Unitary method — direct and inverse proportion

1.1.20 · Maths › Arithmetic & Number Systems

Unitary method KYA hai?

YE KAAM KYU KARTA HAI? Kyunki real duniya ki zyaadatar quantities proportional hoti hain: items double karo toh cost double, workers triple karo toh time ek-tehai ho jaata hai, etc. Jab ek baar "1 ke liye value" pata hai, toh "n ke liye value" simple multiplication se nikal aati hai.

Proportion ke do flavours

Direct proportion — rule derive karna

Maano $y$ (cost) directly proportional hai $x$ (pens ki sankhya) se. "Directly proportional" ka matlab hai ki unka ratio constant hai:

$\frac{y}{x} = k \Rightarrow y = k x$

Constant ratio kyu? Kyunki price per pen nahi badlti. Agar 5 pens ka cost ₹20 hai, toh har pen ₹4 ki value "carry" karti hai. Woh ₹4 hi $k$ hai — ek unit ki cost. Toh:

$k = \frac{y _{1}}{x _{1}} = ek unit ki cost, y_{2} = k x_{2}$

Dono ko combine karne pe cross-multiplication form milta hai: $\frac{y _{1}}{x _{1}} = \frac{y _{2}}{x _{2}}$

Inverse proportion — rule derive karna

Maano $y$ (days) inversely proportional hai $x$ (workers) se. Yahan product constant hota hai:

$x y = k \Rightarrow y = \frac{k}{x}$

Constant product kyu? Kyunki total kaam $k$ fixed hai (maano 60 worker-days). Agar zyada workers $x$ hain, toh har ek kam karta hai, isliye days $y$ kam hote hain — lekin $x \cdot y$ hamesha same total job ko rebuild karta hai.

$x_{1} y_{1} = x_{2} y_{2}$

Figure — Unitary method — direct and inverse proportion

Solve KAISE karte hain — recipe

Do quantities identify karo aur decide karo direct hai ya inverse (up/down test).
"Bahut saare" wali line likho, phir 1 unit ki value nikalo.
- Direct: divide karo value-per-one ke liye.
- Inverse: multiply karo total (constant) product ke liye.
Required number of units ke liye scale karo.

Worked examples

Common mistakes (Steel-manned)

Active recall

Recall Quick self-test (answers chhupa ke dekho)

Direct proportion mein kya constant rehta hai? → ratio $y / x$ .
Inverse mein kya constant rehta hai? → product $x y$ .
Bahut se ek par jaana: multiply karo ya divide? → divide.
6 books ka cost ₹150 hai, 10 ka cost? → ₹25 each → ₹250.
8 pipes 6 h mein bharte hain, 4 pipes mein time? → $48/4 =$ 12 h.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho ek badi bag mein candy hai. Agar 4 doston ko 8 sweets har ek chahiye, toh pehle kahte hain "ek dost = 8" aur phir jitne bhi dost aayein unke liye add karte hain — yahi pehle ek pe jaana hai. Ab socho ek room clean karna: agar 1 baccha 60 minute leta hai, toh 2 bacche same room split karte hain aur aadhe time mein khatam karte hain. Kam helpers = zyada time, zyada helpers = kam time. Total kaam (60 kid-minutes) kabhi nahi badlta; bas hum use alag-alag share karte hain. "Same-direction sharing" direct hai, "opposite-direction sharing" inverse hai.

Connections

Ratio and Proportion — ratio $y / x = k$ literally ek proportion hai.
Percentages — "per cent" = per 100, ek unitary "value per 1" jo 100 tak scale hoti hai.
Speed Distance Time — classic inverse-proportion ka playground.
Time and Work — worker-days constant pure inverse proportion hai.
Linear Equations — $y = k x$ origin se ek straight line hai.

Unitary method ka matlab kya hai?

Pehle EK unit ki value nikalo, phir required number of units ke liye scale karo.

Direct proportion mein kya constant rehta hai?

Ratio y/x = k (per-unit value).

Inverse proportion mein kya constant rehta hai?

Product x·y = k (total fixed work/quantity).

Direct aur inverse mein fark batane ka quick test?

Agar ek quantity UPAR jaaye aur doosri bhi UPAR → direct; agar ek UPAR aur doosri NEECHE → inverse.

Bahut units se ek unit par jaate waqt multiply karte ho ya divide?

Divide.

7 pens ka cost ₹56; 12 pens ka cost?

₹8 per pen × 12 = ₹96 (direct).

15 workers 8 days mein khatam karte hain; 20 workers mein kitne days?

15×8=120 worker-days; 120/20 = 6 days (inverse).

Direct proportion equation form?

y₁/x₁ = y₂/x₂.

Inverse proportion equation form?

x₁·y₁ = x₂·y₂.

Direct proportion mein ek unit ki value divide karke kyu nikalte hain?

Kyunki total sabhi units mein equally split hota hai, toh total ÷ number = value per one.

Type	Test	Constant	Equation
Direct	same direction	ratio $y / x$	$\frac{y _{1}}{x _{1}} = \frac{y _{2}}{x _{2}}$
Inverse	opposite direction	product $x y$	$x_{1} y_{1} = x_{2} y_{2}$