Average, Alligation and Partnership: Shortcuts That Save Minutes

Average, mixture and partnership questions appear in almost every SSC and banking quant section. They look different, but they all rest on one idea: work with totals, not averages. This guide shows the shortcuts and the traps.

1. Averages: always convert to totals

Total = Average × Number of items. Most average questions become simple subtraction once you switch to totals.

Example: The average age of 30 students is 15. With the teacher included, the average becomes 16. How old is the teacher?

• New total = 31 × 16 = 496
• Old total = 30 × 15 = 450
• Teacher = 496 − 450 = 46 years

The “change in average” shortcut

When one item is replaced and the average of n items changes by d, the total changes by n × d.

Example: The average weight of 8 people rises by 2.5 kg when a new person replaces someone weighing 65 kg. The total rises by 8 × 2.5 = 20 kg, so the newcomer weighs 65 + 20 = 85 kg.

Consecutive numbers

For numbers in arithmetic progression (consecutive numbers, consecutive even numbers and so on), the average is the middle term, or the average of the first and last terms.

• 7 consecutive numbers with average 20 → the middle (4th) is 20, so the numbers run from 17 to 23.
• Average of 1 to 10 = (1 + 10) ÷ 2 = 5.5.

The overlapping-groups trap

If the first 6 and the last 6 of 11 results are averaged separately, the 6th result is counted twice:

6th result = (sum of first 6) + (sum of last 6) − (sum of all 11)

2. Average speed is not the simple average

If a vehicle covers equal distances at speeds x and y:

Average speed = 2xy ÷ (x + y)

Going at 60 km/h and returning at 40 km/h gives 2 × 60 × 40 ÷ 100 = 48 km/h, not 50. The car spends more time at the slower speed, so the average is pulled down. This is the most common trap in this chapter.

If the times are equal instead (for example, 1 hour at each speed), then the simple average (x + y) ÷ 2 is correct.

3. Alligation: the cross method for mixtures

When two items with different prices (or concentrations) are mixed to get a mean value:

   Cheaper (c)        Dearer (d)
            \          /
             Mean (m)
            /          \
     (d − m)            (m − c)

Cheaper : Dearer = (d − m) : (m − c)

Example: Rice at ₹30/kg and ₹40/kg is mixed to sell at ₹36/kg (cost price). Cheaper : Dearer = (40 − 36) : (36 − 30) = 4 : 6 = 2 : 3.

Alligation works for any “average of two groups” question: prices, percentages of milk, average marks of boys and girls, even average speeds over equal times.

Adding water to change a ratio

Keep the quantity that does not change fixed. In a 40-litre mix of milk and water at 3 : 1, milk = 30 L. To make the ratio 3 : 2, water must be 20 L, so add 10 L of water.

4. Partnership: capital × time

Profit is shared in the ratio of capital × months invested.

Example: A invests ₹45,000 for 12 months. B joins after 3 months with ₹30,000 (so for 9 months). A : B = 45,000 × 12 : 30,000 × 9 = 5,40,000 : 2,70,000 = 2 : 1.

Tips:

• Cancel common zeros first: 45 × 12 : 30 × 9 = 540 : 270.
• If everyone invests for the same time, share only by capital.
• Read carefully whether a partner joins late or leaves early; both reduce their months.

Quick revision checklist

Question type	Use
Item added, removed or replaced	Totals: new total − old total
Consecutive numbers	The middle term is the average
Equal distances at two speeds	2xy ÷ (x + y)
Two items mixed at a mean price	Alligation: (d − m) : (m − c)
Business profit	Capital × time ratio

Practise all of these in our average, mixture and partnership set, where every answer shows the working.