Costing
Find what the institution spends: fixed cost, variable cost, total cost and cost per learner.
S4 · Topic 9 · Reasoning & Quant
Costing, Fee & Revenue Word Problems
Notes for UPSC Principal & Vice-Principal examinations only.
01Basics
The Foundation — Build the Arithmetic Language
Identify the base, translate the words, and preserve units before calculating.
AWhat this topic really tests
Fee
Convert fee per student per month, term or year into demand, concession and actual collection.
Revenue
Distinguish amount due from amount received, then add all legitimate income streams.
Percent Base
Profit uses cost price as base; discount uses marked price; collection rate uses fee demand.
BDecode every word before touching the numbers
| Term | Meaning | Mathematical translation | School-context cue |
|---|---|---|---|
| Cost Price (CP) | Amount paid to acquire an item/service. | Base for profit or loss | Purchase price of laboratory kits. |
| Selling Price (SP) | Amount received on sale. | After profit/loss/discount chain | Sale price of old furniture or prospectus. |
| Marked Price (MP) | Displayed or listed price before discount. | Base for discount | Catalogue price of smart boards. |
| Fixed Cost (FC) | Does not change with short-run student volume. | Constant within the relevant range | Building rent, basic software licence. |
| Variable Cost (VC) | Changes with activity or learner count. | Unit variable cost × volume | Worksheets, consumables, meal cost. |
| Fee Demand | Gross amount legally/administratively due. | Learners × applicable fee | Before concessions and non-payment. |
| Concession | Approved reduction from fee demand. | Rate × eligible base | Sibling, merit or need-based relief. |
| Collection | Cash actually received during the period. | Demand − concession − unpaid dues | Not the same as gross fee demand. |
| Arrears / Outstanding | Amount due but not yet collected. | Net demand − collection | May be recovered in a later period. |
| Revenue | Income earned from all stated sources. | Fee + grants + services + other income | Use only sources supplied in the question. |
| Surplus / Deficit | Excess of revenue over cost, or vice versa. | Revenue − total cost | A school surplus is not automatically business profit. |
The golden reading rule
Write the base beside every percentage. For example, “concession of \(15\%\)” is incomplete until you know whether it applies to tuition fee, total fee, eligible students, or every student.
CPercentage backbone — the engine of the chapter
Part, Rate and Base
Core relationship\[\text{Part}=\frac{\text{Rate}}{100}\times\text{Base}\]
Cover the required quantity and divide by the other two to rearrange mentally.
Percentage Change
Always use the original as denominator\[\%\Delta=\frac{\text{New}-\text{Original}}{\text{Original}}\times100\]
A fall is negative; an increase is positive.
Reverse Percentage
Recover the original\[\text{Original}=\frac{\text{Final}}{1\pm r}\]
Use \(+r\) after an increase and \(-r\) after a decrease, with \(r\) in decimal form.
CostWhat is spent?
Marked / Stated FeeWhat is quoted?
Gross DemandWhat becomes due?
Net DemandAfter concessions
CollectionWhat is received?
DFoundation formula bank
| Concept | Formula | Base / interpretation |
|---|---|---|
| Profit | \(P=SP-CP\) | Positive when \(SP\gt CP\) |
| Loss | \(L=CP-SP\) | Positive when \(CP\gt SP\) |
| Profit percent | \(P\%=\dfrac{P}{CP}\times100\) | Cost price is the denominator. |
| Loss percent | \(L\%=\dfrac{L}{CP}\times100\) | Cost price is the denominator. |
| Selling price at profit | \(SP=CP\left(1+\dfrac{p}{100}\right)\) | Multiplier \(1+\frac p{100}\) |
| Selling price at loss | \(SP=CP\left(1-\dfrac{l}{100}\right)\) | Multiplier \(1-\frac l{100}\) |
| Discount | \(D=MP-SP\) | Discount is measured on marked price. |
| Discount percent | \(D\%=\dfrac{D}{MP}\times100\) | Marked price is the denominator. |
| Selling price after discount | \(SP=MP\left(1-\dfrac d{100}\right)\) | Apply the discount multiplier. |
| Total cost | \(TC=FC+VC\) | All costs must cover the same period. |
| Total variable cost | \(VC=vq\) | \(v\): variable cost per unit; \(q\): volume. |
| Average cost | \(AC=\dfrac{TC}{q}=\dfrac{FC}{q}+v\) | Fixed cost per learner falls as enrolment rises. |
| Fee demand | \(G=\sum n_i f_i t_i\) | Group-specific learners × fee × chargeable periods. |
| Net demand | \(N=G-C\) | \(C\): valid concessions/waivers. |
| Collection rate | \(r_c=\dfrac{A}{N}\times100\) | \(A\): amount actually collected. |
| Outstanding amount | \(O=N-A\) | Do not subtract concessions again. |
| Surplus / deficit | \(S=R-TC\) | \(S\gt 0\): surplus; \(S\lt 0\): deficit. |
02Expert
The Deep Dive — Group-A / Group-B Complexity
Layered rates, weighted groups, reverse calculations, capacity and break-even.
ASuccessive percentage changes: never add blindly
Two increases
\[a+b+\frac{ab}{100}\%\]
Example: fee rises by \(10\%\) and then \(20\%\); net rise is \(32\%\), not \(30\%\).
Two decreases
\[a+b-\frac{ab}{100}\%\text{ decrease}\]
Two discounts of \(20\%\) and \(10\%\) equal a single discount of \(28\%\).
Increase then decrease
\[a-b-\frac{ab}{100}\%\]
Keep signs: treat the decrease as \(-b\). Equal rise and fall of \(x\%\) cause a net fall of \(\frac{x^2}{100}\%\).
Universal multiplier method\[F=I\prod_{k=1}^{m}\left(1+\frac{r_k}{100}\right),\quad r_k<0\text{ for a decrease}\]
BMarkup–discount–profit chain
Forward chain
\[MP=CP\left(1+\frac m{100}\right),\qquad SP=MP\left(1-\frac d{100}\right)\]
\[\frac{SP}{CP}=\left(1+\frac m{100}\right)\left(1-\frac d{100}\right)\]
Then \(P\%=\left(\frac{SP}{CP}-1\right)100\).
Required markup for a target profit
\[1+\frac m{100}=\frac{1+\frac p{100}}{1-\frac d{100}}\]
If a supplier gives \(20\%\) discount but the vendor needs \(25\%\) profit, markup must satisfy the ratio above.
CFee models with heterogeneous groups
Group-wise demand
\[G=\sum_{i=1}^{k}n_i f_i t_i\]
Use separate rows when classes, fee rates, months or admission dates differ.
Weighted effective fee
\[\bar f=\frac{\sum n_i f_i}{\sum n_i}\]
Do not take a simple average unless groups are equally large.
Weighted concession rate
\[r=\frac{\sum B_i r_i}{\sum B_i}\]
Weight each concession rate by its fee base \(B_i\), not merely by headcount.
Headcount is not always the correct weight
If one group pays \(\mathrm{Rs.}\,1{,}000\) and another pays \(\mathrm{Rs.}\,2{,}000\), equal numbers of students do not create equal fee bases. Compute rupee concessions group by group.
DTime dimension: months, terms, joining and leaving
| Situation | Correct treatment | Typical error |
|---|---|---|
| Monthly fee, annual demand | Multiply by chargeable months: \(nf\times12\), unless stated otherwise. | Using the academic-session months without reading the question. |
| Quarterly / term fee | Use number of chargeable quarters/terms, not months. | Multiplying a quarterly fee by \(12\). |
| Mid-session admission | Charge only stated applicable periods plus one-time charges. | Applying full-year recurring fee. |
| One-time admission fee | Apply only to new admissions. | Charging every continuing student. |
| Withdrawal / dropout | Use the number present during each fee period if relevant. | Using year-end strength for the full year. |
| Arrear recovered this year | Add to cash collection only if the question asks current receipts. | Treating it as current-year fee demand. |
| Advance fee | Follow the asked measure: cash receipt or earned revenue. | Assuming both are identical without wording support. |
EContribution, break-even and capacity utilisation
Contribution per learner
\[c=f-v\]
Fee per learner minus variable cost per learner.
Total contribution
\[C=q(f-v)\]
Contribution first covers fixed cost; the remainder is surplus.
Break-even enrolment
\[q_{BE}=\frac{FC}{f-v}\]
Round up when learners must be whole.
Capacity utilisation
\[U=\frac{q}{Q}\times100\]
\(Q\) is maximum capacity, \(q\) is actual enrolment.
Surplus with collection leakage
\[S=r_c qf-qv-FC\]
Here \(r_c\) is the collection rate in decimal form. If concessions exist, apply collection rate to net demand unless the question specifies a different base.
Enrolment for a target surplus
\[q=\frac{FC+S^*}{r_cf-v}\]
\(S^*\) is target surplus. The denominator is effective contribution per learner.
FReverse, ratio and invariance models
Same revenue after fee change
\[q_1f_1=q_2f_2\]
If fee rises by \(x\%\), the permissible percentage fall in enrolment for unchanged revenue is \(\frac{100x}{100+x}\%\).
Same revenue after enrolment fall
\[\text{Required fee rise}=\frac{100x}{100-x}\%\]
Use when enrolment falls by \(x\%\) and revenue must remain constant.
Revenue ratio
\[R_1:R_2=(q_1:q_2)(f_1:f_2)\]
Multiply ratios component-wise; reduce only after multiplication.
GAdvanced costing extensions
Shared cost allocation
When a common cost \(K\) is allocated in ratio \(a:b:c\):
\[K_A=K\frac{a}{a+b+c},\quad K_B=K\frac{b}{a+b+c},\quad K_C=K\frac{c}{a+b+c}\]
Allocate only the shared amount; add direct costs separately.
Weighted average unit cost
For batches bought at different unit costs:
\[\bar c=\frac{q_1c_1+q_2c_2+\cdots+q_kc_k}{q_1+q_2+\cdots+q_k}\]
This is the correct average cost of consumables or books.
Wastage / loss of quantity
If \(w\%\) units are unusable, saleable units are:
\[q_s=q\left(1-\frac w{100}\right)\]
To recover total cost, break-even price per usable unit is \(\frac{TC}{q_s}\).
Commission or collection charge
If an agency retains \(k\%\) of collection:
\[\text{Net receipt}=A\left(1-\frac k{100}\right)\]
If a target net receipt is known, divide by the retention multiplier.
03Accuracy
The UPSC Trap / Watch Out
Classic distractors, false shortcuts and errors made under exam pressure.
ATwelve traps examiners repeatedly exploit
Profit denominator trap
Profit percent is on \(CP\), not \(SP\). If profit is \(\mathrm{Rs.}\,200\) on cost \(\mathrm{Rs.}\,800\), it is \(25\%\), not \(20\%\).
Discount denominator trap
Discount percent is on \(MP\), not \(CP\) or \(SP\).
Demand vs collection
Fee demand is what is due; collection is what arrives. A \(90\%\) collection rate creates outstanding dues.
Concession counted twice
Once net demand is computed, do not subtract concession again while calculating outstanding.
Percentages simply added
Successive changes act on changing bases. Use multipliers.
Average of rates
The simple mean of group rates is valid only when their monetary bases are equal.
Wrong time unit
Convert months, quarters, terms and years before multiplication. Never mix monthly fee with annual fixed cost untreated.
New admissions vs total strength
Admission fee usually applies to new entrants only; tuition applies to the chargeable enrolled population.
Whole-person rounding
Break-even enrolment of \(312.1\) means at least \(313\) learners, not \(312\).
Surplus vs profit language
Mathematically both may be revenue minus cost, but answer with the institutional term used in the question.
Arrears classification
Opening arrears collected now affect cash received, but not necessarily current-period demand.
Irrelevant real-world assumptions
Do not invent taxes, refunds, regulations or exemptions. Use only data stated in the problem.
BHigh-speed shortcuts and mental maths
Multiplier strip
Increase by \(p\%\): multiply by \(1+\frac p{100}\). Decrease by \(p\%\): multiply by \(1-\frac p{100}\).
Fraction–percent pairs
Memorise \(\frac18=12.5\%\), \(\frac16=16\frac23\%\), \(\frac15=20\%\), \(\frac14=25\%\), \(\frac13=33\frac13\%\), \(\frac38=37.5\%\), \(\frac25=40\%\), \(\frac58=62.5\%\), \(\frac34=75\%\).
Equal rise and fall
A rise and fall by the same \(x\%\) always gives loss \(\frac{x^2}{100}\%\). For \(20\%\), net fall is \(4\%\).
Reverse by ratio
After \(20\%\) reduction, final : original is \(4:5\). So divide by \(4\), multiply by \(5\).
Revenue invariance
If learners fall \(20\%\), fee must rise \(\frac{20}{80}\times100=25\%\) to preserve revenue.
Use assumed base \(100\)
For percentage-only questions, assume \(CP=100\), fee \(=100\), or enrolment \(=100\), whichever removes arithmetic.
Contribution lens
Each extra learner contributes fee minus variable cost. Divide the fixed gap by that contribution.
Weighted rate from rupees
Find total concession rupees first; then divide by total gross demand. This avoids choosing the wrong weights.
Option elimination
Check direction, magnitude, units and denominator. Many options fail before full calculation.
Pressure protocol: B–U–R–C
Base → Units → Rate order → Check. Mark the base of each percentage, align time units, apply changes in order, and test whether the result’s direction is sensible.
04Revise
Last-Minute Master Sheet
Recall these relationships before entering the examination hall.
Profit / loss\[P=SP-CP,\quad L=CP-SP\]\[P\%=\frac{P}{CP}100,\quad L\%=\frac{L}{CP}100\]
Discount\[D=MP-SP,\quad D\%=\frac{D}{MP}100\]\[SP=MP\left(1-\frac d{100}\right)\]
Successive rates\[F=I\prod\left(1+\frac r{100}\right)\]
Fee flow\[G=\sum n_i f_i t_i,\quad N=G-C\]\[A=r_cN,\quad O=N-A\]
Costing\[TC=FC+vq,\quad AC=\frac{FC}{q}+v\]
Break-even\[q_{BE}=\frac{FC}{f-v}\qquad q_{S^*}=\frac{FC+S^*}{f-v}\]
Collection-adjusted target\[q=\frac{FC+S^*}{r_cf-v}\]
Weighted average\[\bar x=\frac{\sum w_ix_i}{\sum w_i}\]
Revenue invariance\[q_1f_1=q_2f_2\]
One-line memory map
Cost creates the base → fee creates demand → concession creates net demand → payment creates collection → revenue minus cost creates surplus or deficit.