Cost Price (CP)
Effective cost incurred to acquire and prepare an article for sale. Include stated transport, repair or overhead expenses.
6 SSC CGL · Mathematics
Profit, Loss, Marked Price & Discount
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
Track the price path and apply every percentage to the correct base.
CP
→ MP
→ SP
→ MP
→ SP
1. Costpurchase plus allowed expenses
2. Mark-upcost to marked/list price
3. Discountmarked price to selling price
4. Profit/Lossselling price against cost price
5. Special Modelssame SP, false weight and tax
6. Reverserecover CP, MP or target SP
\[\mathrm{CP}\xrightarrow{\text{mark-up}}\mathrm{MP}\xrightarrow{\text{discount}}\mathrm{SP}\xrightarrow{\text{compare with CP}}\text{profit or loss}\]
02FOUNDATION
The Foundation (Basics) — Core Concepts
First identify CP, MP and SP; then attach each percentage to its own base.
AThe price vocabulary
Marked Price (MP)
Label/list price before discount. Mark-up is normally calculated on CP.
Selling Price (SP)
Amount charged after discount, before or after tax as the wording specifies.
Profit or Loss
\[\text{profit}=\mathrm{SP}-\mathrm{CP}\]
\[\text{loss}=\mathrm{CP}-\mathrm{SP}\]
The three-tag price path
Mark-up takes CP to MP. Discount takes MP to the pre-tax SP. Profit/loss compares SP with CP. Jumping across the wrong tags changes the denominator.
BProfit and loss percentages
Profit percentage
\[p\%=\frac{\mathrm{SP}-\mathrm{CP}}{\mathrm{CP}}\times100\%\]
CP is the standard base.
Loss percentage
\[\ell\%=\frac{\mathrm{CP}-\mathrm{SP}}{\mathrm{CP}}\times100\%\]
Price multipliers
\[\mathrm{SP}=\mathrm{CP}\left(1+\frac p{100}\right)\]
\[\mathrm{SP}=\mathrm{CP}\left(1-\frac\ell{100}\right)\]
Recover CP from profit
\[\mathrm{CP}=\mathrm{SP}\frac{100}{100+p}\]
Recover CP from loss
\[\mathrm{CP}=\mathrm{SP}\frac{100}{100-\ell}\]
Target SP
For target profit \(t\%\):
\[\mathrm{SP}_{t}=\mathrm{CP}\frac{100+t}{100}\]
CP is the ground floor
Profit and loss percentages are measured from CP. SP above CP creates profit; SP below CP creates loss.
\[\frac{\text{difference}}{\mathrm{CP}}\times100\%\]
CMark-up and profit margin are different
Mark-up on cost
\[m\%=\frac{\mathrm{MP}-\mathrm{CP}}{\mathrm{CP}}\times100\%\]
Profit margin on sales
\[q\%=\frac{\mathrm{SP}-\mathrm{CP}}{\mathrm{SP}}\times100\%\]
Margin uses SP, not CP.
Convert mark-up and margin
\[q=\frac{100p}{100+p},\qquad p=\frac{100q}{100-q}\]
Here \(p\) is profit on CP and \(q\) is margin on SP.
DDiscount and marked price
Discount amount
\[D=\mathrm{MP}-\mathrm{SP}\]
Discount percentage
\[d\%=\frac{\mathrm{MP}-\mathrm{SP}}{\mathrm{MP}}\times100\%\]
MP is the base.
Selling price after discount
\[\mathrm{SP}=\mathrm{MP}\left(1-\frac d{100}\right)\]
Recover marked price
\[\mathrm{MP}=\mathrm{SP}\frac{100}{100-d}\]
Discount is not loss
Discount compares MP with SP; loss compares CP with SP. An article may be sold at a discount and still yield profit.
The discount cutter trims MP, not CP
Imagine the marked tag being cut down. Whether the remaining SP produces profit or loss can be decided only after comparing it with CP.
ESuccessive discounts and equivalent discount
Multiplier method
\[\mathrm{SP}=\mathrm{MP}\prod_i\left(1-\frac{d_i}{100}\right)\]
Two discounts
\[d_{\rm eq}=a+b-\frac{ab}{100}\%\]
Three or more
Multiply remaining-price factors; do not add discount rates.
\[d_{\rm eq}=100\left[1-\prod_i(1-d_i/100)\right]\%\]
\[20\%\text{ and }10\%\Rightarrow1-(0.8)(0.9)=0.28\Rightarrow28\%\text{ equivalent discount}\]
Every discount acts on what survives
Convert each discount to its remaining fraction. Their product gives the final fraction of MP.
FMark-up followed by discount
Net factor from CP to SP
\[\frac{\mathrm{SP}}{\mathrm{CP}}=\left(1+\frac m{100}\right)\left(1-\frac d{100}\right)\]
Profit after mark-up and discount
\[p=100\left[\left(1+\frac m{100}\right)\left(1-\frac d{100}\right)-1\right]\]
Required mark-up
For discount \(d\%\) and target profit \(p\%\):
\[m=100\left(\frac{100+p}{100-d}-1\right)\]
\[\mathrm{MP}=\mathrm{CP}\frac{100+p}{100-d}\quad\text{for target profit }p\%\text{ after discount }d\%\]
GTax, commission and expenses
Discount then tax
If tax rate is \(t\%\) on discounted price:
\[\text{bill}=\mathrm{MP}(1-d/100)(1+t/100)\]
Tax is not seller profit
Unless stated otherwise, tax collected for the government is not part of the seller's revenue for profit calculation.
Cost expenses
Add allowed carriage, repair, loading or production expenses to purchase price before computing profit percentage.
\[\mathrm{CP}_{\rm effective}=\text{purchase cost}+\text{expenses}\]
| Sequence | Multiplier from MP | Warning |
|---|---|---|
| Discount then tax | \((1-d/100)(1+t/100)\) | Tax generally applies to discounted price. |
| Two discounts then tax | \((1-d_1/100)(1-d_2/100)(1+t/100)\) | Apply in the stated order; multiplication makes order irrelevant only for pure factors on the same base chain. |
| Commission on SP | Net receipt \(=\mathrm{SP}(1-c/100)\) | Use stated commission base. |
| Overheads added to purchase | Effective CP \(=\) purchase plus overheads | Profit denominator becomes effective CP. |
The checkout conveyor has ordered gates
Price passes through discount, then tax or commission as stated. Apply one multiplier at each gate and keep tax separate from profit unless the question explicitly combines them.
HSame SP and same CP comparison cases
Same SP: equal gain and loss rates
One article gains \(x\%\), another loses \(x\%\), both sold at the same SP.
\[\text{net loss}=\frac{x^2}{100}\%\]
Same CP: equal gain and loss rates
If their CPs are equal, equal \(x\%\) profit and loss cancel in amount.
\[\text{net result}=0\]
General same SP case
For common SP \(S\), gain \(p\%\), loss \(\ell\%\):
\[\mathrm{CP}_1=\frac{100S}{100+p},\quad\mathrm{CP}_2=\frac{100S}{100-\ell}\]
Same destination can hide different costs
With equal selling prices, the profitable item had a lower CP and the loss-making item a higher CP. Reconstruct CPs before combining.
IDishonest dealer and false-weight models
Short weight only
Charges for \(W\) units but gives \(w\) units at nominal CP rate:
\[\text{gain}=\frac{W-w}{w}\times100\%\]
False weight plus price gain
If nominal selling rate has gain \(p\%\):
\[\text{effective factor}=\left(1+\frac p{100}\right)\frac{W}{w}\]
Claimed loss but short weight
Use signed price factor and weight factor together.
\[\text{net }\%=100\left[\left(1-\frac\ell{100}\right)\frac{W}{w}-1\right]\]
Money is collected for the promised weight
The dealer's revenue corresponds to \(W\), but cost corresponds only to actual weight \(w\). Effective gain compares revenue with cost of what was truly delivered.
JQuantity offers and mixed commercial models
Buy \(x\), get \(y\) free
\[\text{equivalent discount}=\frac{y}{x+y}\times100\%\]
Extra quantity \(q\%\)
Pay for \(100\) units and receive \(100+q\).
\[\text{effective discount}=\frac{100q}{100+q}\%\]
Price reduction vs quantity increase
For constant spending, a price fall of \(d\%\) permits quantity increase
\[\frac{100d}{100-d}\%\]
03SAVE TIME
Short Tricks & Magic Formulas
Use a base of \(100\) and multiply price factors instead of rebuilding amounts repeatedly.
1Assume CP or MP as \(100\)
Profit/loss question
Set \(\mathrm{CP}=100\). Then profit \(p\%\) gives \(\mathrm{SP}=100+p\); loss \(\ell\%\) gives \(\mathrm{SP}=100-\ell\).
Discount question
Set \(\mathrm{MP}=100\). Discount \(d\%\) leaves \(\mathrm{SP}=100-d\).
Mixed mark-up and discount
Set CP to \(100\), mark to \(100+m\), then apply the discount factor.
2One-line net percentage
Successive changes
\[a+b+\frac{ab}{100}\%\]
Use negative signs for decreases.
Equal gain/loss with same SP
\[-\frac{x^2}{100}\%\]
Two discounts
\[a+b-\frac{ab}{100}\%\]
3Target-price formulas
Change profit from \(p\%\) to \(q\%\)
If old SP is known:
\[\mathrm{SP}_{q}=\mathrm{SP}_{p}\frac{100+q}{100+p}\]
Change loss to profit
\[\mathrm{SP}_{p}=\mathrm{SP}_{\ell}\frac{100+p}{100-\ell}\]
Profit amount identifies CP
\[\mathrm{CP}=\frac{100\times\text{profit amount}}{p}\]
4Option-first checks
The commercial arithmetic dashboard
Identify the percentage base, translate every rate to a multiplier, and trace CP → MP → SP in order. Estimate whether the result should exceed CP.
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Wrong options usually use the wrong percentage base or add successive rates directly.
Trap 1: discount is not loss
Discount uses MP as its base. Loss uses CP. A discounted article may still be profitable if SP remains above CP.
Trap 2: equal gain and loss need a condition
Equal rates cancel only when CPs are equal. With equal SPs, they produce a net loss of \(x^2/100\%\).
Trap 3: tax and discount use sequential bases
Tax is normally calculated after discount. Add/subtract rates only when they truly share the same base.
05LOCK IT IN
Memory Hooks & Mnemonics
Use the price-tag story to recall every formula under exam pressure.
“Profit stands on Cost”
Profit and loss percentages use CP as the denominator unless another base is explicitly named.
“Discount cuts the Mark”
Discount amount and discount percentage are measured from marked price.
“Successive means multiply”
Each discount or price change acts on what remains after the previous step.
“Trace the price staircase”
Move CP → MP → SP, one multiplier at a time, then compare SP with CP.
“False weight compares promised revenue with actual cost”
Revenue is for full stated weight; cost is for the smaller delivered weight.
“When only rates matter, make the base 100”
Choose CP \(=100\) for profit/loss and MP \(=100\) for discount questions.
Profit %Profit ÷ CP
Loss %Loss ÷ CP
Discount %Discount ÷ MP
MarginProfit ÷ SP
SuccessiveMultiply factors
Target priceReverse the multiplier