Averages, weighted averages and mixtures

Average questions look easy and are where many candidates lose marks, because they work with the averages themselves instead of the totals behind them. Mixture questions are the same idea: two groups with different values combined into one. This lesson gives one method that handles both, with worked examples you can check by hand.

Rule one: convert every average to a total

Average equals total divided by count, so total equals average times count. Before doing anything else, write the total.

Example: the average of 5 numbers is 24. One number is removed and the average of the remaining 4 is 22. What was removed? Total of 5 is 24 times 5, which is 120. Total of 4 is 22 times 4, which is 88. Removed number: 120 minus 88, which is 32. Check: 88 plus 32 is 120, and 120 divided by 5 is 24.

Rule two: a new member shifts the average by a known amount

When one person joins or replaces another, the change in total is spread across everyone.

Example: the average weight of 8 people rises by 2 kg when a new person replaces one weighing 60 kg. Find the new person's weight. The total rose by 2 times 8, which is 16 kg. So the new person weighs 60 plus 16, which is 76 kg. No need to know anyone else's weight.

Rule three: weighted averages and mixtures

When two groups combine, the combined average sits between the two group averages, closer to the larger group. Compute it from totals.

Example: 20 litres of milk at 40 per litre is mixed with 30 litres at 50 per litre. Average price: total cost is 800 plus 1500, which is 2300, over 50 litres, which is 46 per litre. It is closer to 50 because the 50-rupee milk is the larger part.

The reverse question, how much of each to mix to reach a target, uses the same idea. To get a 46 average from 40 and 50, the distances from the target are 6 and 4. The quantities are in the ratio 4 to 6, which is 2 to 3, with the cheaper item getting the ratio part from the dearer item's distance. Check with 20 and 30 litres above: 2 to 3. Correct.

Common mistakes

• Averaging two averages directly, as in (40 plus 50) over 2 equals 45, which ignores the quantities.
• Forgetting that the count changes when someone joins versus when someone is replaced.
• Mixing units, such as hours in one group and minutes in another.

Practice habit: Do five average questions a day and write the total for every group before you calculate anything. Check each answer by rebuilding the total from your result. When you are fast with totals, add five mixture questions using the ratio of distances.