Ratio, Proportion and Partnership
What you must know
A ratio a:b compares like quantities and is scale-free; a proportion a:b = c:d states equality of two ratios, with the cross-product rule ad = bc. Learn the derived rules: if a/b = c/d, then each equals (a+c)/(b+d) (componendo), and invertendo, alternando, and dividendo follow from it. Direct proportion means the ratio stays constant (more workers, more work); inverse proportion means the product stays constant (more workers, less time).
Partnership problems apply ratio to money: in a simple partnership, profit shares follow capital × time. So if A invests ₹5000 for 12 months and B invests ₹6000 for 8 months, shares are 60000:48000 = 5:4. Distinguish working partners (draw a salary before splitting profit) from sleeping partners.
Traps
- Units must match before forming a ratio: convert 40 minutes to hours before comparing with 2 hours.
- In inverse proportion, do not form a ratio the same way as direct proportion; invert one side.
- Partnership: forgetting to multiply capital by the period of investment.
- Ratio of a:b:c from a:b and b:c requires making the common term equal (LCM of the b-values).
- "Divide ₹x in the ratio 2:3:5" - the total number of parts is 10, not 5.
Rule of thumb and example
In ratio-sharing problems, let one part equal k and express every share as a multiple of k; the sum of coefficients equals the total.
Worked example: A, B, C share ₹1800 in the ratio 2:3:4. Total parts = 9, so k = 200. Shares are ₹400, ₹600, ₹800. Check: largest share minus smallest = 400, which matches (4−2)k.