Percentage, Profit and Loss: SSC CGL Quantitative Aptitude

Formulas (3)

• Percentage of a quantity and finding the whole · Part and whole
  $$x\% \text{ of } N = \frac{x}{100} \times N \qquad \text{whole} = \frac{\text{part} \times 100}{x}$$
• Equal percentages of two numbers give their ratio · Equal percentages and differences
  $$a\% \text{ of } A = b\% \text{ of } B \implies A : B = b : a \qquad x\% \text{ of } N - y\% \text{ of } N = (x - y)\% \text{ of } N$$
• More than, less than and the changing base · Turning a comparison around
  $$\text{less by } \frac{r}{100 + r} \times 100\% \qquad \text{more by } \frac{r}{100 - r} \times 100\%$$

Reference tables (1)

Watch out for (6)

• Answering the part that was given
  → Percentage of a quantity and finding the whole
• Mixing units
  → Percentage of a quantity and finding the whole
• Rounding a recurring percentage too early
  → Common percentages as fractions
• Writing the ratio the wrong way round
  → Equal percentages of two numbers give their ratio
• Using the same percentage both ways
  → More than, less than and the changing base
• Comparing against the wrong number
  → More than, less than and the changing base

Formulas (4)

• Successive changes as a product of factors · Net effect of two changes
  $$\text{net}\% = a + b + \frac{ab}{100} \qquad \text{equal rise and fall: } -\frac{r^2}{100}\%$$
• Spending a share of what is left · Amount left after spending out of the remainder
  $$\text{left} = S\left(1 - \frac{x}{100}\right)\left(1 - \frac{y}{100}\right)\left(1 - \frac{z}{100}\right)$$
• Working back to the original value · Back to the start, and growth over years
  $$\text{original} = \frac{\text{final}}{\left(1 + \frac{a}{100}\right)\left(1 + \frac{b}{100}\right)} \qquad P_n = P_0\left(1 + \frac{r}{100}\right)^n$$
• Price and consumption when spending is fixed · Cut in consumption to keep spending fixed
  $$\text{cut} = \frac{r}{100 + r} \times 100\% \qquad \text{spending} = \text{price} \times \text{quantity}$$

Watch out for (6)

• Adding the percentages
  → Successive changes as a product of factors
• Dropping the sign in the product term
  → Successive changes as a product of factors
• Adding shares of different amounts
  → Spending a share of what is left
• Undoing a change with the same percentage
  → Working back to the original value
• Cutting consumption by the price rise
  → Price and consumption when spending is fixed
• Mixing up the new price and the old price
  → Price and consumption when spending is fixed

Formulas (4)

• Elections: voters, votes cast, valid votes and the margin · Margin and valid votes
  $$\text{margin} = (2w - 100)\% \text{ of valid votes} \qquad \text{valid} = N \times \frac{p}{100} \times \frac{q}{100}$$
• Pass marks and marks scored · Pass mark
  $$\frac{p}{100} \times M = \text{score} + \text{shortfall}$$
• Income, expenditure and savings · Savings and their change
  $$\text{income} = \text{expenditure} + \text{savings} \qquad s = \frac{100\,i - E\,e}{S}$$
• Combined percentage of several groups · Weighted percentage and alligation
  $$\text{overall} = \frac{A\,a + B\,b}{A + B} \qquad A : B = (b - \text{overall}) : (\text{overall} - a)$$

Watch out for (6)

• Taking the winner's share as the margin
  → Elections: voters, votes cast, valid votes and the margin
• Taking invalid votes out of the whole list
  → Elections: voters, votes cast, valid votes and the margin
• Subtracting the shortfall
  → Pass marks and marks scored
• Averaging percentages from papers of different sizes
  → Pass marks and marks scored
• Averaging the two changes
  → Income, expenditure and savings
• Taking a plain average of the rates
  → Combined percentage of several groups

Formulas (4)

• Cost price, selling price and profit percentage · Profit percentage and the two prices
  $$\text{profit}\% = \frac{SP - CP}{CP} \times 100 \qquad SP = CP\left(1 + \frac{p}{100}\right) \qquad CP = \frac{SP}{1 + \frac{p}{100}}$$
• Two selling prices of the same article · Swing between two selling prices
  $$SP_2 - SP_1 = \frac{p_2 - p_1}{100} \times CP \qquad CP = \frac{SP_1 + SP_2}{2} \text{ (equal profit and loss)}$$
• An article sold through several hands · Price after two sales
  $$\text{final} = C\left(1 + \frac{p_1}{100}\right)\left(1 + \frac{p_2}{100}\right)$$
• Overall profit on several items or lots · Overall profit
  $$\text{overall}\% = \frac{\sum SP - \sum CP}{\sum CP} \times 100$$

Watch out for (6)

• Taking the profit on the selling price
  → Cost price, selling price and profit percentage
• Leaving out the overheads
  → Cost price, selling price and profit percentage
• Subtracting a loss from a profit
  → Two selling prices of the same article
• Adding the dealers' profits
  → An article sold through several hands
• Averaging percentages on different costs
  → Overall profit on several items or lots
• Leaving the spoilt stock out of the cost
  → Overall profit on several items or lots

Formulas (4)

• Discount on the marked price · Selling price after a discount
  $$SP = MP\left(1 - \frac{d}{100}\right) \qquad d\% = \frac{MP - SP}{MP} \times 100$$
• Profit after a mark-up and a discount · Profit after mark-up and discount
  $$\text{profit}\% = m - d - \frac{m\,d}{100}$$
• Finding the cost or marked price from the profit · The two routes to the selling price
  $$CP\left(1 + \frac{p}{100}\right) = MP\left(1 - \frac{d}{100}\right) \qquad \frac{CP}{MP} = \frac{100 - d}{100 + p}$$
• Several stages and fixed rupee amounts · Price after several stages
  $$\text{final} = C\left(1 + \frac{m_1}{100}\right)\left(1 - \frac{d_1}{100}\right)\left(1 + \frac{m_2}{100}\right)\left(1 - \frac{d_2}{100}\right)$$

Watch out for (6)

• Adding the discount back on the selling price
  → Discount on the marked price
• Treating a rupee discount like a percentage
  → Discount on the marked price
• Subtracting the discount from the mark-up
  → Profit after a mark-up and a discount
• Marking up by profit plus discount
  → Finding the cost or marked price from the profit
• Taking the discount as a share of the cost
  → Finding the cost or marked price from the profit
• Putting a rupee discount into the product
  → Several stages and fixed rupee amounts

Formulas (3)

• Single discount equal to successive discounts · Equal single discount
  $$D = a + b - \frac{ab}{100}$$
• Finding a missing discount or the bill · Bill from the gap between two schemes
  $$\text{bill} = \frac{\text{rupee gap}}{(D - D_{eq}) / 100}$$
• Comparing schemes and free-item offers · Discount from free items
  $$\text{buy } x \text{ get } y \text{ free: discount} = \frac{y}{x + y} \times 100\%$$

Watch out for (5)

• Adding successive discounts
  → Single discount equal to successive discounts
• Giving the share paid as the discount
  → Single discount equal to successive discounts
• Taking the second discount on the marked price
  → Finding a missing discount or the bill
• Dividing the free items by the items paid for
  → Comparing schemes and free-item offers
• Answering the opposite of the question
  → Comparing schemes and free-item offers

Formulas (3)

• Faulty weight at the cost price · Gain from a short weight
  $$\text{gain}\% = \frac{\text{true} - \text{false}}{\text{false}} \times 100$$
• Faulty weight with a profit or loss on price · Real profit with a short weight
  $$1 + \frac{\text{gain}}{100} = \left(1 + \frac{p}{100}\right) \times \frac{\text{true}}{\text{false}}$$
• Cost price of some articles equal to the selling price of others · Profit when two counts are equal in value
  $$\text{profit}\% = \frac{x - y}{y} \times 100 \text{ when CP of } x = \text{SP of } y$$

Watch out for (5)

• Dividing by the true weight
  → Faulty weight at the cost price
• Adding the two gains
  → Faulty weight with a profit or loss on price
• Reading 'weight 20% less' as a 20% gain
  → Faulty weight with a profit or loss on price
• Dividing by the larger count
  → Cost price of some articles equal to the selling price of others
• Mixing up 'gains the SP of n' and 'gains the CP of n'
  → Cost price of some articles equal to the selling price of others