Ratio, Proportion and Variation — CDS Elementary Mathematics

Formulas (2)

• Chaining ratios · Joining two ratios
  $$A : B = p : q,\ B : C = r : s \;\Rightarrow\; A : B : C = pr : qr : qs$$
• Dividing a quantity in a ratio · Share in a ratio
  $$\text{share} = \dfrac{p}{p + q + r}\times T$$

Watch out for (2)

• Scale, do not add
  → Chaining ratios
• Answer the share that was asked
  → Dividing a quantity in a ratio

Formulas (2)

• Income, expenditure and savings · Savings
  $$\text{saving} = \text{income} - \text{expenditure}$$
• Ages in a ratio · Ages then and now
  $$\dfrac{M - t}{D - t} = \dfrac pq$$

Watch out for (2)

• Two ratios do not decide who saves more
  → Income, expenditure and savings
• Shift both ages
  → Ages in a ratio

Formulas (2)

• Put each ratio equal to k · Equal ratios
  $$\dfrac ab = \dfrac cd = k \;\Rightarrow\; a = kb,\ c = kd$$
• Cross-multiply and factor · Cross-multiplication
  $$\dfrac PQ = \dfrac RS \iff PS = QR$$

Watch out for (2)

• Mixed degrees cannot be determined
  → Put each ratio equal to k
• Keep both signs
  → Cross-multiply and factor

Formulas (2)

• Finding the constant of variation · Direct and inverse
  $$y = kx^n, \qquad x^n y = k$$
• Which quantities vary together · Test for variation
  $$A \propto B \iff \dfrac AB = \text{constant}$$

Watch out for (2)

• Inverse means the product is constant
  → Finding the constant of variation
• Count the degree
  → Which quantities vary together

Formulas (1)

• Capital × time · Profit share
  $$\text{share} \propto \text{capital}\times\text{time}$$

Watch out for (1)

• Count each period separately
  → Capital × time

Formulas (2)

• Combining mixtures · Component in a mixture
  $$\text{amount of first} = \dfrac{a}{a + b}\times\text{quantity}$$
• Alligation · Alligation
  $$\dfrac{\text{first}}{\text{second}} = \dfrac{c_2 - c}{c - c_1}$$

Watch out for (2)

• Do not average the ratios
  → Combining mixtures
• Cross the distances
  → Alligation