Numerical Reasoning: Solving Everyday Number Problems

Formulas (1)

• Linked relations: substituting one into another · Chaining two relations
  $$a = kb \ \text{and}\ b = mc \;\Rightarrow\; a = kmc, \quad c = \frac{a}{km}$$

Watch out for (4)

• Solving for the wrong quantity
  → Setting up and solving an equation from a word problem
• "5 less than x" is x − 5
  → Setting up and solving an equation from a word problem
• Chained multipliers multiply
  → Linked relations: substituting one into another
• Undo the steps in reverse order
  → Working backwards and testing the options

Formulas (3)

• Percentage change and the multiplier · Percentage change
  $$\text{percentage change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$$
• Successive changes and reverse percentages · Successive changes and the original value
  $$\text{final} = \text{original} \times m_1 \times m_2 \qquad \text{original} = \frac{\text{final}}{m}$$
• Percentages of a part, and percentage points · Part of a part
  $$\text{fraction of the whole} = \frac{p_1}{100} \times \frac{p_2}{100}$$

Watch out for (5)

• Percentage change divides by the original
  → Percentage change and the multiplier
• Successive percentages do not add
  → Successive changes and reverse percentages
• To find the original, divide by the multiplier
  → Successive changes and reverse percentages
• Percentage points are not percent
  → Percentages of a part, and percentage points
• "Of the rest" means of the remainder, not of the whole
  → Percentages of a part, and percentage points

Formulas (4)

• Sharing a quantity in a given ratio · Share in a ratio
  $$\text{share}_i = \text{total} \times \frac{r_i}{r_1 + r_2 + \dots}$$
• Direct and inverse proportion, and value for money · Direct and inverse proportion
  $$\text{direct: } \frac{y_1}{x_1} = \frac{y_2}{x_2} \qquad \text{inverse: } x_1 y_1 = x_2 y_2$$
• Speed, distance and time, and average speed · Speed and average speed
  $$v = \frac{d}{t} \qquad \bar{v} = \frac{d_{\text{total}}}{t_{\text{total}}}$$
• Work and flow rates: adding rates, not times · Two workers together
  $$\frac{1}{T} = \frac{1}{t_1} + \frac{1}{t_2}$$

Watch out for (4)

• 2 : 5 means 2 parts out of 7
  → Sharing a quantity in a given ratio
• More workers means less time
  → Direct and inverse proportion, and value for money
• Average speed is not the average of the speeds
  → Speed, distance and time, and average speed
• Rates add; times do not
  → Work and flow rates: adding rates, not times

Formulas (3)

• Mean and weighted mean in word problems · Weighted mean
  $$\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$$
• Unit and currency conversions · Energy used by an appliance
  $$E\,(\text{kWh}) = P\,(\text{kW}) \times t\,(\text{h})$$
• Time, clocks, calendars and repeating cycles · Day of the week after n days
  $$\text{shift} = n \bmod 7$$

Watch out for (5)

• Do not average two averages of unequal groups
  → Mean and weighted mean in word problems
• Area and volume factors are squared and cubed
  → Unit and currency conversions
• Exchange rates work in one direction at a time
  → Unit and currency conversions
• Hours are not decimals
  → Time, clocks, calendars and repeating cycles
• Counting both ends
  → Time, clocks, calendars and repeating cycles

Formulas (3)

• Systematic listing and the multiplication rule · Choosing r items from n
  $$\text{ordered: } \frac{n!}{(n-r)!} \qquad \text{unordered: } \binom{n}{r} = \frac{n!}{r!\,(n-r)!}$$
• Overlapping groups and Venn reasoning · Two overlapping groups
  $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$
• Probability in context · Two events in turn
  $$P(A \text{ then } B) = P(A) \times P(B \text{ given } A)$$

Watch out for (4)

• Does order matter?
  → Systematic listing and the multiplication rule
• Adding two groups counts the overlap twice
  → Overlapping groups and Venn reasoning
• Without replacement, the second fraction changes
  → Probability in context
• A positive test is not the same as having the condition
  → Probability in context

Formulas (1)

• Repeated growth, doubling and half-life · Growth and decay
  $$N = N_0\, r^{\,n} \qquad N = N_0 \left(\tfrac{1}{2}\right)^{t/T}$$

Watch out for (3)

• Repeated percentages compound
  → Repeated growth, doubling and half-life
• Two half-lives leave a quarter, not nothing
  → Repeated growth, doubling and half-life
• "How many are needed" rounds up
  → Estimation, bounds and checking the options