Partnership · Topic 3 of 5

🔍 Reverse: Find the Unknown

2 exam question types, fully solved

Sometimes the profit ratio is handed to you and the unknown is hidden inside — a missing investment, a missing number of months, or the total profit. The same one rule still applies; you just run it backwards.

Run the formula in reverse

Everything starts from the master equation:

• Missing time or capital: put in everything you know and cross-multiply for the one blank.

• Capital from profit and time: rearrange to capital , so . Divide profit by time — do NOT multiply.

• Find the total from one share: a share . Get one part from the known share, then multiply by the sum of all parts.

Shortcut: if the profit ratio already equals the (simplified) time ratio, the capitals are equal — , no working needed.

🤝Profit SplitterEnter capital & time for each partner

A120.0K

B120.0K

C—

Partner A50.0%

Partner B50.0%

Ratio : A: 1 : B: 1

Total Profit (₹)

Equal capital, invested for 6 and 9 months. The profit ratio is the time ratio: .

Q1A and B invest in the ratio 4 : 3. A's profit share is ₹8,000. The total profit is:

Q2X invests for 6 months and Y for 9 months, with equal capital. The profit ratio X : Y is:

Q3A, B and C invest in the ratio 2 : 3 : 5. C's share is ₹10,000. The total profit is:

📚

Real exam questions — Find the Unknown

2 question types · 8 solved examples from real SSC papers

Run the formula backwards: from the profit ratio recover a missing time, a missing capital, or the total profit. Cancel a common factor before you solve.

How to solve this type

Start from

\dfrac{C_A \times T_A}{C_B \times T_B} = \dfrac{\text{profit}_A}{\text{profit}_B}

and rearrange for the unknown.
• Missing capital

= \dfrac{\text{profit ratio} \times \text{time of other}}{\text{own time}}

style — just cross-multiply.
• Capital ratio from profit and time: capital

\propto \dfrac{\text{profit}}{\text{time}}

, so

C_A : C_B = \dfrac{P_A}{T_A} : \dfrac{P_B}{T_B}

.
Shortcut: if profit ratio equals the (simplified) time ratio, the capitals are equal, 1 : 1.

P and Q invest ₹40,000 each, for 6 months and 9 months respectively. Find the profit ratio P : Q.

A4 : 3B3 : 2C1 : 2D2 : 3

X and Y invest ₹25,000 and ₹30,000 for 8 months and 6 months respectively. Find the profit ratio X : Y.

A10 : 9B4 : 5C5 : 4D9 : 10

A invested ₹30,000 and B invested ₹20,000. The profit at year end is shared equally. For how many months did B invest (A invested for 12 months)?

A12 monthsB15 monthsC9 monthsD18 months

A and B invest for 6 months and 9 months respectively. The profit ratio is 1 : 2. Find the capital ratio A : B.

A3 : 4B3 : 2C2 : 3D4 : 3