Cross-multiplication in proportions
Scenario
Aap market mein `3` pen `15` rupees ke laaye. Agar aapko `5` pen chahiye hote, toh kitne paise lagte?
Jab do ratios equal hote hain, toh missing value ko jaldi kaise nikalein?
Mental model
X banane ki ninja technique.
Jab do cheezein fraction mein barabar hoti hain (`a/b = c/d`), toh hum cross mein multiply karke unhe straight equation bana sakte hain. Ye light ke magnification aur mirror formulas mein unknown values nikalne ke liye ek universal tool hai.
Explanation
Physics numericals mein, khaaskar Light chapter mein, humein aksar aise situations milte hain jahan do fractions equal hote hain. Jaise magnification ke formulas mein (`m = h'/h = -v/u`). Aisi equations se unknown variable nikalne ka sabse reliable tarika cross-multiplication hai.
Cross-multiplication ka rule simple hai: `a/b = c/d` equation mein, `a` ko `d` se aur `b` ko `c` se multiply karein. Isse humein `a × d = b × c` milta hai. Is trick se fraction seedha ek normal linear equation ban jata hai.
Sign mistakes yahan bahut aam hain. Agar aapko `(-5)/v = 2/(-3)` jaisa fraction mile, toh cross multiply karte time minus signs ko dhyan se numbers ke sath hi rakhna chahiye (Neeche worked example 2 mein isko solve karke dekho).
Ek aur common galti hai addition aur multiplication ko mix karna. Cross-multiplication sirf tab kaam karta hai jab equal to (`=`) ke dono taraf sirf ek hi fraction ho. Agar `a/b + c = d/e` hai, toh pehle LCM lekar left side ko single fraction banana padega.
Worked example
Problem: Solve for `x` in the equation: `4/x = 12/15`
- Cross multiply terms.
- Simplify the left side.
- Divide by `12` to isolate `x`.
Answer: `x = 5`
Problem: If `(-v)/20 = (-3)/(-5)`, find the value of `v`.
- Cross multiply while keeping the negative signs attached.
- Multiply out both sides.
- Divide by `5`.
Answer: `v = -12`
Key points
- Cross-Multiply Rule: For an equation `a/b = c/d`, it simplifies to `a × d = b × c`.
- Single Fraction Condition: Don't use it if there's addition or subtraction outside the fraction. First simplify to a single fraction on both sides.
Common mistakes
- Ignoring negative signs during multiplication: Bachhe cross multiply toh kar dete hain, par negative sign bhool jate hain. Always carry the minus sign with the numerator or denominator.
Recall questions
- What does a/b = c/d become after cross-multiplication?
- If a/b = c/d, what is the cross-multiplied form?
- When is cross-multiplication most useful?
Questions & answers
Solve for y: (y+2)/3 = 4/6
y = 0
Approach: Cross-multiply to get 6(y+2) = 12. Expand and solve for y.
Solve for x: x/4 = 3/2
x = 6
Approach: 2x = 12 => x = 6.
Continue learning
Next: Calculate linear magnification (mirrors)