Ratio and Proportion: AFCAT Numerical Ability

Formulas (3)

• A ratio as equal parts · A ratio as parts
  $$a : b = p : q \;\text{means}\; a = pk,\ b = qk$$
• Simplest form of a ratio · Scaling both terms
  $$a : b = ka : kb \quad (k \neq 0)$$
• Working back from a duplicate or triplicate ratio · Undoing a duplicate ratio
  $$a^2 : b^2 = p : q \;\Rightarrow\; a : b = \sqrt p : \sqrt q$$

Reference tables (1)

Watch out for (8)

• The ratio is not the amount
  → A ratio as equal parts
• Order matters
  → A ratio as equal parts
• Units first
  → Simplest form of a ratio
• A ratio answer keeps both terms
  → Simplest form of a ratio
• Duplicate is not double
  → Inverse, duplicate, triplicate and sub-duplicate ratios
• Sub-duplicate is not half
  → Inverse, duplicate, triplicate and sub-duplicate ratios
• One operation per name
  → Working back from a duplicate or triplicate ratio
• Roots do not always come out whole
  → Working back from a duplicate or triplicate ratio

Formulas (5)

• Joining ratios through a common term · Joining two ratios
  $$A : B = p : q,\ B : C = r : s \;\Rightarrow\; A : B : C = pr : qr : qs$$
• Ratios from equal multiples · Equal multiples
  $$pA = qB = rC \;\Rightarrow\; A : B : C = \tfrac 1p : \tfrac 1q : \tfrac 1r$$
• Compound ratio: multiplying ratios · Compound ratio
  $$(a : b),\ (c : d) \;\Rightarrow\; ac : bd$$
• Finding a missing ratio from a compound ratio · The missing ratio
  $$\dfrac ab \times \dfrac cd = \dfrac mn \;\Rightarrow\; \dfrac cd = \dfrac mn \div \dfrac ab$$
• Comparing two ratios · Cross-multiplication test
  $$\dfrac ab > \dfrac cd \;\text{exactly when}\; ad > bc \quad (b, d > 0)$$

Watch out for (7)

• Scale, do not stitch
  → Joining ratios through a common term
• Read which term comes first
  → Joining ratios through a common term
• Equal multiples flip the ratio
  → Ratios from equal multiples
• Compounding multiplies, it does not add
  → Compound ratio: multiplying ratios
• A ratio of fractions is one division
  → Compound ratio: multiplying ratios
• Divide, do not subtract
  → Finding a missing ratio from a compound ratio
• Bigger terms are not a bigger ratio
  → Comparing two ratios

Formulas (5)

• Dividing a total in a ratio · Share in a ratio
  $$\text{share of } p = \dfrac{p}{p + q + r} \times T$$
• Dividing in a ratio of fractions · Clearing fractions
  $$\tfrac 1a : \tfrac 1b : \tfrac 1c = \tfrac La : \tfrac Lb : \tfrac Lc \quad (L = \text{LCM of } a, b, c)$$
• Numbers in a ratio with one more condition · Numbers from a ratio
  $$x : y : z = p : q : r \;\Rightarrow\; x = pk,\ y = qk,\ z = rk$$
• A changed ratio, or shares with an amount taken off · Gain or loss from a changed ratio
  $$\text{change} = \left(\dfrac{q}{p + q} - \dfrac{s}{r + s}\right) T$$
• Partnership: profit in the ratio of capital times time · Profit ratio
  $$P_1 : P_2 = C_1 t_1 : C_2 t_2$$

Watch out for (6)

• Answer in the unit asked
  → Dividing a total in a ratio
• A share of 'the other two together'
  → Dividing a total in a ratio
• The denominators are not the ratio
  → Dividing in a ratio of fractions
• A sum of squares gives k squared
  → Numbers in a ratio with one more condition
• Add the deduction back
  → A changed ratio, or shares with an amount taken off
• Time counts as much as money
  → Partnership: profit in the ratio of capital times time

Formulas (4)

• Four numbers in proportion, and the fourth proportional · Product of extremes equals product of means
  $$a : b :: c : d \;\text{means}\; ad = bc, \qquad d = \dfrac{bc}{a}$$
• Third proportional · Third proportional
  $$a : b = b : c \;\Rightarrow\; c = \dfrac{b^2}{a}$$
• Mean proportional · Mean proportional
  $$a : x = x : b \;\Rightarrow\; x = \sqrt{ab}$$
• Continued proportion · Continued proportion
  $$a : b = b : c \;\text{means}\; b^2 = ac$$

Watch out for (4)

• Extremes times extremes
  → Four numbers in proportion, and the fourth proportional
• Square the second number, not the first
  → Third proportional
• A square root, not an average
  → Mean proportional
• Continued proportion is a G.P., not an A.P.
  → Continued proportion

Formulas (4)

• Evaluating an expression from a ratio · Substituting the ratio
  $$x : y = p : q \;\Rightarrow\; \dfrac{ax + by}{cx + dy} = \dfrac{ap + bq}{cp + dq}$$
• Finding x : y from an equation · Collecting x and y
  $$ax + by = cx + dy \;\Rightarrow\; \dfrac xy = \dfrac{d - b}{a - c}$$
• Componendo and dividendo · Componendo and dividendo
  $$\dfrac ab = \dfrac cd \;\Rightarrow\; \dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}$$
• Equal ratios: sums and the k method · Sum of equal ratios
  $$\dfrac ab = \dfrac cd = \dfrac ef = k \;\Rightarrow\; \dfrac{a + c + e}{b + d + f} = k$$

Watch out for (5)

• Keep the minus sign
  → Evaluating an expression from a ratio
• Only when every term has the same degree
  → Evaluating an expression from a ratio
• Collect before you read off the ratio
  → Finding x : y from an equation
• Do the same on both sides
  → Componendo and dividendo
• The multipliers must match
  → Equal ratios: sums and the k method

Formulas (4)

• Adding or subtracting the same number from both terms · A changed ratio
  $$\dfrac{pk + x}{qk + x} = \dfrac rs$$
• Age problems with ratios · Ages after n years
  $$\dfrac{pk + n}{qk + n} = \dfrac rs$$
• When each term changes by a percentage · Terms scaled by percentages
  $$p : q \;\to\; p\left(1 + \tfrac{a}{100}\right) : q\left(1 + \tfrac{b}{100}\right)$$
• Income, expenditure and savings · Savings
  $$\text{savings} = \text{income} - \text{expenditure}$$

Watch out for (5)

• Add to the quantities, not to the ratio
  → Adding or subtracting the same number from both terms
• Both people age
  → Age problems with ratios
• 'Ago' subtracts
  → Age problems with ratios
• A percentage multiplies the term
  → When each term changes by a percentage
• Two ratios need two letters
  → Income, expenditure and savings

Formulas (5)

• A mixture as a ratio of two quantities · Milk in a mixture
  $$\text{milk} = \dfrac{m}{m + w} \times T$$
• Adding one liquid: the other stays fixed · Water to add
  $$\text{water needed} = \dfrac sr \times \text{milk}, \qquad \text{add} = \text{water needed} - \text{water now}$$
• Replacing part of a mixture, and mixing two mixtures · Mixing two vessels
  $$\text{milk} = \dfrac{m_1}{m_1 + w_1}V_1 + \dfrac{m_2}{m_2 + w_2}V_2$$
• Coins: a ratio of how many coins there are · Total value
  $$\text{value} = \sum (\text{number of coins} \times \text{face value})$$
• Coins when the total value is given · Number of sets
  $$k = \dfrac{\text{total value}}{au + bv + cw}$$

Watch out for (5)

• The ratio is of the parts, not of the total
  → A mixture as a ratio of two quantities
• Adding water never changes the milk
  → Adding one liquid: the other stays fixed
• Mix litres, not ratio terms
  → Replacing part of a mixture, and mixing two mixtures
• Count first, then value
  → Coins: a ratio of how many coins there are
• Divide by the value of a set, not the sum of the ratio
  → Coins when the total value is given

Formulas (4)

• Direct variation · Direct variation
  $$x \propto y \;\text{means}\; x = ky, \text{ so } \dfrac{x_1}{y_1} = \dfrac{x_2}{y_2}$$
• Inverse variation · Inverse variation
  $$x \propto \dfrac 1y \;\text{means}\; xy = k, \text{ so } x_1 y_1 = x_2 y_2$$
• Joint and combined variation · Combined variation
  $$x \propto \dfrac{y}{z^2} \;\text{means}\; x = \dfrac{ky}{z^2}$$
• Variation in word problems: work, speed and wages · Man-days
  $$M_1 D_1 = M_2 D_2$$

Watch out for (4)

• A square doubles twice
  → Direct variation
• Inverse means dividing, not subtracting
  → Inverse variation
• One relation, one constant
  → Joint and combined variation
• More workers, fewer days
  → Variation in word problems: work, speed and wages