Foizlar: matematika darsi

Maosh $4\,000\,000$ so'm va u $15\%$ ga oshsa, yangi maosh $4\,600\,000$ so'm. Bir narsa $60\%$ ga qimmat bo'lsa, ikkinchisi undan $60\%$ emas, $37{,}5\%$ ga arzon. Foiz masalalaridagi deyarli barcha xatolar shundan: foiz qaysi sondan olinayotgani aniqlanmaydi. Bu darsda foizning ta'rifi, uch asosiy masala va imtihon turlari beriladi.

Foiz nima

Sonning foizi uning yuzdan bir qismidir.

1\%=\frac{1}{100}=0{,}01

p\%=\frac{p}{100}

Foiz	O'nli va oddiy kasr
$1\%$	$0{,}01=\dfrac{1}{100}$
$5\%$	$0{,}05=\dfrac{1}{20}$
$10\%$	$0{,}1=\dfrac{1}{10}$
$12{,}5\%$	$0{,}125=\dfrac{1}{8}$
$20\%$	$0{,}2=\dfrac{1}{5}$
$25\%$	$0{,}25=\dfrac{1}{4}$
$40\%$	$0{,}4=\dfrac{2}{5}$
$50\%$	$0{,}5=\dfrac{1}{2}$
$75\%$	$0{,}75=\dfrac{3}{4}$
$100\%$	$1$ , butun son

Uch asosiy masala

So'ralgani	Formula	Misol
So'ralgani $a$ ning $p\%$ i	Formula $a\cdot\dfrac{p}{100}$	Misol $60$ ning $35\%$ i: $60\cdot 0{,}35=21$
So'ralgani $p\%$ i $b$ bo'lgan son	Formula $b:\dfrac{p}{100}$	Misol $35\%$ i $21$ bo'lgan son: $21:0{,}35=60$
So'ralgani $a$ soni $b$ ning necha foizi ( $b\ne 0$ )	Formula $\dfrac{a}{b}\cdot 100\%$	Misol $21$ soni $60$ ning: $\dfrac{21}{60}\cdot 100\%=35\%$

Ikkinchi va uchinchi masalada avval qaysi son butun, ya'ni $100\%$ ekanini aniqlaymiz. " $a$ soni $b$ ning necha foizi" gapida butun son $b$ : "ning" qo'shimchasi olgan so'z.

Oshirish va kamaytirish

a\cdot\left(1+\frac{p}{100}\right)

a

p\%

ga oshirish:

100\%

ga yana

p\%

qo'shiladi

a\cdot\left(1-\frac{p}{100}\right)

a

p\%

ga kamaytirish:

100\%

dan

p\%

ayriladi

O'zgarishdan keyingi son berilib, oldingisi so'ralsa, ko'paytuvchiga bo'lamiz. Masala: $15\%$ ga arzonlashgach, narx $2550$ so'm bo'ldi. Noma'lum $x$ : dastlabki narx. Tenglama: $0{,}85x=2550$ , $x=3000$ . Tekshiruv: $3000$ ning $15\%$ i $450$ , $3000-450=2550$ .

Xato yo'l: $2550$ ga uning $15\%$ ini qo'shish, chunki $15\%$ yangi narxdan emas, dastlabkisidan olingan.

Matnda	Formulada
$a$ soni $b$ dan $25\%$ ga katta	$a=1{,}25b$
$a$ soni $b$ dan $25\%$ ga kichik	$a=0{,}75b$
$a$ soni $b$ ning $25\%$ i	$a=0{,}25b$
$a$ soni $b$ dan $100\%$ ga katta	$a=2b$ , ya'ni $2$ marta katta
$a$ soni $b$ dan $200\%$ ga katta	$a=3b$ , ya'ni $3$ marta katta
$a$ soni $b$ ning $200\%$ i	$a=2b$

Necha foiz ko'p, necha foiz kam

Taqqoslashda foiz "-dan" qo'shimchasi turgan sondan olinadi. Shuning uchun " $A$ soni $B$ dan necha foiz ko'p" va " $B$ soni $A$ dan necha foiz kam" savollarining javobi har xil.

\frac{A-B}{B}\cdot 100\%

A>B>0

A

soni

B

dan shuncha foiz ko'p

\frac{A-B}{A}\cdot 100\%

A>B>0

B

soni

A

dan shuncha foiz kam

Masala: A ishchi B ishchidan $60\%$ ko'p haq oladi. B ishchi A dan necha foiz kam oladi? Noma'lum: B ning haqi $B$ , u har qanday musbat son bo'lishi mumkin va javobda qisqaradi.

Miqdor	Ifoda
B ning haqi	$B$
A ning haqi	$1{,}6B$
farq	$0{,}6B$
farqning A dagi ulushi	$\dfrac{0{,}6B}{1{,}6B}=0{,}375$

Tekshiruv: $B=100$ da A $160$ oladi. $160$ ning $37{,}5\%$ i $60$ , $160-60=100$ . Javob: $37{,}5\%$ .

$A$ soni $B$ dan ko'p	$B$ soni $A$ dan kam	Hisob
$A$ soni $B$ dan ko'p $25\%$	$B$ soni $A$ dan kam $20\%$	Hisob $\dfrac{0{,}25}{1{,}25}=0{,}2$
$A$ soni $B$ dan ko'p $50\%$	$B$ soni $A$ dan kam $33\dfrac{1}{3}\%$	Hisob $\dfrac{0{,}5}{1{,}5}=\dfrac{1}{3}$
$A$ soni $B$ dan ko'p $60\%$	$B$ soni $A$ dan kam $37{,}5\%$	Hisob $\dfrac{0{,}6}{1{,}6}=0{,}375$
$A$ soni $B$ dan ko'p $100\%$	$B$ soni $A$ dan kam $50\%$	Hisob $\dfrac{1}{2}=0{,}5$

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