📊 SSC CPO – Quantitative Aptitude
Topic: Average
Average (also called mean) is a measure of the central tendency of a set of numbers.
It is obtained by adding all the numbers and dividing by the count of numbers.
Formula
[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} ]
In SSC CPO, average problems may involve:
- Simple averages
- Weighted averages
- Average of averages
- Solving for unknowns using average conditions
- Comparing averages
📌 Key Points
| Concept | What It Means | Typical Question Type |
|---|---|---|
| Simple Average | Sum of given numbers ÷ count | “If 5 numbers have an average of 12, what is their sum?” |
| Weighted Average | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) where (w_i) = weight | “The first 3 hours of a course had an average score of 70, the next 2 hours 80. What is the overall average?” |
| Average of Averages | (\displaystyle \frac{\sum (n_i \times \text{avg}_i)}{\sum n_i}) | “Class A avg = 75 (20 students), Class B avg = 80 (30 students). Overall avg?” |
| Finding Unknowns | Use the average equation to set up a linear equation | “Five students score 70, 75, 80, 85 and one unknown. The overall average is 80. Find the unknown.” |
| Comparing Averages | Compare two averages by cross‑multiplication or difference | “Average of set X is 30% higher than set Y. If Y’s average is 50, what is X’s?” |
| Integer Constraints | Often averages are whole numbers → Sum must be divisible by count | “If the average of 7 numbers is 15, the sum is 105 (an integer).” |
| Use of Proportions | Average problems often reduce to simple proportions | “If 6 students average 90 and 4 students average 80, what is the overall average?” |
📚 Worked Example
Problem (Typical SSC style):
The average score of 8 students in a class is 70. Five of them scored 80, 75, 70, 65, and 60 respectively. What is the score of the remaining student?
Step‑by‑Step Solution
-
Write the average formula
[ \text{Average} = \frac{\text{Sum of all scores}}{\text{Number of students}} ]
Given:
(\text{Average} = 70)
(\text{Number of students} = 8) -
Compute the total sum of all scores
[ \text{Sum}_{\text{all}} = \text{Average} \times \text{Number of students} = 70 \times 8 = 560 ]
-
Sum the known scores
[ 80 + 75 + 70 + 65 + 60 = 350 ]
-
Find the unknown score
[ \text{Unknown} = \text{Sum}{\text{all}} - \text{Sum}{\text{known}} = 560 - 350 = 210 ]
-
Answer
The remaining student scored 210.
Quick Check
- Average of 8 numbers = 70 → Total = 560
- Known sum = 350 → Remaining = 560 – 350 = 210, which is plausible.
🚀 Quick Revision Summary
- Average = Sum ÷ Count
- Weighted average: use weights (w_i)
- Average of averages: (\displaystyle \frac{\sum (n_i \times \text{avg}_i)}{\sum n_i})
- Unknown from average: set up equation ( \frac{(\text{sum of known} + x)}{n} = \text{avg} )
- Comparisons: cross‑multiply or use percentage differences
- Integer check: Total sum must be divisible by number of items
- Common tricks:
- Multiply average by count to get total sum quickly.
- Replace unknowns with variables and solve a simple linear equation.
- For weighted problems, treat “hours” or “units” as weights.
Practice Problem
In a class of 12 students, the average of the first 6 students is 68 and the average of the remaining 6 students is 73. What is the overall average of the class?
(Try solving before reading the answer below.)
Answer:
[
\text{Overall Avg} = \frac{6\times68 + 6\times73}{12}
= \frac{408 + 438}{12}
= \frac{846}{12}
= 70.5
]
Good luck on your SSC CPO exam!
Remember: practice quick mental calculations for sums and averages; this will save precious time during the test.