Percentage: AFCAT Numerical Ability

Formulas (4)

• x% of a number, and what percent one number is of another · Percent of, and as a percent
  $$x\% \text{ of } y = \dfrac{xy}{100}, \qquad \dfrac{a}{b} \times 100 = \text{a as a percentage of b}$$
• A percentage of a percentage · Percent of a percent
  $$a\% \text{ of } b\% \text{ of } N = \dfrac{ab}{10000}\,N$$
• Finding the whole from a part · The whole from a part
  $$N = \dfrac{100 \times \text{part}}{p}$$
• Two percentages of the same number · Difference of two percentages
  $$(a - b)\% \text{ of } N = D \;\Rightarrow\; N = \dfrac{100D}{a - b}$$

Reference tables (1)

Watch out for (6)

• Convert the units first
  → x% of a number, and what percent one number is of another
• The base follows 'of'
  → x% of a number, and what percent one number is of another
• One ninth is not 11%
  → Fractions and their percentages
• Multiply, do not add
  → A percentage of a percentage
• Use the percentage that is left
  → Finding the whole from a part
• Answer the last line
  → Two percentages of the same number

Formulas (5)

• Percentage change measured on the old value · Percentage change
  $$\%\text{ change} = \dfrac{\text{new} - \text{old}}{\text{old}} \times 100$$
• The value before a percentage change · Undoing a change
  $$\text{old} = \dfrac{\text{new}}{1 \pm \tfrac{r}{100}}$$
• More than and less than: the base changes · Swapping the base
  $$\text{more by } r\% \;\to\; \text{less by } \dfrac{100r}{100 + r}\%, \qquad \text{less by } r\% \;\to\; \text{more by } \dfrac{100r}{100 - r}\%$$
• Two quantities measured against a third · Set the reference at 100
  $$\text{third} = 100 \;\Rightarrow\; \text{first} = 100 \pm a, \quad \text{second} = 100 \pm b$$
• When x% of one equals y% of another · Equal percentages
  $$\dfrac{x}{100}A = \dfrac{y}{100}B \;\Rightarrow\; A : B = y : x$$

Watch out for (5)

• Divide by the starting value
  → Percentage change measured on the old value
• Do not take the percentage off the new value
  → The value before a percentage change
• The two percentages are never equal
  → More than and less than: the base changes
• Percentages along a chain multiply
  → Two quantities measured against a third
• The ratio crosses over
  → When x% of one equals y% of another

Formulas (5)

• The net effect of successive changes · Two successive changes
  $$\text{net } \% = a + b + \dfrac{ab}{100}$$
• Undoing a change: the percentage needed to get back · The change that undoes another
  $$\text{after a fall of } r\%: \; \dfrac{100r}{100 - r}\% \text{ rise}, \qquad \text{after a rise of } r\%: \; \dfrac{100r}{100 + r}\% \text{ fall}$$
• When a quantity is a product: area, volume, takings · Change in a product
  $$Q = XY \;\Rightarrow\; \text{factor}_Q = \left(1 + \tfrac{a}{100}\right)\left(1 + \tfrac{b}{100}\right)$$
• Keeping expenditure fixed when the price changes · Consumption cut for a fixed budget
  $$\text{cut} = \dfrac{r}{100 + r} \times 100\%$$
• A cheaper price buys more for the same money · Reduced price from the extra quantity
  $$\text{reduced price} = \dfrac{\tfrac{r}{100} \times M}{\text{extra quantity}}$$

Watch out for (7)

• Up and down by the same percentage is a loss
  → The net effect of successive changes
• Do not add the percentages
  → The net effect of successive changes
• The same percentage does not undo it
  → Undoing a change: the percentage needed to get back
• Area squares the change
  → When a quantity is a product: area, volume, takings
• The cut is smaller than the rise
  → Keeping expenditure fixed when the price changes
• The saving method gives the reduced price
  → A cheaper price buys more for the same money
• Per dozen, per kg, per item
  → A cheaper price buys more for the same money

Formulas (5)

• Growth and depreciation year after year · Compound growth
  $$P_n = P\left(1 + \dfrac{r}{100}\right)^n$$
• Going back in time · The starting value
  $$P = \dfrac{P_n}{\left(1 \pm \tfrac{r}{100}\right)^n}$$
• A different rate each year · Changing rates
  $$P_{\text{final}} = P\left(1 + \tfrac{r_1}{100}\right)\left(1 + \tfrac{r_2}{100}\right)\left(1 + \tfrac{r_3}{100}\right)$$
• Finding the rate of growth · Rate from two figures
  $$1 + \dfrac{r}{100} = \left(\dfrac{P_n}{P}\right)^{1/n}$$
• The make-up of a population · A share of the whole
  $$\text{total} = \dfrac{\text{size of a part} \times 100}{\text{its percentage}}$$

Watch out for (5)

• Compound, not simple
  → Growth and depreciation year after year
• Divide, do not subtract
  → Going back in time
• The second loss is on the survivors
  → A different rate each year
• Do not divide the total growth by the years
  → Finding the rate of growth
• Weight each part's change
  → The make-up of a population

Formulas (6)

• Parts of a whole, and the rest · The rest, and the whole
  $$\text{rest} = 100\% - \sum \text{parts}, \qquad \text{whole} = \dfrac{100 \times \text{part}}{p}$$
• Income, spending and savings · Savings from spending
  $$\text{income} = \dfrac{\text{savings} \times 100}{100 - \text{spending}\%}$$
• A share of what is left · Amount left after shares of the rest
  $$\text{left} = S\left(1 - \tfrac{a}{100}\right)\left(1 - \tfrac{b}{100}\right)$$
• When income and spending both change · New savings
  $$\text{new savings} = 100\left(1 + \tfrac{a}{100}\right) - s\left(1 + \tfrac{b}{100}\right)$$
• Combining groups with different percentages · Weighted percentage
  $$\text{overall } \% = \dfrac{n_1 p_1 + n_2 p_2}{n_1 + n_2}$$
• Overlapping groups: failed in one, both or neither · Two overlapping groups
  $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

Watch out for (6)

• Answer the part asked
  → Parts of a whole, and the rest
• Divide by the savings share, not the spending
  → Income, spending and savings
• 'Of the rest' is not of the whole
  → A share of what is left
• Savings swing more than income
  → When income and spending both change
• Do not average the percentages
  → Combining groups with different percentages
• Take the overlap off once
  → Overlapping groups: failed in one, both or neither

Formulas (5)

• Scores, questions and negative marking · Score with negative marking
  $$\text{score} = m \times \text{right} - p \times \text{wrong}$$
• The pass mark from one student's score · Pass mark
  $$\text{pass mark} = \text{score} + \text{shortfall} = \dfrac{p}{100} \times \text{maximum}$$
• Two scores either side of the pass mark · Maximum from two candidates
  $$\dfrac{b - a}{100}\,M = f + s$$
• Overall percentage across papers · Weighted overall percentage
  $$\text{overall} = \dfrac{\sum w_i p_i}{\sum w_i}$$
• Elections: shares, margins and invalid votes · Total from the margin
  $$\text{valid votes} = \dfrac{\text{margin} \times 100}{\text{winner}\% - \text{loser}\%}$$

Watch out for (5)

• Wrong answers cost twice
  → Scores, questions and negative marking
• Failing by means add, passing by means subtract
  → The pass mark from one student's score
• Add the shortfall and the surplus
  → Two scores either side of the pass mark
• Unequal maxima are not averaged
  → Overall percentage across papers
• The margin is a share of the votes cast
  → Elections: shares, margins and invalid votes

Formulas (5)

• Profit and loss on the cost price · Selling price from cost price
  $$SP = CP\left(1 + \dfrac{p}{100}\right), \qquad CP = \dfrac{SP}{1 + p/100}$$
• Articles bought and sold in different lots · Cost of a equals selling price of b
  $$\text{profit } \% = \dfrac{a - b}{b} \times 100$$
• Marked price, discount and profit · From cost to marked to selling price
  $$SP = CP\left(1 + \dfrac{m}{100}\right)\left(1 - \dfrac{d}{100}\right)$$
• Successive discounts and chains of profit · Two successive discounts
  $$\text{single discount} = a + b - \dfrac{ab}{100}$$
• Overall profit when parts sell at different rates · Same selling price, gain and loss of r%
  $$\text{loss } \% = \dfrac{r^2}{100}$$

Watch out for (6)

• Profit is on cost, not on selling price
  → Profit and loss on the cost price
• A loss and a gain add
  → Profit and loss on the cost price
• Divide by the articles sold
  → Articles bought and sold in different lots
• Discount on marked price, profit on cost
  → Marked price, discount and profit
• Successive discounts are less than their sum
  → Successive discounts and chains of profit
• Equal gain and loss is not break-even
  → Overall profit when parts sell at different rates

Formulas (5)

• Strength of a solution when water is added or removed · New strength
  $$\text{new } \% = \dfrac{\text{unchanged amount}}{\text{new total}} \times 100$$
• How much of the pure component to add · Pure component to add
  $$\dfrac{a + x}{T + x} = \dfrac{\text{target}}{100}$$
• Changing a ratio by adding one component · Amount to add
  $$\text{added} = \text{fixed} \times \dfrac{\text{new term of the other}}{\text{new term of the fixed}} - \text{other now}$$
• Mixing two strengths, and alligation · Alligation
  $$\dfrac{q_1}{q_2} = \dfrac{s_2 - m}{m - s_1}$$
• Drawing off and replacing · Repeated replacement with water
  $$\text{left} = V\left(1 - \dfrac{x}{V}\right)^n$$

Watch out for (5)

• Keep the right thing fixed
  → Strength of a solution when water is added or removed
• The total grows too
  → How much of the pure component to add
• Find the parts before adding
  → Changing a ratio by adding one component
• The ratio is crossed
  → Mixing two strengths, and alligation
• The second draw takes less milk
  → Drawing off and replacing

Formulas (4)

• Simple and compound interest · Simple and compound interest
  $$SI = \dfrac{PRT}{100}, \qquad A = P\left(1 + \dfrac{R}{100}\right)^T$$
• Work done as a percentage of a job · Rates add
  $$\dfrac{1}{T} = \dfrac{1}{a} + \dfrac{1}{b}$$
• Speed changed by a percentage · Time after a change in speed
  $$t_{\text{new}} = \dfrac{t_{\text{old}}}{1 + r/100}$$
• Speed, distance and time: the basics used here · Speed, distance and time
  $$d = s \times t, \qquad \text{average speed} = \dfrac{\text{total distance}}{\text{total time}}$$

Watch out for (4)

• Amount is not interest
  → Simple and compound interest
• More efficient does not mean that much less time
  → Work done as a percentage of a job
• Time does not fall by the same percentage
  → Speed changed by a percentage
• Average speed is not the average of speeds
  → Speed, distance and time: the basics used here