Percentage, ratio and proportion, averages, and mixtures and alligation, with worked examples and practice
This is the single most reused toolkit in the paper. Percentage logic powers profit loss and interest and parts of data interpretation; ratio and average appear in nearly every word-problem family. Build fluency here before moving on.
| Concept | Formula |
|---|---|
| Percentage to fraction | x% = x / 100 |
| What percent A is of B | (A / B) times 100 |
| Percentage change | (change / original) times 100 |
| Increase by x% | multiply by (1 + x/100) |
| Decrease by x% | multiply by (1 - x/100) |
| Successive change a% then b% | net = a + b + (ab/100) |
| Ratio a : b means | parts a and b of total (a + b) |
| Proportion a:b = c:d | a times d = b times c (product of means = product of extremes) |
| Average | (sum of all values) / (number of values) |
| Sum from average | average times count |
When two ingredients of prices (or concentrations) C (cheaper) and D (dearer) are mixed to give mean M: (quantity of cheaper) : (quantity of dearer) = (D - M) : (M - C).
| Percent | Fraction | Percent | Fraction |
|---|---|---|---|
| 50% | 1/2 | 12.5% | 1/8 |
| 33.33% | 1/3 | 20% | 1/5 |
| 25% | 1/4 | 16.66% | 1/6 |
| 10% | 1/10 | 6.25% | 1/16 |
Knowing these by sight turns many percentage sums into one-step mental arithmetic.
A candidate scored 165 marks out of 250 in Paper I. What percentage is this?
(165 / 250) times 100 = 66%.
A shopkeeper raises a price by 20% and then gives a 20% discount. What is the net change?
Net = 20 + (-20) + (20 times -20)/100 = 20 - 20 - 4 = -4%. So the final price is 4% below the original. (Up-then-down by the same percent always loses.)
Rs 6,300 is to be divided among A, B, and C in the ratio 2 : 3 : 4. Find each share.
Total parts = 2 + 3 + 4 = 9. One part = 6300 / 9 = 700. A = 1400, B = 2100, C = 2800. (Check: 1400 + 2100 + 2800 = 6300.)
The average age of 10 recruits is 22 years. A new recruit joins and the average becomes 22.5. Find the new recruit's age.
Old sum = 10 times 22 = 220. New sum = 11 times 22.5 = 247.5. New recruit's age = 247.5 - 220 = 27.5 years.
In what ratio must rice at Rs 30 per kg be mixed with rice at Rs 45 per kg to get a mixture worth Rs 35 per kg?
Cheaper C = 30, dearer D = 45, mean M = 35. Ratio = (D - M) : (M - C) = (45 - 35) : (35 - 30) = 10 : 5 = 2 : 1. So 2 parts of the Rs 30 rice to 1 part of the Rs 45 rice.
A jeep covers a stretch at 60 km/h and returns over the same stretch at 40 km/h. Find the average speed.
The average speed for equal distances is the harmonic mean, not the simple mean. Average = 2ab / (a + b) = 2 times 60 times 40 / (60 + 40) = 4800 / 100 = 48 km/h. (Note it is below 50, the simple average, which is why the simple mean is wrong here.)