60-second revision
- Total = position from left + position from right - 1.
- Position from one end = total - position from the other end + 1.
- People between two positions counted from the same end = difference - 1.
- After an interchange, use the position that is known from both ends.
- With 'k people between' and an unknown order, test both cases; one gives the minimum, the other the maximum.
- Middle of a row of N (N odd) = (N + 1) ÷ 2.
- Class total = passed + failed + absent; ranks are counted among those who passed.
The core formulas
Order and ranking questions are about positions in a line or ranks in a list. Draw a short line, mark the people you know, and use the three formulas below. Never add both positions without subtracting 1 — the person is counted twice.
| You know | Formula | Example |
|---|---|---|
| Position of one person from both ends | Total = L + R - 1 | 9th from left, 14th from right: 22 |
| Total and position from one end | Other end = N - position + 1 | N = 50, 18th from top: 33rd from bottom |
| Two positions from the same end | Between = difference - 1 | 6th and 21st from left: 14 between |
| Positions from opposite ends, total known | Convert one to the other end first | N = 30, 8th from left and 10th from right: 21st from left, 12 between |
Interchange questions
When two people swap places, each takes over the other's old position. Find the one position that is now known from both ends and apply Total = L + R - 1. Example: P is 7th from the left and Q 12th from the right. After they swap, P is 15th from the left. P now stands where Q stood, so that spot is 15th from the left and 12th from the right: total = 26.
Minimum and maximum (two cases)
If the question gives two people's positions from opposite ends and the number between them, but not who is ahead, there are two arrangements. In the first, they do not overlap and the total is large; in the second, the one counted from the back is ahead and the total is smaller.
Worked example: X is 10th from the front, Y is 15th from the back, and 4 people are between them. Case 1 (X ahead): Y is 10 + 5 = 15th from the front, total = 15 + 15 - 1 = 29. Case 2 (Y ahead): Y is 10 - 5 = 5th from the front, total = 5 + 15 - 1 = 19. Minimum 19, maximum 29.
Middle of a row
A row of N people (N odd) has its middle at (N + 1) ÷ 2. If someone is k-th from one end, the places to move to reach the middle = middle - k (towards the centre). If a person is 'n-th from either end', the row has 2n - 1 people.
Pass, fail and absent
When ranks are given 'among those who passed', the formula gives only the number who passed. Add the students who failed, were absent or did not take the test to get the class strength. Example: 20th from the top and 25th from the bottom among those who passed, 5 failed and 2 absent: passed = 44, class = 51.
Comparison ranking
Some questions rank people by height, weight or marks using statements like 'A is taller than B but shorter than C'. Write the people in a vertical chain from highest to lowest and add each clue. 'Taller than only two' means exactly two people are below; 'third tallest' is counted from the top.
Test yourself: 10 questions, 50 s each
Every answer is explained. Free, no sign-up.
Start the quiz