Profit and Loss — AFCAT Numerical Ability Formulas and Traps

Cost Price, Selling Price and Profit Per Cent

Profit, loss and their percentages

Profit per cent

\text{Profit}\% = \dfrac{SP - CP}{CP} \times 100

Selling price and cost price as multiples

Selling price from cost price

SP = CP\left(1 \pm \dfrac{x}{100}\right)

Profit from a ratio of cost price to selling price

Profit from a ratio

CP : SP = a : b \;\Rightarrow\; \text{Profit}\% = \dfrac{b - a}{a} \times 100

Common traps

Divide by the cost price

Bought for ₹400 and sold for ₹500: the profit of 100 is

25\%

of the cost. Dividing by the selling price gives

20\%

, which is the wrong base.

Find the cost before the percentage

When a question gives the selling price and the gain in rupees, the cost is SP minus gain. Taking the gain as a share of the selling price is the usual slip.

The percentage is of the cost, not of the selling price

Sold for ₹1,320 at a

10\%

profit: the cost is

1320 \div 1.1 = 1200

, not

1320 - 132 = 1188

.

Undo a fall by dividing

After a

25\%

fall to ₹36, the old price is

36 \div 0.75 = 48

. Adding

25\%

to 36 gives 45, which is too little.

The base is the cost

CP : SP = 3 : 2

is a loss of 1 on 3, that is

33\tfrac 13\%

. Dividing by the 2 gives

50\%

.

A loss on the selling price is a smaller loss on the cost

A loss of

10\%

of the selling price is

9\tfrac{1}{11}\%

of the cost. The option that repeats the

10\%

answers a different question.

Finding the Cost Price, Then Repricing

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Through the cost price to a new selling price

Repricing in one line

SP_2 = SP_1 \times \dfrac{100 \pm y}{100 \pm x}

When two prices differ by a known sum

Cost from a price gap

CP = \dfrac{\text{difference in rupees}}{\text{difference in per cent}} \times 100

Setting up an equation in the cost price

Equal profit and loss

SP_1 - c = c - SP_2 \;\Rightarrow\; c = \dfrac{SP_1 + SP_2}{2}

Common traps

The percentages do not add

From a

10\%

loss to a

20\%

gain is not 'add

30\%

to the selling price':

1890 \times 1.3 = 2457

, while the right answer is 2520. Both percentages are on the cost.

Divide to reach the cost

₹1,890 at a

10\%

loss means a cost of

1890 \div 0.9 = 2100

, not

1890 + 189 = 2079

.

A loss and a gain add

From a

6\%

loss to a

12\%

gain is

18\%

of the cost, not

6\%

. Count the loss as below zero.

The gap is a share of the cost

The rupee gap is a percentage of the cost price, so divide it by that percentage. Dividing it by either selling price gives nothing useful.

The new percentage is on the new cost

If both prices rise by ₹100, the profit in rupees is the same but its percentage is taken on

c + 100

. Dividing by the old cost gives the wrong equation.

Midway only for equal rupee amounts

The cost is the average of two prices only when the profit at one equals the loss at the other in rupees.

Profit Counted in Articles, and Rates

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Cost of x articles equals the selling price of y

Cost of x equals sale of y

\text{Profit}\% = \dfrac{x - y}{y} \times 100

A gain equal to the cost of n articles

Gain worth the cost of n articles

\text{Profit}\% = \dfrac{n}{N} \times 100

A gain equal to the selling price of n articles

Gain worth the sale of n articles

\text{Profit}\% = \dfrac{n}{N - n} \times 100

Bought and sold at so many for a price

Profit from two rates

\text{Profit}\% = \left(\dfrac{\text{SP per article}}{\text{CP per article}} - 1\right) \times 100

Common traps

Divide by the number sold

The cost of 21 equals the sale of 15: the gain is 6 on 15,

40\%

. Dividing by 21 gives

28.6\%

, the wrong base.

Which side is larger decides profit or loss

If the cost of fewer articles equals the sale of more, each article sells below cost: a loss, however the question words it.

Scale the count, do not add to it

If 60 pens gain the cost of 12, then 120 pens gain the cost of 24. The gain grows with the number sold; it is not 12 more each time.

The cost of n is not the selling price of n

A gain worth the cost of n articles and one worth their selling price give different answers. Read which one the question names; the next concept handles the second.

Subtract before dividing

A gain of the selling price of 6 on 30 articles is 6 on a cost of 24, that is

25\%

. Dividing 6 by 30 gives

20\%

, which is the profit as a share of the selling price.

A loss adds to the count

A loss of the selling price of

n

means the cost is the selling price of

N + n

articles, not

N - n

.

Compare prices per article, not counts

'8 for ₹10' against '4 for ₹6' compares nothing until each is a price for one article (or for the same number of articles).

Keep the fractions

A cost of

\tfrac 23

rupee rounded to 0.67 can turn an exact

20\%

into

19.4\%

. Work in fractions, or with an LCM number of articles.

Marked Price, Discount and Successive Discounts

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Discount on the marked price

Selling price after a discount

SP = MP\left(1 - \dfrac{d}{100}\right)

Successive discounts

Single equivalent of two discounts

a + b - \dfrac{ab}{100}

Markup, then discount

Profit after a markup and a discount

\text{Profit}\% = m - d - \dfrac{md}{100}

Marked price for a target profit

Marked price for a target

MP = CP \times \dfrac{100 + p}{100 - d}

Common traps

The discount is on the marked price

A jacket marked ₹2,400 sold for ₹1,980 has a discount of 420 on 2400,

17.5\%

. Dividing by 1980 gives

21.2\%

, which no shop means.

Divide to undo a discount

A price of ₹960 after a

20\%

discount comes from

MP = 960 \div 0.8 = 1200

. Adding

20\%

to 960 gives 1152.

Discounts do not add

25\%

and

12\%

together are

34\%

, not

37\%

: the

12\%

is taken on a price already cut.

A rupee figure is a share of the marked price

If the equivalent discount comes to a rupee amount, that amount is the single percentage of the MP. Divide by the percentage, not by what the buyer paid.

Subtracting the rates overstates the profit

50\%

up and

12\%

off is a profit of

32\%

, not

38\%

: the discount is taken on the raised price.

Same rate up and down is a loss

30\%

up and

30\%

off gives

1.3 \times 0.7 = 0.91

of the cost, a

9\%

loss, not 'no profit, no loss'.

Divide by the discount factor

A selling price of ₹612 after

15\%

off means

MP = 612 \div 0.85 = 720

. Multiplying by 1.15 gives 703.80.

The profit is on the cost, not on the marked price

A

20\%

profit is

20\%

of the cost. Taking it on the marked price gives a selling price that is too high.

Two Articles at One Price, and Costs with Overheads

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Equal gain and loss on equal selling prices

Equal gain and loss

\text{Loss}\% = \dfrac{x^2}{100}

One selling price, different percentages

Total cost of two articles at one price

CP_1 + CP_2 = \dfrac{SP}{1 + \frac{a}{100}} + \dfrac{SP}{1 - \frac{b}{100}}

Overheads, repairs and broken stock

Cost with overheads

CP = \text{purchase price} + \text{overheads}

Common traps

Not 'no gain, no loss'

Equal percentages on equal selling prices always lose

\tfrac{x^2}{100}\%

. The option 'no profit, no loss' is the one set for students who let the two cancel.

Equal costs are a different question

If two articles cost the same and one gains

x\%

while the other loses

x\%

, the result is no gain, no loss. Check which price the question says is equal.

Do not average the percentages

A

50\%

gain and a

25\%

loss on equal selling prices is not a

12.5\%

gain. The costs are different, and here they balance exactly.

The percentage is on the total cost

A gain of ₹200 on costs of 4000 and 7000 is

\tfrac{200}{11000}

, about

1.8\%

, not

\tfrac{200}{11200}

.

Overheads are part of the cost

The repair money goes into the cost price, so the percentage is taken on purchase plus repairs. Leaving it out overstates the profit.

Broken items still cost money

If 12 of 60 cups break, the cost is still that of all 60; only 48 bring money in.

Lots Sold at Several Prices, and Mixtures

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Total cost against total takings

Profit on a lot

\text{Profit}\% = \dfrac{\text{total takings} - \text{total cost}}{\text{total cost}} \times 100

Cost price of a mixture

Mean cost of a mixture

\bar c = \dfrac{q_1 c_1 + q_2 c_2}{q_1 + q_2}

Part at one profit, the rest at another

Splitting by two rates

\dfrac{\text{part at } p_1}{\text{part at } p_2} = \dfrac{p_2 - P}{P - p_1}

Common traps

Add the totals, do not average the prices

Selling 20 kg at ₹105 and 30 kg at ₹85 is not an average price of ₹95 a kg; it is

4650 \div 50 = 93

.

The base is the total cost

A profit of ₹150 on a lot costing ₹4,500 is

3\tfrac 13\%

. Dividing by the takings of ₹4,650 gives a smaller, wrong figure.

Weight the average

Mixed

2 : 1

, tea at ₹180 and ₹240 averages ₹200 a kg, not ₹210. The plain average is right only for equal quantities.

Alligation pairs each grade with the other's gap

The share of the cheap grade is (dear minus mean), not (mean minus cheap). Swapping them mixes the grades the wrong way round.

Weight by cost

Three quarters at

20\%

and a quarter at a

12\%

loss is not the plain average of

4\%

; the bigger part counts three times, giving

12\%

.

Each rate is on its own part

A part's profit percentage is on that part's cost, not on the whole stock. Work part by part in rupees when in doubt.

False Weights, Short Measures and Adulteration

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Selling at cost price with a short weight

Gain from a short weight

\text{Gain}\% = \dfrac{\text{error}}{\text{true value} - \text{error}} \times 100

A short weight with a profit, a loss or a discount

Actual profit with a short weight

\text{Profit}\% = \left(\dfrac{1 \pm \frac{s}{100}}{w} - 1\right) \times 100

Different false weights for buying and selling

Two false weights

\text{Profit}\% = \left(\dfrac{b}{s} - 1\right) \times 100

Mixing in water, sand or a cheaper grade

Gain from a free filler at cost price

\text{Gain}\% = \dfrac{\text{filler}}{\text{real part}} \times 100

Common traps

Divide by what was given

960 g for a kilogram gains 40 on 960, that is

4\tfrac 16\%

. Dividing by 1000 gives

4\%

, the option set for that slip.

Two gains multiply

A

25\%

profit together with 800 g for a kilogram is

1.25 \times 1.25 = 1.5625

, a profit of

56.25\%

, not

50\%

.

The cost is of what he hands over

With 800 g given, the cost to compare against is 80, not 100. Comparing the price with a full kilogram's cost leaves only the stated profit.

Do not add the percentages

20\%

more when buying and

20\%

less when selling is

\tfrac{1.2}{0.8} = 1.5

, a

50\%

profit, not

40\%

.

One balance used both ways cancels

If the same faulty balance is used for buying and selling, the gain on one side is lost on the other: no profit, no loss.

Water's share is of the mixture

A

20\%

gain at cost price means 1 part water to 5 parts milk, which is

\tfrac 16

of the mixture, not

20\%

of it.

Free filler adds weight, not cost

Sand mixed in raises the quantity sold but not the cost, so divide the target takings by the whole weight, sand included.

Sharing Profit in a Partnership

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Same period: share by capital

Share of profit

\text{share of A} = \dfrac{a}{a + b + c} \times P

Different periods: capital times time

Capital times time

\text{A} : \text{B} = a\,t_a : b\,t_b

Common traps

Take out the deduction first

If a share of the profit goes to charity, split only what is left. Applying the partners' ratio to the whole profit overstates every share.

Chained equalities need the LCM

A = 2B = 3C gives

6 : 3 : 2

, not

1 : 2 : 3

. Give A the LCM of 2 and 3 and divide down.

Time counts

Equal capitals in for 12 and 8 months share the profit

3 : 2

, not equally.

A withdrawal splits the period

A partner who takes out money midway has two products, one before and one after. Using the starting capital for the whole year overstates his share.

Buying on Instalments, and Two Special Cases

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Extra paid on an instalment plan

Extra paid on instalments

\text{extra} = \text{down payment} + n \times \text{instalment} - \text{cash price}

Cash price from instalments with interest

Cash price from n instalments

\text{cash price} = \sum_{k=1}^{n} \dfrac{x}{\left(1 + \frac{r}{100}\right)^k}

Drying fruit and a broken stone

Situation	What stays fixed or sets the price	Worked case
Fresh fruit is dried	The dry pulp	Fruit with $75\%$ water has $25\%$ pulp, so 16 kg holds 4 kg of pulp. Dried to $60\%$ water, it weighs $4 \div 0.4 = 10$ kg. The water share changes; the pulp does not.
A stone priced by the square of its weight breaks	Price $= k \times \text{weight}^2$	Pieces $1 : 3$ of a stone weighing 4: value $4^2 = 16$ before, $1 + 9 = 10$ after, a loss of $\tfrac{6}{16} = 37.5\%$ .
The same stone breaks into equal halves	Price $= k \times \text{weight}^2$	Value $2^2 = 4$ before, $1 + 1 = 2$ after: a $50\%$ loss.

Common traps

Include the down payment

The total paid is the down payment plus every instalment. Comparing only the instalments with the cash price makes the plan look cheaper than cash.

Each payment has its own factor

The second annual instalment is divided by

\left(1 + \tfrac{r}{100}\right)^2

, not by

1 + \tfrac{r}{100}

again.

Use the rate per period

Quarterly instalments at

20\%

a year are discounted at

5\%

a quarter, over four quarters, not at

20\%

once.

Drying does not cut the weight by the water percentage

Going from

75\%

water to

60\%

water does not remove

15\%

of the weight: 16 kg becomes 10 kg. Hold the pulp fixed and divide.

Profit and Loss formulas

Cost Price, Selling Price and Profit Per Cent

Profit, loss and their percentages

Selling price and cost price as multiples

Profit from a ratio of cost price to selling price

Common traps

Finding the Cost Price, Then Repricing

Through the cost price to a new selling price

When two prices differ by a known sum

Setting up an equation in the cost price

Common traps

Profit Counted in Articles, and Rates

Cost of x articles equals the selling price of y

A gain equal to the cost of n articles

A gain equal to the selling price of n articles

Bought and sold at so many for a price

Common traps

Marked Price, Discount and Successive Discounts

Discount on the marked price

Successive discounts

Markup, then discount

Marked price for a target profit

Common traps

Two Articles at One Price, and Costs with Overheads

Equal gain and loss on equal selling prices

One selling price, different percentages

Overheads, repairs and broken stock

Common traps

Lots Sold at Several Prices, and Mixtures

Total cost against total takings

Cost price of a mixture

Part at one profit, the rest at another

Common traps

False Weights, Short Measures and Adulteration

Selling at cost price with a short weight

A short weight with a profit, a loss or a discount

Different false weights for buying and selling

Mixing in water, sand or a cheaper grade

Common traps

Sharing Profit in a Partnership

Same period: share by capital

Different periods: capital times time

Common traps

Buying on Instalments, and Two Special Cases

Extra paid on an instalment plan

Cash price from instalments with interest

Drying fruit and a broken stone

Common traps

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