Ratio, Proportion and Variation: SSC CGL Quantitative Aptitude

Formulas (4)

• Turning fractions, percentages and equal parts into a ratio · Scaling and equal parts
  $$a : b = ka : kb \qquad pA = qB \Rightarrow A : B = q : p \qquad pA = qB = rC \Rightarrow A : B : C = \frac{1}{p} : \frac{1}{q} : \frac{1}{r}$$
• Sharing a quantity in a ratio with the ratio unit · Share and ratio unit
  $$\text{share of A} = \frac{a}{a + b + c} \times \text{total} \qquad k = \frac{\text{difference}}{c - a}$$
• Coins in a ratio: number of coins against their value · Value of coins in a ratio
  $$\text{total value} = k(av_1 + bv_2 + cv_3)$$
• Value of an expression when a ratio is given · Rules for given ratios
  $$\frac{x + y}{x - y} = \frac{p}{q} \Rightarrow \frac{x}{y} = \frac{p + q}{p - q} \qquad \frac{a}{b} = \frac{c}{d} = k \Rightarrow \frac{pa + qc}{pb + qd} = k$$

Watch out for (7)

• Putting the multipliers straight into the ratio
  → Turning fractions, percentages and equal parts into a ratio
• Adding the percentage to the ratio term
  → Turning fractions, percentages and equal parts into a ratio
• Dividing a gap by the total parts
  → Sharing a quantity in a ratio with the ratio unit
• Reading a part-to-whole ratio as part to part
  → Sharing a quantity in a ratio with the ratio unit
• Sharing the money in the coin ratio
  → Coins in a ratio: number of coins against their value
• Substituting when the powers differ
  → Value of an expression when a ratio is given
• Stopping at the given ratio
  → Value of an expression when a ratio is given

Formulas (3)

• Proportion and the fourth proportional · Proportion and fourth proportional
  $$a : b :: c : d \Leftrightarrow ad = bc \qquad d = \frac{bc}{a}$$
• Third proportional · Third proportional
  $$a : b = b : c \Leftrightarrow b^2 = ac \qquad c = \frac{b^2}{a}$$
• Mean proportional · Mean proportional
  $$a : m = m : b \Leftrightarrow m = \sqrt{ab} \qquad b = \frac{m^2}{a}$$

Watch out for (4)

• Taking the numbers in the wrong order
  → Proportion and the fourth proportional
• Squaring the wrong term
  → Third proportional
• Halving the sum
  → Mean proportional
• Misplacing the decimal in a square root
  → Mean proportional

Formulas (2)

• Direct proportion and the unitary method · Direct variation
  $$y = kx \qquad \frac{y_2}{y_1} = \frac{x_2}{x_1}$$
• Inverse and joint variation · Inverse and joint variation
  $$xy = k \qquad y = \frac{k x^a}{w^c} \qquad y_2 = y_1 \left(\frac{x_2}{x_1}\right)^a \left(\frac{w_1}{w_2}\right)^c$$

Watch out for (3)

• Turning a direct proportion upside down
  → Direct proportion and the unitary method
• Treating an inverse variation as direct
  → Inverse and joint variation
• Forgetting the power
  → Inverse and joint variation

Formulas (2)

• Joining ratios through a common term · Joining ratios
  $$A : B : C = a_1b_2 : b_1b_2 : b_1c_2 \qquad \frac{A}{D} = \frac{A}{B} \times \frac{B}{C} \times \frac{C}{D}$$
• A ratio after items are added or removed · Changed ratio
  $$\frac{ak + x}{bk + y} = \frac{m}{n}$$

Watch out for (3)

• Writing the ratios side by side
  → Joining ratios through a common term
• Reading parts before matching the unchanged term
  → A ratio after items are added or removed
• Answering for the wrong moment
  → A ratio after items are added or removed

Formulas (2)

• Income, expenditure and savings in a ratio · Income identity
  $$I = E + S \qquad S_{\text{new}} = I(1 + r_I) - E(1 + r_E)$$
• Ages in a ratio · Ages after t years
  $$\frac{ak + t}{bk + t} = \frac{m}{n} \qquad bk - ak \text{ stays fixed}$$

Watch out for (4)

• Combining the percentages instead of the amounts
  → Income, expenditure and savings in a ratio
• Confusing the saving rate with the spending rate
  → Income, expenditure and savings in a ratio
• Adding the years to the ratio terms
  → Ages in a ratio
• Letting the age gap change
  → Ages in a ratio

Formulas (4)

• Profit in the ratio of capital times time · Profit sharing
  $$P_1 : P_2 : P_3 = C_1t_1 : C_2t_2 : C_3t_3$$
• Capital that changes during the year · Capital-months with changes
  $$\text{capital-months} = C_1t_1 + C_2t_2 + \cdots$$
• Charity, salary or interest taken out before sharing · Share after a deduction
  $$\text{share} = \text{own extra} + \frac{c}{c_1 + c_2} \times (P - \text{deductions})$$
• Rent for a shared field in animal-months · Share of the rent
  $$\text{share} = \frac{\text{own animal-months}}{\text{total animal-months}} \times \text{rent}$$

Watch out for (7)

• Counting the whole year for a late partner
  → Profit in the ratio of capital times time
• Taking the year when the stem says two years
  → Profit in the ratio of capital times time
• Using the final capital for the whole year
  → Capital that changes during the year
• Reading "2 times more than"
  → Capital that changes during the year
• Sharing before deducting
  → Charity, salary or interest taken out before sharing
• Forgetting the working partner's capital share
  → Charity, salary or interest taken out before sharing
• Adding animals of different kinds
  → Rent for a shared field in animal-months

Formulas (4)

• Adding one ingredient while the other stays fixed · Adding to a mixture
  $$\text{new total} = \frac{\text{fixed amount}}{\text{its new fraction}} \qquad \frac{A + x}{T + x} = p$$
• Combining two mixtures · Combined amount of an ingredient
  $$A_{\text{new}} = \frac{a_1}{a_1 + b_1}V_1 + \frac{a_2}{a_2 + b_2}V_2$$
• Taking out part of a mixture and topping it up · Amount left after a removal
  $$\text{left} = \text{original} \times \left(1 - \frac{x}{T}\right)$$
• Repeated replacement · Repeated replacement
  $$A_n = A\left(1 - \frac{x}{T}\right)^n$$

Watch out for (6)

• Working from the total instead of the fixed ingredient
  → Adding one ingredient while the other stays fixed
• Adding litres to a ratio term
  → Adding one ingredient while the other stays fixed
• Adding the ratio terms of unequal amounts
  → Combining two mixtures
• Taking out only one ingredient
  → Taking out part of a mixture and topping it up
• Tracking the ingredient that is poured back
  → Taking out part of a mixture and topping it up
• Subtracting the same amount every round
  → Repeated replacement

Formulas (3)

• Mean price of a mixture · Mean price
  $$m = \frac{q_1p_1 + q_2p_2}{q_1 + q_2} \qquad q_1(p_1 - m) + q_2(p_2 - m) = 0$$
• The alligation cross for a mixing ratio · Rule of alligation
  $$\frac{q_{\text{cheaper}}}{q_{\text{dearer}}} = \frac{p_2 - m}{m - p_1}$$
• Mixtures sold at a profit or a loss · Cost price of the mixture
  $$CP = \frac{100 \times SP}{100 + r} \qquad \text{gain \%} = \frac{SP - CP}{CP} \times 100$$

Watch out for (6)

• Averaging the prices without the quantities
  → Mean price of a mixture
• Mixing price units
  → Mean price of a mixture
• Giving each kind its own distance
  → The alligation cross for a mixing ratio
• A mean outside the two prices
  → The alligation cross for a mixing ratio
• Putting the selling price in the cross
  → Mixtures sold at a profit or a loss
• Taking the profit off the selling price
  → Mixtures sold at a profit or a loss