Qudurat (GAT) Complete Review Guide 2026–2027

4. An arithmetic sequence has a constant difference: \(a_n = a_1 + (n-1)d\). A geometric one has a constant ratio: \(a_n = a_1 \cdot r^{n-1}\) (u5.1).
5. To compare two fractions, use a common denominator or cross-multiply: \(\frac{a}{b} < \frac{c}{d}\) when \(ad < bc\) (for positive denominators) (u5.2).
6. \(p\%\) of \(x\) equals \(\frac{p}{100} \times x\). Mental tip: 10% of a number means shifting the decimal point one place, and 5% is half of that (u5.2).
7. Percentage change = \(\frac{\text{change}}{\text{original value}} \times 100\), always dividing by the original value (u5.2).
8. To split a quantity in the ratio \(a:b\), add the parts \(a+b\); the first share is \(\frac{a}{a+b}\) of the quantity (u5.2).
9. Direct proportion: the quotient is constant (\(\frac{y}{x} = k\)); inverse proportion: the product is constant (\(xy = k\)) (u5.2).
10. Exponent laws: \(a^m \cdot a^n = a^{m+n}\), \((a^m)^n = a^{mn}\), \(a^0 = 1\), \(a^{-n} = \frac{1}{a^n}\) (u5.3).
11. Roots: \(\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}\) for non-negative numbers, so \(\sqrt{50} = 5\sqrt{2}\). Memorize squares up to 15² (u5.3).
12. Speed = distance ÷ time, so distance = speed × time and time = distance ÷ speed (u5.4).
13. Average speed = total distance ÷ total time. When two objects move toward each other, add their speeds to find the meeting time (u5.4).
14. To convert km/h to m/s multiply by \(\frac{5}{18}\), and for the reverse multiply by \(\frac{18}{5}\). A map scale of 1:100000 means 1 cm represents 1 km (u5.4).
15. Mean = sum of values ÷ number of values, so sum = mean × count. The median is the middle value after sorting, the mode is the most frequent, and range = largest − smallest (u5.5).
16. Probability = favorable outcomes ÷ total outcomes, its value lies between 0 and 1, and P(not happening) = 1 − P(happening) (u5.5).
17. Comparison choices: first is greater, second is greater, equal, or the data are insufficient. Pick the first, second or equal only if it holds in every possible case (u5.6).

Common mistakes in this chapter

• ✕ A 20% increase followed by a 20% decrease does not return the original number: the result is 96% of it, because the second percentage is taken from the new number.
• ✕ Successive discounts of 20% and 10% are not a 30% discount; the real discount is 28%, because the price is multiplied by 0.8 and then by 0.9.
• ✕ Average speed is not the mean of two speeds when the time for each leg differs; always use total distance ÷ total time.
• ✕ \(\sqrt{a+b}\) does not equal \(\sqrt{a}+\sqrt{b}\), and \((a+b)^2\) does not equal \(a^2+b^2\).
• ✕ A ratio of 2:3 does not mean the first part is \(\frac{2}{3}\) of the whole; it is \(\frac{2}{5}\), because the whole = 2 + 3.
• ✕ The median is taken after sorting the values in ascending order; taking the middle value of an unsorted list is a common mistake.
• ✕ In a comparison, testing one value is not enough to pick "first is greater"; try zero, fractions and negatives, and if the results differ the data are insufficient.
• ✕ Unit-conversion slips: multiplying by 60 instead of dividing, or forgetting to square the factor for area (1 m² = 10000 cm²).