WHY does it work? Because most real quantities are proportional: doubling the items doubles the cost, tripling the workers cuts the time to a third, etc. Once you know "the value for 1", you can rebuild "the value for n" by simple multiplication.
Suppose y (cost) is directly proportional to x (number of pens). "Directly proportional" means their ratio is constant:
xy=k⇒y=kx
Why constant ratio? Because the price per pen doesn't change. If 5 pens cost ₹20, each pen "carries" ₹4 of value. That ₹4 is k — the cost of one unit. So:
k=x1y1=cost of one unit,y2=kx2
Combining both, we get the cross-multiplication form:
x1y1=x2y2
Suppose y (days) is inversely proportional to x (workers). Here the product is constant:
xy=k⇒y=xk
Why constant product? Because the total amount of workk is fixed (say 60 worker-days). If you have more workers x, each does less, so days y shrink — but x⋅y always rebuilds the same total job.
Direct proportion keeps what constant? → the ratioy/x.
Inverse keeps what constant? → the productxy.
Many → one: multiply or divide? → divide.
6 books cost ₹150, cost of 10? → ₹25 each → ₹250.
8 pipes fill in 6 h, time for 4 pipes? → 48/4=12 h.
Recall Feynman: explain to a 12-year-old
Imagine sharing a big bag of candy. If 4 friends need 8 sweets each, we first say "one friend = 8" and then just add up for however many friends come — that's going to one first. Now imagine cleaning a room: if 1 kid takes 60 minutes, then 2 kids split the same room and finish in half the time. Fewer helpers = more time, more helpers = less time. The total job (60 kid-minutes) never changes; we just share it differently. "Same-direction sharing" is direct, "opposite-direction sharing" is inverse.
Unitary method ka matlab bilkul simple hai: pehle ek unit ki value nikaalo, phir jitne chahiye utne se multiply/divide kar do. Jaise 7 pen ka ₹56 hai, to pehle 1 pen = 56÷7 = ₹8, phir 12 pen = 8×12 = ₹96. Yeh "ek pe aao" wala step hi poori technique ka dil hai. Zyada items ki value diye ho to divide karke ek pe aao, phir multiply karke required number tak jaao.
Ab do type hote hain. Direct proportion mein dono cheezein ek hi taraf jaati hain — zyada pen to zyada paisa. Yahan ratio y/x constant rehta hai (per unit price fix). Inverse proportion mein ek badhta hai to doosra ghatta hai — zyada workers to kam din. Yahan product x⋅y constant rehta hai (total kaam fix, jaise 120 worker-days). Bas ek line yaad rakho: "up-up = direct, up-down = inverse".
Sabse badi galti students ki yeh hoti hai ki har jagah cross-multiply (ratio) laga dete hain, chahe problem inverse ho. Isliye pehle up/down test karo, phir formula chuno. Direct ke liye y1/x1=y2/x2, inverse ke liye x1y1=x2y2. Speed-distance-time aur time-and-work ke saare sawaal isi se ban jaate hain, isliye yeh concept exam mein bahut kaam ka hai — thoda practice karo aur direction pehchaanna seekh lo.
