Mixture and Alligation: AFCAT Numerical Ability

Formulas (5)

• Splitting a mixture, then adding to it · Milk in a mixture
  $$\text{milk} = \dfrac{m}{m + w} \times T$$
• Adding water to reach a ratio: the milk is fixed · Water to add
  $$\text{add} = \dfrac{s}{r} \times M - W$$
• Adding milk to reach a ratio: the water is fixed · Milk to add
  $$\text{add} = \dfrac{r}{s} \times W - M$$
• Unknown amounts: when the unchanged liquid keeps its term · One part
  $$k = \dfrac{\text{litres added}}{\text{extra parts}}$$
• When the unchanged liquid's term differs: rescale first · One part after rescaling
  $$k = \dfrac{\text{litres added}}{\text{extra parts after matching the fixed term}}$$

Watch out for (7)

• Add litres to litres, not to ratio terms
  → Splitting a mixture, then adding to it
• Do not split the old total in the new ratio
  → Adding water to reach a ratio: the milk is fixed
• Answer with the litres added
  → Adding water to reach a ratio: the milk is fixed
• Adding milk, scale from the water
  → Adding milk to reach a ratio: the water is fixed
• A reversed ratio is a fraction of the water, not of the new total
  → Adding milk to reach a ratio: the water is fixed
• k is not the answer
  → Unknown amounts: when the unchanged liquid keeps its term
• Match the liquid that did not change
  → When the unchanged liquid's term differs: rescale first

Formulas (2)

• When the percentage is of the liquid that stays · New total
  $$\text{new total} = \dfrac{\text{fixed amount}}{\text{its new share}}$$
• When the percentage is of the liquid you add · New total
  $$\text{new total} = \dfrac{\text{other liquid}}{1 - p}$$

Watch out for (3)

• Divide by the new share; do not subtract percentages
  → When the percentage is of the liquid that stays
• A percentage is of the whole mixture
  → When the percentage is of the liquid that stays
• Using the target share for the liquid that changes
  → When the percentage is of the liquid you add

Formulas (2)

• Mean price of a mixture · Mean price
  $$\bar p = \dfrac{ap + bq + cr}{a + b + c}$$
• Mean strength: add up the pure part · Mean strength
  $$c = \dfrac{c_1 V_1 + c_2 V_2}{V_1 + V_2}$$

Watch out for (3)

• Weight the prices; do not average them
  → Mean price of a mixture
• Divide by the sum of the ratio
  → Mean price of a mixture
• Do not add the ratio terms
  → Mean strength: add up the pure part

Formulas (2)

• The alligation cross · Rule of alligation
  $$\dfrac{\text{cheaper}}{\text{dearer}} = \dfrac{d - m}{m - c}$$
• From the ratio to a quantity, a price, or any average · Scaling the ratio
  $$\text{cheaper} = \dfrac{d - m}{m - c} \times \text{dearer}$$

Watch out for (3)

• The cheaper kind gets the dearer kind's distance
  → The alligation cross
• The mean must lie between the two
  → The alligation cross
• Scale from the right side
  → From the ratio to a quantity, a price, or any average

Formulas (2)

• Water sold at the price of milk · Cost of one litre of the mixture
  $$\text{CP} = \text{SP} \times \dfrac{100}{100 + \text{gain}}$$
• Profit on a mixture, and mixed profit rates · Profit on a mixture
  $$\text{profit} = \dfrac{\text{SP} - \text{mean cost}}{\text{mean cost}} \times 100\%$$

Watch out for (3)

• Gain is measured on the milk, not on the mixture
  → Water sold at the price of milk
• The selling price is not the mean
  → Water sold at the price of milk
• Weight the profit rates
  → Profit on a mixture, and mixed profit rates

Formulas (3)

• One draw and one top-up: track each liquid · What a draw leaves
  $$\text{left} = \left(1 - \dfrac{x}{T}\right) \times \text{amount before}$$
• Topping up with water: the milk keeps a fixed fraction · Fraction the milk keeps
  $$\dfrac{\text{share after}}{\text{share before}} = \dfrac{T - x}{T}$$
• Repeated replacement · Milk after n rounds
  $$\text{milk left} = V\left(1 - \dfrac{x}{V}\right)^n$$

Watch out for (4)

• A draw takes some of every liquid
  → One draw and one top-up: track each liquid
• The percentage is multiplied, not reduced by a subtraction
  → Topping up with water: the milk keeps a fixed fraction
• Count the rounds
  → Repeated replacement
• Do not subtract the same amount each round
  → Repeated replacement