Proportion and Partnership

Proportion

Four numbers a, b, c, d are in proportion if a : b = c : d, i.e. a × d = b × c (product of extremes = product of means).

• Fourth proportional of a, b, c: d = (b × c)/a.
• Third proportional of a, b: c = b²/a.
• Mean proportional of a, b: √(a × b).

Partnership

Profit is shared in the ratio of capital × time invested.

• A invests ₹20,000 for 12 months, B invests ₹30,000 for 8 months. Ratio = 240000 : 240000 = 1 : 1.
• If time is the same, profit ratio = capital ratio.
• A working partner may first receive a salary or a fixed % of profit; the rest is shared by capital ratio.

Dividing an amount in a ratio

Divide ₹1,200 in the ratio 2 : 3 : 5 → total parts 10, one part ₹120 → ₹240, ₹360, ₹600.

Exam tip

Always reduce ratios to simplest form and check that the parts add up to the total.

Analogy

Partnership is like sharing a taxi fare fairly. If one friend rides 10 km and another 5 km, the first pays twice as much. In business, money is the "seat" and months are the "distance" — whoever keeps more money in for longer gets the bigger share of profit.