Average, Ages and Partnership Formula Guide: Important Formulas, Tricks and Solved Examples for SSC Maths
Rahul Kumar
SSC Exam Expert & Content Editor
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Average, ages and partnership are three chapters of arithmetic that depend entirely on a few formulas. This average, ages and partnership formula guide for SSC gives the formulas in tables, with the standard tricks and solved examples, for a quick revision before the exam.
Average
| Item | Formula |
|---|---|
| Average | Sum of the observations / number of observations |
| Sum | Average × number of observations |
| Average of the first n natural numbers | (n + 1)/2 |
| Average of the first n even numbers | n + 1 |
| Average of the first n odd numbers | n |
| Sum of the first n natural numbers | n(n + 1)/2 |
| Sum of the first n even numbers | n(n + 1) |
| Sum of the first n odd numbers | n² |
| Average of consecutive numbers | The middle number (for an odd count) or the mean of the two middle numbers |
| Average of an arithmetic progression | (First term + last term)/2 |
| Combined average of two groups | (n1A1 + n2A2)/(n1 + n2) |
| Weighted average | Σ(weight × value)/Σweights |
Changes in an Average
| Case | Formula |
|---|---|
| A new member is added and the average changes by x | New member = old average + (n + 1) × x (if the average rises) |
| A member leaves and the average changes by x | Leaving member = old average − (n − 1) × x (if the average rises) |
| A member is replaced and the average changes by x | New member − old member = n × x (rise) |
| Every number is increased by k | The average is increased by k |
| Every number is multiplied by k | The average is multiplied by k |
Average Speed
| Case | Formula |
|---|---|
| Equal distances at speeds x and y | 2xy/(x + y) |
| Equal times at speeds x and y | (x + y)/2 |
| Equal distances at speeds x, y, z | 3xyz/(xy + yz + zx) |
| General | Total distance / total time |
Age Problems
- Let the present ages be in terms of x; for n years ago subtract n from each age and for n years later add n.
- If the ratio of the ages is a : b, take the ages as ax and bx.
- The difference between the ages of two persons remains the same for ever.
- The sum of the ages of n persons increases by n every year.
- The average age of a family after t years = the present average + t (if no member is added or removed).
- If a person is x times as old as another, then after n years the relation changes, so form an equation for the given time.
Partnership
| Item | Formula |
|---|---|
| Profit sharing ratio (all the capitals invested for the same time) | Ratio of the capitals |
| Profit sharing ratio (capitals for the different times) | (Capital × time) of A : (Capital × time) of B |
| Capital for the same time | Ratio of the profits = ratio of the capitals |
| Working partner and sleeping partner | The working partner first gets a commission or a salary from the profit; the rest is divided in the ratio of the investments |
| Entry of a partner after t months | His capital × (12 − t) months |
Solved Examples
Example 1. The average of 5 numbers is 20. When a sixth number is added, the average becomes 22. Find the sixth number.
Sum of 6 numbers = 132; sum of 5 numbers = 100. The new number = 32.
Example 2. A car goes from A to B at 40 km/h and returns at 60 km/h. Find the average speed.
2 × 40 × 60/100 = 48 km/h.
Example 3. The average of the first 20 natural numbers is?
(20 + 1)/2 = 10.5.
Example 4. A father is four times as old as his son. After 5 years he will be three times as old. Find their present ages.
Let the son be x. 4x + 5 = 3(x + 5), so x = 10. Son = 10 years and father = 40 years.
Example 5. A invests 6,000 rupees for 12 months and B invests 8,000 rupees for 9 months. Find the ratio of the profit.
6000 × 12 : 8000 × 9 = 72,000 : 72,000 = 1 : 1.
Example 6. A invests 5,000 rupees for 12 months and B invests 7,000 rupees for 6 months. If the total profit is 17,000 rupees, find their shares.
60,000 : 42,000 = 10 : 7. A = 10,000 rupees and B = 7,000 rupees.
Quick Tips
- Use the sum method: sum = average × number.
- Use 2xy/(x + y) for equal distances.
- Convert the ratio of ages into algebra.
- Use capital × time for partnership.
Quick Revision Points
- Average = sum/number.
- Average speed for equal distances = 2xy/(x + y).
- The age difference stays constant.
- Profit ratio = capital × time.
- Sum of the first n odd numbers = n².
Practice Questions With Answers
- The average of 4 numbers is 15. Find their sum.
Answer: 60 - Find the average speed if a man goes at 30 km/h and returns at 20 km/h.
Answer: 24 km/h - Find the average of the first 10 even numbers.
Answer: 11 - The ratio of the ages of two persons is 3 : 5 and the sum is 40. Find the ages.
Answer: 15 and 25 - A invests 4,000 and B invests 6,000 for the same time. What is the ratio of the profits?
Answer: 2 : 3 - The average of 3 numbers is 12 and two of them are 10 and 14. Find the third.
Answer: 12 - The sum of the first 15 natural numbers is?
Answer: 120 - A and B invest 3,000 and 4,000 for 12 and 6 months. The ratio of the profit is?
Answer: 3 : 2 - A person is 25 years older than his son. After 5 years he will be 3 times as old. What is the son’s age?
Answer: 10 - The average of 11 numbers is 30. If one number 40 is removed, find the average of the remaining.
Answer: 29
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Formula Guide notes page and the next topic Trains, Boats and Streams Formula Guide.
Frequently Asked Questions
What is the average?
The sum of the observations divided by their number.
What is the average speed for equal distances?
2xy/(x + y).
What is the average of the first n natural numbers?
(n + 1)/2.
What is the sum of the first n natural numbers?
n(n + 1)/2.
What is the combined average?
The total sum of the two groups divided by the total number of the observations.
How do I solve the age problems?
Form equations with the ages as variables and solve them.
What is the profit ratio in a partnership?
The ratio of capital × time of the partners.
What is a sleeping partner?
A partner who invests money but does not take part in the business.
Is this topic asked in SSC?
Yes. Average, ages and partnership are asked in every SSC maths paper.
How much time should a question take?
About 30 to 40 seconds.
What happens to the average if each number is increased by 5?
The average also increases by 5.
What is a weighted average?
An average in which each value has a different importance (weight).
Previous chapter
Simple and Compound Interest Formula Guide: Effective Rate Table, Difference and Half-Yearly Formulas for SSC
Next chapter
Trains, Boats and Streams Formula Guide: Speed, Crossing Time and Upstream Downstream Formulas for SSC Maths
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.