CDS Mathematics · Ratio, Proportion and Variation

Partnership

Profit is shared in the ratio of capital × time; when capital changes during the year, add up capital × months for each period.

All Ratio, Proportion and Variation notes CDS Mathematics strategy

Why this matters

Five PYQs. One rule — capital multiplied by the months it was invested — handles withdrawals, additions and unequal periods. The 'who gets most' versions pair an increasing capital with a decreasing time; the product is largest in the middle.

Concept 1 of 1: Capital × time

Rs. 100 invested for two months earns the same share as Rs. 200 for one month. So each partner's claim is capital × months, and profit is divided in the ratio of those claims.

Definition

Share $\propto$ capital $\times$ time.
If capital changes, add the pieces: Rs. $700$ for $3$ months then Rs. $500$ for $9$ months is $700\times 3 + 500\times 9$ .
Capitals in a fractional ratio ( $\tfrac13 : \tfrac14 : \tfrac15$ ): convert to whole numbers first.
Capitals $1, 2, \ldots, n$ with times $n, \ldots, 1$ : the product $k(n + 1 - k)$ is largest in the middle.

Profit share

\text{share} \propto \text{capital}\times\text{time}

Worked example

A invests Rs. 6,000 for the year; B invests Rs. 8,000 but withdraws Rs. 4,000 after 6 months. The profit is Rs. 5,200. Find B's share.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (II) 2020 — Elementary Mathematics · Q25Hard

Example 1 · Ratio, Proportion and Variation · Partnership

Three persons start a business with capitals in the ratio

\frac{1}{3} : \frac{1}{4} : \frac{1}{5}

. The first person withdraws half his capital after 4 months. What is his share of profit if the business fetches an annual profit of Rs. 96,800 ?

Count each period separately

A partner who withdraws part of the capital after four months holds the full amount for four months and the rest for eight. Using the final capital for the whole year understates the share.

Drill 4 more on capital × time

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Capital × time
Profit share
$\text{share} \propto \text{capital}\times\text{time}$

Watch out for (1)

Count each period separately
→ Capital × time

Test yourself on Ratio, Proportion and Variation

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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