Reverse average, often encountered in problems where you know the overall mean and need to find a missing value or total, is calculated by working backwards from the given mean. Here's how it's done, incorporating the approach seen in the provided reference:
The core idea behind reverse average problems is to first calculate the total sum of all values and then use it to find any unknown values.
Here is how to work out a typical reverse average problem, such as that explained in the referenced YouTube short:
-
Calculate the Total Sum:
-
The video explains, using the given class average of 60 and the total number of students of 30. So first, calculate the total combined score by multiplying the mean by the total number of values (i.e., total students).
- Example: Mean score (60) * Total students (30) = Total score (1800)
-
-
Determine the Sum of Known Values:
- If you have subgroups with their own averages or totals, calculate their sums separately.
-
Find the Missing Value or Group Sum:
-
Subtract the sum of known values from the total sum you calculated in step 1. This difference is the missing sum.
-
Example: If we knew, for example, that the boys average mark was 50 then the total marks for the boys would be 50 * 10 = 500. Subtracting this from the total class marks gives you 1800 - 500 = 1300 total marks for the girls. We can then work out their average if needed by dividing this figure by their number (1300 / 20 = 65).
-
Practical Insight
Reverse average problems often involve scenarios where you're given an overall mean and some subgroup information. The key is always to find the total sum first, then break that total down into parts.
Example Summary From Reference
| Steps | Calculation |
|---|---|
| 1. Total students | 10 boys + 20 girls = 30 |
| 2. Total class score | 30 students * mean score of 60 = 1800 |
| 3. Total score example for sub group (boys) | If mean score for boys is 50, then 10 * 50 = 500 |
| 4. Total score example for sub group (girls) | Total score (1800) - total score of boys (500) = 1300 |
| 5. Mean score example for sub group (girls) | total score for girls(1300) / total number of girls (20) = 65 |
By following these steps, you can effectively tackle reverse average calculations.
