Ratio, Proportion, Mixture and Partnership Formula Guide for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
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PDF Notes: Ratio, Proportion, Mixture and Partnership Formula Guide for SSC Exams
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Ratio and proportion is a base topic that returns inside mixture, partnership, average and age problems. This ratio, proportion, mixture and partnership formula guide for SSC gives the formulas for each of these topics in tables, with the shortcuts that save time and solved examples.
Ratio and Proportion
| Item | Formula or rule |
|---|---|
| Ratio | a : b = a/b |
| Proportion | If a : b = c : d, then a × d = b × c |
| Mean proportion of a and b | √(a × b) |
| Third proportion of a and b | b² / a |
| Fourth proportion of a, b and c | (b × c) / a |
| Compounded ratio of a:b and c:d | ac : bd |
| Duplicate ratio of a:b | a² : b² |
| Sub-duplicate ratio of a:b | √a : √b |
| Dividing a total T in the ratio a:b | First part = [a/(a + b)] × T and second part = [b/(a + b)] × T |
To combine A : B and B : C into A : B : C, make the common term B equal in both ratios.
Mixture and Alligation
| Item | Formula |
|---|---|
| Rule of alligation | Quantity of the cheaper : Quantity of the dearer = (d − m) : (m − c) |
| Meaning of the letters | c = price of the cheaper, d = price of the dearer and m = price of the mixture |
| Mean price of a mixture | (Total cost of all the items) / (Total quantity) |
| Replacement of a liquid | Quantity left after n operations = V × (1 − x/V)^n |
In the replacement formula, V is the volume of the container, x is the volume that is drawn out and replaced each time, and n is the number of operations.
Partnership
- The profit is divided in the ratio of (capital × time) of the partners.
- If the time is the same, the profit ratio is the ratio of the capitals. If the capital is the same, it is the ratio of the times.
- A working partner first gets a fixed commission, and the rest of the profit is divided in the ratio of the capitals.
Average
| Item | Formula |
|---|---|
| Average | Sum of the observations / 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 |
| Weighted average | (n1 × a1 + n2 × a2) / (n1 + n2) |
| When a person is replaced | Change in the total = number of persons × change in the average |
| Average speed for equal distances | 2xy / (x + y) |
Ages
- Write the ages as ratios (for example, 3x and x), and change them by adding or subtracting the years in the question.
- The difference between the ages of two persons never changes.
- The sum of the ages of n persons after t years = present sum + n × t.
Solved Examples
Example 1. Divide 1200 rupees in the ratio 3 : 5.
The first part = (3/8) × 1200 = 450, and the second part = (5/8) × 1200 = 750.
Example 2. Find the mean proportion of 4 and 9.
√(4 × 9) = 6.
Example 3. In what ratio must rice at 30 rupees a kg be mixed with rice at 50 rupees a kg to get a mixture worth 35 rupees a kg?
(50 − 35) : (35 − 30) = 15 : 5 = 3 : 1.
Example 4. A container has 60 litres of milk. 6 litres of milk are taken out and replaced by water. The same is done once more. Find the milk left.
60 × (1 − 6/60)² = 60 × 0.9 × 0.9 = 48.6 litres.
Example 5. A invests 5000 rupees for 12 months and B invests 6000 rupees for 6 months. Find the ratio of the profits.
5000 × 12 : 6000 × 6 = 60000 : 36000 = 5 : 3.
Example 6. 20 students have an average of 60 marks and 30 students have an average of 70 marks. Find the average of all 50 students.
(20 × 60 + 30 × 70) / 50 = 3300 / 50 = 66.
Example 7. A father is 3 times as old as his son. After 10 years he will be twice as old. Find the present age of the father.
Let the son be x years old. Then 3x + 10 = 2(x + 10), so x = 10, and the father is 30 years old.
Quick Tips
- Convert every ratio into one common term before you combine the ratios.
- Draw the cross diagram for alligation and read the answer from the differences.
- In partnership, always multiply the capital by the time.
- Use the weighted average formula when two groups have different averages.
Quick Revision Points
- Mean proportion = √(ab) and third proportion = b²/a.
- Alligation: cheaper : dearer = (d − m) : (m − c).
- Replacement: left = V(1 − x/V)^n.
- Partnership profit ratio = capital × time.
- Weighted average = (n1a1 + n2a2)/(n1 + n2).
Practice Questions With Answers
- Divide 900 rupees in the ratio 2 : 3.
Answer: 360 and 540 rupees - Find the mean proportion of 4 and 25.
Answer: 10 - Find the third proportion of 9 and 12.
Answer: 16 - In what ratio must rice at 30 rupees a kg be mixed with rice at 50 rupees a kg to get a mixture worth 35 rupees a kg?
Answer: 3 : 1 - A container has 80 litres of milk. 8 litres are taken out and replaced by water. Find the milk left.
Answer: 72 litres - A invests 4000 rupees for 12 months and B invests 6000 rupees for 8 months. The profit is 3000 rupees. Find the share of A.
Answer: 1500 rupees - Find the average of 10, 20, 30 and 40.
Answer: 25 - Find the average of the first 9 natural numbers.
Answer: 5 - 15 students have an average of 40 marks and 25 students have an average of 60 marks. Find the average of all.
Answer: 52.5 - If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
Answer: 8 : 12 : 15
How to Practise This Chapter
Write each formula on a card, with a solved example on the back, and revise the cards every day. Then test yourself with our practice papers and the previous year papers page. See all guides on the Formula Guide page, and also read the Mathematics notes and Quantitative Aptitude notes. The previous guide is Percentage, Profit and Loss, and Discount and the next guide is Time, Speed and Distance.
Frequently Asked Questions
What is the mean proportion of two numbers?
It is the square root of their product.
What is the third proportion of a and b?
It is b squared divided by a.
What is the rule of alligation?
The quantity of the cheaper to the quantity of the dearer is (d minus m) to (m minus c).
What is the replacement formula for a mixture?
The quantity left after n operations is V times (1 minus x/V) to the power n.
How is the profit divided in a partnership?
It is divided in the ratio of capital times time of each partner.
What is the weighted average?
It is the sum of (number times average) of each group, divided by the total number.
What is the average of the first n natural numbers?
It is (n + 1)/2.
Does the difference of the ages of two persons change?
No. The difference of ages is always the same.
How do I combine two ratios A:B and B:C?
Make the value of B the same in both ratios, and then write A : B : C.
How do I divide a total in a given ratio?
Find the sum of the ratio parts, and multiply the total by each part divided by this sum.
Is this topic asked in SSC exams?
Yes. Ratio, mixture, partnership, average and age questions are asked in every SSC maths paper.
How many questions come from this topic in SSC exams?
Usually three to six questions, but the number changes from exam to exam.
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Percentage, Profit and Loss, and Discount Formula Guide for SSC Exams
Next chapter
Time, Speed and Distance Formula Guide for SSC Exams
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.