SSC CGL Maths · Ratio, Proportion and Variation

Partnership

Partners share a profit in the ratio of capital multiplied by the time it stays in the business, after taking out first any amount the agreement sets aside.

All Ratio, Proportion and Variation notes

Why this matters

Forty-two questions in the bank, the biggest page in this chapter, with twenty of them MODERATE. Twenty-one need only capital times time, nine have capital added or withdrawn during the year, eight take out charity, a salary or interest before sharing, and four share the rent of a field among animals.

Concept 1 of 4: Profit in the ratio of capital times time

₹1,000 working for 2 months earns the business as much as ₹2,000 working for 1 month. So a partner's claim on the profit is capital multiplied by months, and the profit is shared in the ratio of those products.

Definition

Partners share profit in the ratio of capital $\times$ time, often called capital-months.
Same time for everyone: the ratio of capitals. Same capital: the ratio of times.
Joining late: a partner who joins after m months of a 12-month year is in for $12 - m$ months. Read the period: two years is 24 months.
Unknown time or capital: set $\frac{C_1t_1}{C_2t_2}$ equal to the given profit ratio and solve.
Wages for days worked follow the same rule: shares are in the ratio daily wage $\times$ days.
Fast route: drop the thousands and cancel common factors before multiplying. Then a share is its fraction of the profit.

Profit sharing

P_1 : P_2 : P_3 = C_1t_1 : C_2t_2 : C_3t_3

Ca partner's capital
tthe months that capital stays in

Worked example

R starts a business with ₹45,000. S joins 4 months later with ₹60,000. Out of the year's profit of ₹34,000, what is S's share?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 1 · Q55Easy

Example 1 · Ratio, Proportion and Variation · Partnership

Arvind started a business by investing ₹80,000. After 4 months, Bhavin joined with ₹1,20,000. At the end of 8 months from the start, Chandan joined with ₹1,60,000. If the total profit is ₹1,05,000 at the end of the year, find the share of Chandan.

Counting the whole year for a late partner

A partner who joins after 5 months has money in for 7 months of a 12-month year. Multiplying the capital by 12 overstates the share.

Taking the year when the stem says two years

If the profit is shared after two years, the time is 24 months, and a partner who joined after 10 months was in for 14 months, not 2.

Drill 20 more on profit in the ratio of capital times time

Concept 2 of 4: Capital that changes during the year

When a partner adds or takes out money, the year breaks into stretches, and in each stretch the capital is fixed. Each stretch earns capital times its months, and the partner's claim is the sum of the stretches.

Definition

Split each partner's year at every change. Capital-months $= C_1t_1 + C_2t_2 + \cdots$ .
A withdrawal lowers the capital from that month on; an addition raises it. Months before the change keep the old capital.
Fast route: work in thousands or lakhs, and when two partners share a stretch length, compare the bracket sums: $6(100 + 80)$ against $6(80 + 120)$ .
If a stem gives the profit share and asks for a starting capital, write the capital-months in terms of x and use the share ratio.

Capital-months with changes

\text{capital-months} = C_1t_1 + C_2t_2 + \cdots

Worked example

M starts a business with ₹50,000. After 4 months N joins with ₹40,000. Four months after that, M takes out ₹20,000. Find the ratio of their profits at the end of the year.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 2 · Q55Moderate

Example 2 · Ratio, Proportion and Variation · Partnership

Sohan started a business with ₹90,000. After 5 months, Rohan joined with ₹1,50,000. After another 3 months, Sohan withdrew ₹30,000. What is the ratio of their profits at the end of the year?

Using the final capital for the whole year

If ₹30,000 is taken out after 8 months, the lower capital counts only for the last 4 months. The first 8 months still earn on the full amount.

Reading "2 times more than"

In the bank, SSC's key reads "A invests 2 times more than B" as A = 3 times B (B's amount plus twice more). "Twice as much as B" is 2 times B. When the stem uses the first wording, test both readings against the options.

Drill 8 more on capital that changes during the year

Concept 3 of 4: Charity, salary or interest taken out before sharing

Some agreements pay out a slice of the profit before the partners split it: a gift to charity, a working partner's fee, or interest on each capital. That slice must come off first, because only what is left is shared by capital.

Definition

Order of work: take out what the stem sets aside, then share the remainder in the capital ( $\times$ time) ratio.
Charity of r%: the shared amount is $(1 - r)$ of the profit. A partner's share is $\frac{a}{a + b} \times (1 - r)P$ ; solve this for P when a share is given.
Working or managing partner: gets the extra first, PLUS a capital share of the rest.
Interest on capital: each partner receives interest on their own capital first; the remainder splits in the capital ratio.
Effective contribution: labour valued in money is added to that partner's capital before forming the ratio.

Share after a deduction

\text{share} = \text{own extra} + \frac{c}{c_1 + c_2} \times (P - \text{deductions})

cthe partner's capital (times time)
Pthe total profit

Worked example

A and B invest in the ratio 2 : 3. 20% of the profit goes to charity and the rest is shared. B gets ₹14,400. What is the total profit?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 1 · Q55Moderate

Example 3 · Ratio, Proportion and Variation · Partnership

A and B invest ₹60,000 and ₹90,000 respectively, in a business. After one year, the profit is distributed, including simple interest at

10\%

per annum on the capital. Total profit, including interest, is ₹25,000. What is A's share?

Sharing before deducting

If a managing partner takes 10% first, only the other 90% is shared by capital. Sharing the full profit and then adding 10% counts that 10% twice.

Forgetting the working partner's capital share

A working partner gets the fee AND a capital share of what remains. Answering with the fee alone, or the capital share alone, is a printed wrong option.

Drill 7 more on charity, salary or interest taken out before sharing

Concept 4 of 4: Rent for a shared field in animal-months

Rent for a grazing field is paid for the grass eaten. A horse eats more than a sheep, and an animal kept longer eats more, so every animal is first turned into one common kind, and then animals times months plays the part of capital times time.

Definition

Turn every animal into one kind using the given rates. If 3 horses eat as much as 5 cows, 1 horse $= \frac{5}{3}$ cow.
Each person's grazing $=$ (number of animals in the common unit) $\times$ months, summed over their animals.
The rent is shared in the ratio of grazing. A fraction of the rent is that person's grazing over the total.
Pick the smallest animal as the unit; the numbers usually stay whole.

Share of the rent

\text{share} = \frac{\text{own animal-months}}{\text{total animal-months}} \times \text{rent}

Worked example

X grazes 10 cows for 6 months and Y grazes 40 sheep for 3 months. 2 cows eat as much as 5 sheep. What fraction of the rent does X pay?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 3 · Q57Moderate

Example 4 · Ratio, Proportion and Variation · Partnership

Two individuals, A and B rent a pasture. A uses 16 cows for a duration of 3 months and 20 sheep for 4 months and B uses 30 sheep for 6 months. If 4 cows are equivalent to 8 sheep, determine A's portion of the rent.

Adding animals of different kinds

20 horses and 30 cows are not 50 units of grazing. Convert every animal to one kind with the given rate before multiplying by months.

Drill 3 more on rent for a shared field in animal-months

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 (4)

Profit in the ratio of capital times time
Profit sharing
$P_1 : P_2 : P_3 = C_1t_1 : C_2t_2 : C_3t_3$
Capital that changes during the year
Capital-months with changes
$\text{capital-months} = C_1t_1 + C_2t_2 + \cdots$
Charity, salary or interest taken out before sharing
Share after a deduction
$\text{share} = \text{own extra} + \frac{c}{c_1 + c_2} \times (P - \text{deductions})$
Rent for a shared field in animal-months
Share of the rent
$\text{share} = \frac{\text{own animal-months}}{\text{total animal-months}} \times \text{rent}$

Watch out for (7)

Counting the whole year for a late partner
→ Profit in the ratio of capital times time
Taking the year when the stem says two years
→ Profit in the ratio of capital times time
Using the final capital for the whole year
→ Capital that changes during the year
Reading "2 times more than"
→ Capital that changes during the year
Sharing before deducting
→ Charity, salary or interest taken out before sharing
Forgetting the working partner's capital share
→ Charity, salary or interest taken out before sharing
Adding animals of different kinds
→ Rent for a shared field in animal-months

Test yourself on Ratio, Proportion and Variation

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