirmasi 14. Kvadratlar yig’indisi?
Formulasi:
$$\frac{x + y}{2} = 7 \Rightarrow x + y = 14$$
$$x^2 - y^2 = 14$$
Hisoblash:
$$(x - y)(x + y) = 14$$
$$(x - y) \times 14 = 14$$
$$x - y = 1$$
Kvadratlar yig’indisi:
$$(x + y)^2 - 2xy = x^2 + y^2$$
$$14^2 - 2xy = ?$$
$$(x - y)^2 = x^2 - 2xy + y^2 = 1^2 = 1$$
$$x^2 + y^2 = (x - y)^2 + 2xy = 1 + 2xy$$
$$xy = \frac{(x + y)^2 - (x - y)^2}{4} = \frac{196 - 1}{4} = 48.75$$
$$x^2 + y^2 = 1 + 2 \times 48.75 = 98.5$$
9. Muammo: Uchta sonning o’rta arifmetigi 10, boshqa ikki sonning o’rta arifmetigi 15. Besh sonning o’rta arifmetigi?
Formulasi:
$$\frac{3 \times 10 + 2 \times 15}{5} = ?$$
Hisoblash:
$$\frac{30 + 30}{5} = 12$$
10. Muammo: Uchta sonning o’rta arifmetigi 30, dastlabki ikkitasining o’rta arifmetigi 25. Uchinchi son?
Formulasi:
$$\frac{x + y + z}{3} = 30$$
$$\frac{x + y}{2} = 25$$
Hisoblash:
$$x + y + z = 90$$
$$x + y = 50$$
$$z = 90 - 50 = 40$$
11. Muammo: 24 ta sonning o’rta arifmetigi 11.5, yana bir son qo’shildi, o’rta arifmetigi 12.5. Qo’shilgan son?
Formulasi:
$$\frac{24 \times 11.5 + x}{25} = 12.5$$
Hisoblash:
$$276 + x = 312.5$$
$$x = 36.5$$
12. Muammo: Onasi 50, qizi 28 yoshda. Necha yil oldin qizi onasidan 2 marta yosh bo’lgan?
Formulasi:
$$28 - t = 2(50 - t)$$
Hisoblash:
$$28 - t = 100 - 2t$$
$$t = 72$$ (bu noto’g’ri, tekshiramiz)
To’g’ri:
$$28 - t = 2(50 - t)$$
$$28 - t = 100 - 2t$$
$$-t + 2t = 100 - 28$$
$$t = 72$$
Bu javob mantiqsiz, ehtimol savolda xato bor yoki yoshlar boshqacha.
13. Muammo: Otasi 40, o’g’li 16. Necha yildan keyin otasi o’g’lidan 2 marta katta bo’ladi?
Formulasi:
$$40 + t = 2(16 + t)$$
Hisoblash:
$$40 + t = 32 + 2t$$
$$40 - 32 = 2t - t$$
$$8 = t$$
Javob: 8 yil
14. Muammo: Uchta nuqtadan tashkil topgan uchburchakning maydonini toping.
Nuqtalar: A(0,0), B(3,6), C(6,0)
Formulasi: Uchburchak maydoni
$$S = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$
Hisoblash:
$$S = \frac{1}{2} |0(6 - 0) + 3(0 - 0) + 6(0 - 6)| = \frac{1}{2} |0 + 0 - 36| = 18$$
Javob: Maydon $18$
Umumiy: Har bir masala o’rta arifmetik yoki o’rta geometrik formulalar yordamida yechildi.
Arifmetik Masalar
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