60-second revision
- Average = sum of observations / number of observations; sum = average × number.
- Average of evenly spaced numbers = middle term = (first + last)/2.
- Average of first n natural numbers = (n + 1)/2; of first n even numbers = n + 1; of first n odd numbers = n.
- A new member raises the average of n people by d: new member = old average + (n + 1)d.
- Replacing one of n people changes the average by d: newcomer = person replaced + nd.
- Average speed over equal distances = 2xy/(x + y), not (x + y)/2.
- The age difference between two people never changes; their age ratio does.
The basic idea
Average = total / count, so total = average × count. Almost every averages question is solved by working with totals, not averages. If 8 numbers average 15, their total is 120.
Ready formulas
| Set of numbers | Average |
|---|---|
| First n natural numbers | (n + 1)/2 |
| First n even numbers | n + 1 |
| First n odd numbers | n |
| Any evenly spaced (consecutive, AP) list | (first + last)/2 = the middle term |
| Squares of first n natural numbers | (n + 1)(2n + 1)/6 |
Example: five consecutive even numbers average 36, so the middle one is 36 and the numbers are 32, 34, 36, 38, 40.
Change in a group: the deviation method
| Situation | Shortcut | Worked example |
|---|---|---|
| A person joins n people; average rises by d | Newcomer = old average + (n + 1)d | 24 students average 36 kg; with the teacher it rises by 1 kg. Teacher = 36 + 25 = 61 kg |
| One of n people is replaced; average rises by d | Newcomer = person replaced + nd | 10 people; one of 62 kg replaced; average up 1.5 kg. Newcomer = 62 + 15 = 77 kg |
| A person leaves n people; average falls by d | Leaver = old average + (n - 1)d | 9 people averaging 30; one leaves and the average of the other 8 falls by 1. Leaver = 30 + 8 = 38 |
Check the first example with totals: 25 × 37 - 24 × 36 = 925 - 864 = 61.
Batsman and innings
In the n-th innings a batsman scores S and raises his average by d. Then new average = S - (n - 1)d. Example: in his 12th innings he scores 95 and the average rises by 4, so the new average = 95 - 11 × 4 = 51 (old average 47; check 11 × 47 + 95 = 612 = 12 × 51).
Overlapping groups (the middle value)
Example: 9 numbers average 40. The first five average 38 and the last five average 43. The 5th number is counted twice, so 5th = 5 × 38 + 5 × 43 - 9 × 40 = 190 + 215 - 360 = 45.
Weighted average and average speed
- Weighted average: section A has 30 students averaging 60, section B 20 averaging 70. Combined = (30 × 60 + 20 × 70)/50 = 64.
- Average speed = total distance / total time. Over equal distances at x and y km/h it is 2xy/(x + y). At 30 and 45 km/h: 2 × 30 × 45/75 = 36 km/h.
Problems on ages
Let the present ages be the unknowns. "x years ago" subtracts x from each age; "after x years" adds x to each. Both people age by the same number of years, so their difference stays fixed.
- Ratio type: six years ago the ages of A and B were 3:5; four years from now they will be 2:3. Let the ages six years ago be 3x and 5x. Then (3x + 10)/(5x + 10) = 2/3, so 9x + 30 = 10x + 20 and x = 10. Present ages: 36 and 56.
- Times-as-old type: a mother is 4 times her daughter's age; in 5 years she will be 3 times. 4d + 5 = 3(d + 5), so d = 10 and the mother is 40.
- Sum type: if a family's total age is known now, the total after t years rises by t × (number of members).
- Children born at equal gaps: the ages form an evenly spaced list, so the average age is the middle child's age.
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