Profit and Loss: AFCAT Numerical Ability

Formulas (3)

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

Watch out for (6)

• Divide by the cost price
  → Profit, loss and their percentages
• Find the cost before the percentage
  → Profit, loss and their percentages
• The percentage is of the cost, not of the selling price
  → Selling price and cost price as multiples
• Undo a fall by dividing
  → Selling price and cost price as multiples
• The base is the cost
  → Profit from a ratio of cost price to selling price
• A loss on the selling price is a smaller loss on the cost
  → Profit from a ratio of cost price to selling price

Formulas (3)

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

Watch out for (6)

• The percentages do not add
  → Through the cost price to a new selling price
• Divide to reach the cost
  → Through the cost price to a new selling price
• A loss and a gain add
  → When two prices differ by a known sum
• The gap is a share of the cost
  → When two prices differ by a known sum
• The new percentage is on the new cost
  → Setting up an equation in the cost price
• Midway only for equal rupee amounts
  → Setting up an equation in the cost price

Formulas (4)

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

Watch out for (8)

• Divide by the number sold
  → Cost of x articles equals the selling price of y
• Which side is larger decides profit or loss
  → Cost of x articles equals the selling price of y
• Scale the count, do not add to it
  → A gain equal to the cost of n articles
• The cost of n is not the selling price of n
  → A gain equal to the cost of n articles
• Subtract before dividing
  → A gain equal to the selling price of n articles
• A loss adds to the count
  → A gain equal to the selling price of n articles
• Compare prices per article, not counts
  → Bought and sold at so many for a price
• Keep the fractions
  → Bought and sold at so many for a price

Formulas (4)

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

Watch out for (8)

• The discount is on the marked price
  → Discount on the marked price
• Divide to undo a discount
  → Discount on the marked price
• Discounts do not add
  → Successive discounts
• A rupee figure is a share of the marked price
  → Successive discounts
• Subtracting the rates overstates the profit
  → Markup, then discount
• Same rate up and down is a loss
  → Markup, then discount
• Divide by the discount factor
  → Marked price for a target profit
• The profit is on the cost, not on the marked price
  → Marked price for a target profit

Formulas (3)

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

Watch out for (6)

• Not 'no gain, no loss'
  → Equal gain and loss on equal selling prices
• Equal costs are a different question
  → Equal gain and loss on equal selling prices
• Do not average the percentages
  → One selling price, different percentages
• The percentage is on the total cost
  → One selling price, different percentages
• Overheads are part of the cost
  → Overheads, repairs and broken stock
• Broken items still cost money
  → Overheads, repairs and broken stock

Formulas (3)

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

Watch out for (6)

• Add the totals, do not average the prices
  → Total cost against total takings
• The base is the total cost
  → Total cost against total takings
• Weight the average
  → Cost price of a mixture
• Alligation pairs each grade with the other's gap
  → Cost price of a mixture
• Weight by cost
  → Part at one profit, the rest at another
• Each rate is on its own part
  → Part at one profit, the rest at another

Formulas (4)

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

Watch out for (7)

• Divide by what was given
  → Selling at cost price with a short weight
• Two gains multiply
  → A short weight with a profit, a loss or a discount
• The cost is of what he hands over
  → A short weight with a profit, a loss or a discount
• Do not add the percentages
  → Different false weights for buying and selling
• One balance used both ways cancels
  → Different false weights for buying and selling
• Water's share is of the mixture
  → Mixing in water, sand or a cheaper grade
• Free filler adds weight, not cost
  → Mixing in water, sand or a cheaper grade

Formulas (2)

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

Watch out for (4)

• Take out the deduction first
  → Same period: share by capital
• Chained equalities need the LCM
  → Same period: share by capital
• Time counts
  → Different periods: capital times time
• A withdrawal splits the period
  → Different periods: capital times time

Formulas (2)

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

Reference tables (1)

Watch out for (4)

• Include the down payment
  → Extra paid on an instalment plan
• Each payment has its own factor
  → Cash price from instalments with interest
• Use the rate per period
  → Cash price from instalments with interest
• Drying does not cut the weight by the water percentage
  → Drying fruit and a broken stone