Thinking Skills Assessment (TSA) Foundational Logic and Numeracy Flashcards

24 sample cards from the Foundational Logic and Numeracy deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the percentage change formula.

   $\text{percentage change} = \frac{\text{new value} - \text{old value}}{\text{old value}} \times 100\%$. A positive result is an increase, negative a decrease.

2. A price rises by $20\%$ then falls by $20\%$. What is the net percentage change?

   A net decrease of $4\%$. Multiplying factors: $1.20 \times 0.80 = 0.96$, i.e. $96\%$ of the original, a $4\%$ fall. Successive percentage changes multiply, they do not add.

3. How do you reverse a percentage increase to find the original amount? E.g. a price is £60 after a $20\%$ increase.

   Divide by the multiplier, not subtract the percentage. Original $= \frac{60}{1.20} = £50$. (Subtracting $20\%$ of £60 would wrongly give £48.)

4. How do you add the fractions $\frac{2}{3} + \frac{1}{4}$?

   Use a common denominator (LCM of 3 and 4 is 12): $\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$.

5. How do you divide one fraction by another, e.g. $\frac{3}{4} \div \frac{2}{5}$?

   Multiply by the reciprocal of the divisor: $\frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$.

6. How do you simplify the ratio $18:24$ to its lowest terms?

   Divide both parts by their greatest common divisor. $\gcd(18,24)=6$, so $18:24 = 3:4$.

7. To split £200 in the ratio $3:2$, how much does each part receive?

   Total parts $= 3+2 = 5$, so one part $= \frac{200}{5} = £40$. The shares are $3\times40 = £120$ and $2\times40 = £80$.

8. Define direct proportion and write its equation.

   Two quantities are directly proportional if one is a constant multiple of the other: $y \propto x$, i.e. $y = kx$ for constant $k$. Doubling $x$ doubles $y$; their ratio $\frac{y}{x}$ is constant.

9. Define inverse proportion and write its equation.

   Two quantities are inversely proportional if their product is constant: $y \propto \frac{1}{x}$, i.e. $xy = k$ (or $y = \frac{k}{x}$). Doubling $x$ halves $y$.

10. If 4 workers build a wall in 6 days, how long for 3 workers (same rate)? Which proportion applies?

    Inverse proportion (workers $\times$ days $=$ constant). $4 \times 6 = 24$ worker-days, so $\frac{24}{3} = 8$ days.

11. Define the mean, median and mode.

    Mean: sum of values divided by the count, $\bar{x}=\frac{\sum x}{n}$. Median: the middle value when data are ordered (mean of the two middle values if $n$ is even). Mode: the most frequently occurring value.

12. What is the range of a data set, and how do you compute it?

    The range is a measure of spread: $\text{range} = \text{maximum value} - \text{minimum value}$.

13. Find the median of $4, 9, 2, 7, 5$.

    Order the data: $2, 4, 5, 7, 9$. With $n=5$ the middle (3rd) value is the median: $5$.

14. When is the median preferable to the mean as an average?

    When the data are skewed or contain outliers/extreme values. The median is resistant to outliers, whereas the mean is pulled toward extreme values and can misrepresent a typical value.

15. How do you find the quartiles and the interquartile range (IQR)?

    Order the data; the lower quartile $Q_1$ is the median of the lower half and the upper quartile $Q_3$ the median of the upper half. $\text{IQR} = Q_3 - Q_1$, the spread of the middle $50\%$ of the data.

16. Calculate the mean of $12, 15, 9, 14$.

    $\bar{x} = \frac{12+15+9+14}{4} = \frac{50}{4} = 12.5$.

17. What does it mean to round to 1 significant figure, and what is 4,837 to 1 s.f.?

    Keep only the first non-zero digit, rounding based on the next digit and replacing the rest with zeros. $4{,}837 \approx 5{,}000$ to 1 s.f.

18. Estimate $\frac{612 \times 3.9}{19}$ by rounding to 1 significant figure.

    $\approx \frac{600 \times 4}{20} = \frac{2400}{20} = 120$. (Exact value is about $125.6$, so the estimate is close.)

19. What is a quick mental trick to find $15\%$ of a number?

    Find $10\%$ (move the decimal one place left), halve it to get $5\%$, then add the two: $15\% = 10\% + 5\%$. E.g. $15\%$ of $240 = 24 + 12 = 36$.

20. Give the mental shortcut for multiplying a number by 5.

    Multiply by 10 and halve the result (since $5 = \frac{10}{2}$). E.g. $48 \times 5 = \frac{480}{2} = 240$.

21. State the divisibility rules for 3 and for 9.

    A number is divisible by 3 if the sum of its digits is divisible by 3; it is divisible by 9 if the digit sum is divisible by 9. E.g. $171$: digits sum to $9$, so it is divisible by both 3 and 9.

22. When reading a pie chart, how do you convert a sector to an actual quantity, and what does the whole circle represent?

    The whole circle ($360^\circ$ or $100\%$) represents the total. A sector's share $= \frac{\text{sector angle}}{360^\circ}$ of the total; multiply that fraction by the total to get the quantity it represents.

23. What is the basic definition of the probability of an event with equally likely outcomes?

    $P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$, a value between $0$ (impossible) and $1$ (certain).

24. State the addition and multiplication rules for mutually exclusive and independent events respectively.

    For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$. For independent events: $P(A \text{ and } B) = P(A) \times P(B)$. Also, complement: $P(\text{not } A) = 1 - P(A)$.

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