Capital \(C\)
The amount a partner contributes to the business. Capital may remain fixed or change during the accounting period.
8 SSC CGL · Mathematics
Partnership Business
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
A partner’s variable profit share is earned by both the money invested and the time for which it works.
PARTNERSHIP
PROFIT
SHARING
PROFIT
SHARING
1. Capitalmoney placed in the business
2. Active Timeduration for which capital remains
3. Equivalent Capitalcapital multiplied by time
4. Changing Partnersadmission, withdrawal and additions
5. Prior Chargessalary, bonus, interest and commission
6. Missing Datarecover capital, time or total profit
The balance has two arms
Capital alone is incomplete when partners invest for different durations. Convert every contribution into “money-time” before forming the ratio.
\[\boxed{\text{profit share weight}=\text{capital}\times\text{active time}}\]
\[\text{partner's share}=\text{distributable profit}\times\frac{\text{partner's weight}}{\text{sum of all weights}}\]
02ZERO TO EXPERT
The Foundation (Basics) — Complete Concept Build
Separate the fixed payments first; share only the remaining profit in the correct capital-time ratio.
AWhat is an arithmetic partnership?
Active time \(T\)
The duration for which that capital is actually employed. Use one common time unit for all partners.
Profit or loss \(Q\)
The variable business result to be divided after any payments that the question treats as prior charges.
Equivalent capital
\[W=C\times T\]
Also called capital-time product, money-months or investment weight.
Profit-sharing ratio
\[S_1:S_2:\cdots=C_1T_1:C_2T_2:\cdots\]
Loss-sharing ratio
If no different agreement is stated, an arithmetic problem normally allocates loss by the same weight ratio.
\[L_i=L\frac{W_i}{\sum W}\]
Money-months make unlike timelines comparable
Rupees × months and lakhs × months both work if the same capital unit is used throughout. Common units later cancel in the ratio.
\[₹x\text{ for }m\text{ months}\Rightarrow xm\text{ money-months}\]
BThe master sharing formula
Individual share
\[S_i=Q\frac{C_iT_i}{\sum_{j=1}^{n}C_jT_j}\]
Here \(Q\) is the profit left for ratio-based distribution.
Ratio to fraction
If shares are in \(a:b:c\), then
\[S_A=Q\frac{a}{a+b+c}\]
Check sum
\[\sum_{i=1}^{n}S_i=Q\]
The distributed shares must reconstruct the entire distributable pool.
Same time
If all invest for equal duration:
\[S_1:S_2:\cdots=C_1:C_2:\cdots\]
Same capital
If all invest equal capital:
\[S_1:S_2:\cdots=T_1:T_2:\cdots\]
Same share
Equal profit shares require equal weights:
\[C_1T_1=C_2T_2\]
Equal duration removes the clock
When every partner’s time is identical, cancel the common time factor. The capital ratio alone remains.
\[C_1T:C_2T=C_1:C_2\]
CSimple partnership vs compound partnership
| Type | Capital pattern | Time pattern | Required ratio |
|---|---|---|---|
| Simple partnership | Capitals differ or are equal | All active for equal time | Capital ratio |
| Compound partnership | Capitals may differ | Active durations differ | Capital-time ratio |
| Changing-capital partnership | One partner has multiple capital blocks | Each block has its own duration | Sum of capital-time products |
DCapital changes during the year
Segment the timeline
Every time capital changes, close the old interval and begin a new one.
\[W=C_1t_1+C_2t_2+\cdots+C_kt_k\]
Addition after \(m\) months
If starting capital \(C\) increases by \(x\) after \(m\) months of a \(12\)-month year:
\[W=Cm+(C+x)(12-m)\]
Withdrawal after \(m\) months
If \(x\) is withdrawn:
\[W=Cm+(C-x)(12-m)\]
Equivalent capital is the area under the timeline
Capital is the height; time is the width. Each rectangle’s area is a capital-time block. Add all rectangles for that partner.
\[W=\sum(\text{height}\times\text{width})\]
Shortcut: base plus extra
\[W=12C+x(12-m)\]
For an addition \(x\) after \(m\) months. This matches the segmented form.
Shortcut: base minus withdrawn part
\[W=12C-x(12-m)\]
For a withdrawal \(x\) after \(m\) months.
EAdmission, retirement and withdrawal
Admission after \(m\) months
A new partner joining after \(m\) months remains for \(12-m\) months.
\[W_{\text{new}}=C(12-m)\]
Retirement after \(m\) months
A partner leaving after \(m\) months has active time \(m\).
\[W_{\text{leaving}}=Cm\]
Date wording
“At the end of the \(m\)-th month” normally gives \(m\) completed months before the change; the new capital acts for the remaining months.
The door decides active time
Joining late counts the time after the entry door. Leaving early counts the time before the exit door.
\[T_{\text{join}}=12-m,\qquad T_{\text{leave}}=m\]
FWorking and sleeping partners
Working partner
Contributes labour or management in addition to capital. The agreement may award salary, bonus or commission.
Sleeping partner
Contributes capital but does not manage daily operations. The label alone does not change the capital-time ratio.
Agreement controls
First apply every explicitly stated fixed payment or commission rule; then divide the residual profit by the stated ratio or capital-time ratio.
Role and capital share are separate layers
Salary compensates work; ratio share rewards the agreed partnership weight. A working partner may receive both when the question says so.
GSalary, bonus and interest on capital
Fixed salary before sharing
If annual profit before salary is \(Q\) and fixed salary is \(F\):
\[Q_{\text{residual}}=Q-F\]
\[\text{working partner total}=F+\text{ratio share}\]
Monthly salary
\[F=12s\]
Multiply a monthly salary \(s\) by the active salary months stated.
Interest on capital
If separately allowed at rate \(r\), calculate it for the relevant capital-time blocks before sharing the balance.
\[J=\sum\frac{C_i r t_i}{1200}\]
Here \(t_i\) is in months.
Profit waterfall: fixed first, ratio second
Start from the profit named in the question. Remove prior appropriations in their stated order. Only the remainder flows into the ratio buckets.
\[Q\to Q-F-B-J\to\text{divide residual}\]
HCommission rules — identify the base
Commission on profit before commission
If profit \(Q\) is stated before charging commission at \(c\%\):
\[K=\frac{cQ}{100}\]
Commission on profit after commission
If \(Q\) is profit before commission but commission is \(c\%\) of profit remaining after commission:
\[K=\frac{cQ}{100+c}\]
Commission on sales
Apply the stated rate directly to sales \(V\), not to business profit:
\[K=\frac{cV}{100}\]
“Percent of what?” is the whole problem
The words “before commission,” “after commission” and “on sales” create different bases. Translate the sentence into an equation before using a percentage.
\[K=c\%\times\text{stated base}\]
IGuaranteed minimum share and deficiency
Normal share
\[N_i=Q\frac{W_i}{\sum W}\]
Deficiency
If guaranteed minimum \(G_i\) exceeds normal share:
\[D_i=G_i-N_i\quad(G_i\gt N_i)\]
Bearer of deficiency
Deduct the deficiency from the guarantor or guarantors in the ratio explicitly stated. Do not assume a bearer if the wording does not identify one.
A guarantee is a floor, not an extra share
Compare the normal share with the guaranteed minimum. Add only the shortfall, then recover that shortfall from the responsible partner(s).
JFinding missing capital, time, share or total profit
Missing capital
For two partners with share ratio \(a:b\):
\[\frac{C_AT_A}{C_BT_B}=\frac ab\Rightarrow C_B=\frac{bC_AT_A}{aT_B}\]
Missing time
\[T_B=\frac{bC_AT_A}{aC_B}\]
Total profit from one share
If partner \(A\) gets \(S_A\) out of ratio \(a:b:c\):
\[Q=S_A\frac{a+b+c}{a}\]
Share ratio from amounts
\[S_A:S_B:S_C\]
Reduce the actual profit amounts to their lowest integer ratio.
Difference of shares
If \(S_A-S_B=D\) and ratio is \(a:b\):
\[Q=D\frac{a+b}{a-b}\]
One share exceeds another
If \(S_A=kS_B\), then the corresponding weights satisfy
\[C_AT_A=kC_BT_B\]
Build one proportion, then isolate the blank
Almost every missing-value problem is a disguised equation between capital-time weights and profit shares.
\[\frac{C_AT_A}{C_BT_B}=\frac{S_A}{S_B}\]
KComplete decision table
| Wording clue | First operation | Then share |
|---|---|---|
| Same period throughout | Cancel common time | In capital ratio |
| Different joining dates | Count each active duration | In capital-time ratio |
| Capital added or withdrawn | Break the timeline into constant-capital blocks | In total weight ratio |
| Working partner gets salary | Deduct fixed salary from stated profit | Divide residual; add salary to that partner |
| Commission after commission | Use \(cQ/(100+c)\) | Divide residual if required |
| Guaranteed minimum | Find normal share and deficiency | Transfer deficiency from named guarantor(s) |
| Loss instead of profit | Form the same valid weights | Allocate loss in the applicable ratio |
03SAVE TIME
Short Tricks & Magic Formulas
Keep capital in convenient units, measure time consistently and cancel common factors early.
1Ratio compression
Drop common currency units
Capitals \(₹40{,}000\) and \(₹60{,}000\) may be written as \(4:6=2:3\) when time is equal.
Cancel common time unit
\[(40\times12):(60\times8)=6:6=1:1\]
Use thousands and months directly; units cancel.
Divide weights by their HCF
Reduce capital-time products before multiplying by profit. Smaller numbers mean fewer mistakes.
2Timeline shortcuts
Initial capital plus late addition
\[W=12C+x(12-m)\]
Initial capital minus late withdrawal
\[W=12C-x(12-m)\]
Entry/exit complement
\[T_{\text{entry}}+m=12,\qquad T_{\text{exit}}=m\]
Mark the change month on a \(12\)-box strip.
3Difference and fraction shortcuts
Share from ratio
\[a:b\Rightarrow S_A=Q\frac a{a+b}\]
Profit from share difference
\[Q=D\frac{a+b}{|a-b|}\]
Profit from one share
\[Q=S_i\frac{\sum a}{a_i}\]
4Remuneration shortcuts
Fixed-first rule
\[\text{residual}=Q-(\text{salary}+\text{bonus}+\text{fixed interest})\]
After-commission conversion
\[c\%\text{ after commission}\Rightarrow\frac{c}{100+c}\text{ of pre-commission profit}\]
Working partner’s total
\[\text{fixed remuneration}+\text{share of residual}\]
5Fast option filters
Double capital, half time
\[(2C)\left(\frac T2\right)=CT\]
The profit share remains unchanged.
Scale invariance
\[kC_1T_1:kC_2T_2=C_1T_1:C_2T_2\]
A common capital scale does not alter the ratio.
Sanity direction
With all else fixed, more capital or longer active time cannot produce a smaller weight.
SSC partnership speed dashboard
Draw the timeline, convert each block to weight, remove fixed charges, reduce the ratio, then distribute.
\[\text{timeline}\to CT\to\text{residual}\to\text{ratio}\to\text{share}\]
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Distractors usually ignore time, count the wrong side of a date, or share the full profit before fixed payments.
Trap 1: capital ratio used despite unequal time
- Capital ratio is sufficient only when active times are equal.
- Joining after \(m\) months means only \(12-m\) months remain.
- Convert every duration to the same unit.
Trap 2: fixed salary shared again
- Deduct salary, bonus or fixed charge before dividing the residual.
- Add the fixed payment only to its recipient.
- Do not allocate the original profit and then add salary on top.
Trap 3: commission base confused
- \(cQ/100\) is for commission on profit before commission.
- \(cQ/(100+c)\) is for commission on profit after commission.
- Commission on sales uses sales as the base.
Opening–closing average guessed
Do not use \((C_{\text{open}}+C_{\text{close}})/2\) unless capital changes uniformly. Discrete changes need timeline blocks.
Month of change double-counted
Translate the wording into completed months and remaining months. A one-month shift changes the weight.
Sleeping partner ignored
A sleeping partner still earns the agreed capital-based share. Only an explicit agreement changes the allocation.
05LOCK IT IN
Memory Hooks & Mnemonics
Remember partnership as a timeline first and a ratio question second.
“Money shakes hands with months”
Profit weight is capital multiplied by active time, never capital alone when times differ.
“Every change cuts a new rectangle”
Break the year whenever capital changes; add all capital-time rectangle areas.
“Join counts after; leave counts before”
A late entrant uses remaining time; an early retiree uses elapsed time.
“Fixed first, fraction later”
Remove salary, bonus, interest or commission as instructed; divide only the residual pool.
“Commission asks: percent of what?”
Before commission, after commission and sales each create a different percentage base.
“Missing box? Cross the weights”
Set capital-time ratio equal to profit-share ratio, cross-multiply and isolate the unknown.
Master weight\(C\times T\)
Changing capitalsum each block
Join latecount remaining time
Leave earlycount elapsed time
Fixed paymentdeduct before ratio
Missing valueequate weight and share ratios