CMA Foundation · Fundamentals of Business Mathematics and Statistics
Ratios, Variations and Proportions for CMA Foundation
A ratio compares two quantities of the same kind. A proportion says two ratios are equal. A variation says how one quantity changes when another changes: directly, inversely or jointly. To solve MCQs, write the relation, form the equation with a constant k or cross-multiply, then substitute the given values.
What this chapter covers
This chapter in Paper 3 (Fundamentals of Business Mathematics and Statistics) deals with comparing quantities. You start with ratio, which compares two quantities of the same unit, such as ₹600 to ₹400, which is 3 : 2. You then move to proportion, where two ratios are equal, and continued proportion, where a chain of ratios is equal, like a : b = b : c.
Next comes variation. Here one quantity depends on another. In direct variation, if x doubles, y doubles. In inverse variation, if x doubles, y halves. In joint variation, one quantity depends on two or more others at once. The method is the same each time: write the relation with a constant k, find k from the given data, then use it.
The chapter links to much of the paper. Percentages, profit and loss, partnership sharing, mixtures, time and work, and simple and compound interest all use ratio thinking. Averages and index numbers in Statistics also use it. If you are fluent here, many later calculations get shorter.
Questions in this chapter are short and mostly one or two steps, so they suit an objective paper where you have about a minute per question. The skills also carry over to other chapters, so practice here pays back several times. Since there is no negative marking, you can always attempt every question, and fast, clean methods let you secure these marks early and keep time for harder numerical questions.
Ratios, Variations and Proportions: topics in the order to study them
- 1Ratio: Meaning and TypesEverything else is built on ratios, so learn the meaning, simplification, comparison and the types (duplicate, triplicate, sub-duplicate, inverse, compound) first.
- 2Proportion and Continued ProportionProportion is two equal ratios, and it needs ratio skills. You also need it to understand how variation equations are set up.
- 3Direct, Inverse and Joint VariationVariation uses the constant k and proportion ideas together, so it comes after both ratio and proportion are comfortable.
- 4Applications of Ratios and ProportionsWord problems on sharing, ages, mixtures and work need all the earlier tools, so practice them last to combine everything.
How to prepare Ratios, Variations and Proportions
Prepare this chapter through short, repeated practice. The concepts are simple, so marks are won by speed and accuracy.
- Learn the definitions and types of ratio, with one small example for each. Make sure you can say what duplicate, triplicate and sub-duplicate mean without looking.
- Practice simplifying ratios and comparing them by cross-multiplying or converting to decimals. Do this until it takes seconds.
- Learn the proportion rule: if a : b = c : d, then a × d = b × c. For continued proportion a : b = b : c, use b² = a × c.
- For variation, always write the relation first (y = kx, y = k ÷ x, or z = kxy), find k from the given pair of values, then substitute.
- Solve application problems by turning the story into a ratio with a single unit, such as 3x and 5x, and then solving for x.
- Do timed sets of 15 to 20 MCQs, then review each wrong answer and note whether the error was in setup or in arithmetic.
- In the last days, revise only your formula list and your error notes.
Common mistakes in Ratios, Variations and Proportions
Comparing quantities in different units without converting them first.
Fix: Convert both quantities to the same unit before forming the ratio. Here 2 hours is 120 minutes, so the ratio is 120 : 30 = 4 : 1.
Mixing up duplicate and sub-duplicate ratios.
Fix: Remember: duplicate means squares, sub-duplicate means square roots, triplicate means cubes. Test with 4 : 9, which gives 16 : 81 and 2 : 3.
Using direct variation where the variation is inverse.
Fix: Ask whether the other quantity should go up or down. If it goes the opposite way, use x × y = constant.
Treating b² = ac as true for every proportion.
Fix: Use b² = ac only when the middle term repeats, as in a : b = b : c. For a general proportion use ad = bc.
Finding the value of x in a sharing problem but not the final share.
Fix: Reread the question after solving. Substitute x into the part asked for, and check that the parts add up to the total.
Skipping the step of finding k in variation problems.
Fix: Write the equation, plug in the known pair to get k, then use k in the new case. It is slower by a few seconds but much safer.
Last-day revision: Ratios, Variations and Proportions
- A ratio compares two quantities of the same kind and unit; it has no unit itself.
- Ratio a : b is written as the fraction a ÷ b, and b should not be zero.
- Duplicate ratio of a : b is a² : b²; triplicate is a³ : b³; sub-duplicate is √a : √b.
- Inverse ratio of a : b is b : a.
- Compound ratio of a : b and c : d is ac : bd.
- Proportion: a : b = c : d means ad = bc.
- Continued proportion: a : b = b : c means b² = ac, and b is the mean proportional between a and c.
- Third proportional to a and b is b² ÷ a.
- Direct variation: y = kx, so y ÷ x stays constant.
- Inverse variation: y = k ÷ x, so x × y stays constant.
- Joint variation: z = kxy; find k first, then substitute.
- If a quantity is shared in the ratio 3 : 2, take the parts as 3x and 2x and divide the total by 5 to find x.
Ratios, Variations and Proportions practice questions
- If a:b = b:c = 2:3 (a continued proportion), and a = 16, what is the value of c?
- A mixture of 40 litres contains milk and water in the ratio 3 : 1. How many litres of water must be added so that the ratio of milk to water…
- Which of the following is the duplicate ratio of the ratio 3:5, and what is its value when compared with the sub-duplicate ratio of 36:64?
- The number of units of a product demanded in a Surat market varies inversely with its price per unit. At a price of ₹40, the demand is 600 u…
- If 4, x and 25 are in continued proportion with x positive, and the mean proportional x is the second term, what is the value of (x + 4) : (…
- A contractor's 20 men working 8 hours a day can build 120 metres of road in 15 days. How many days will 30 men working 6 hours a day need to…
- The time taken to complete a job by a team varies inversely with the number of workers. If 12 workers complete a job in 15 days, how many da…
- A firm's monthly sales to purchases are in the ratio 7:5. If sales for a month are ₹4,90,000, what is the amount of purchases?
Ratios, Variations and Proportions in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Ratios, Variations and Proportions: frequently asked questions
How many questions can I expect from this chapter in the CMA Foundation exam?
ICMAI does not fix the number of questions per chapter, so do not rely on a number. Paper 3 has 50 MCQs of 2 marks each. Treat this chapter as a scoring area, because its questions are short and direct.
What is the difference between proportion and variation?
A proportion states that two ratios are equal, such as 2 : 3 = 4 : 6. Variation describes how one quantity changes with another using a constant k. Proportion is usually used for a single comparison, and variation is used when a relationship holds across many values.
Do I need to memorise many formulas in this chapter?
No. There are only a few: the types of ratio, ad = bc, b² = ac for continued proportion, and the three variation equations. Understanding them is faster than memorising, and you can test any formula on small numbers.
How should I attempt these questions with no negative marking?
Attempt every question. If you cannot solve one quickly, eliminate options that are clearly off, such as values that do not fit the ratio or that exceed the total, and then pick one. Come back later if time remains.
Is this chapter useful for other chapters in Paper 3?
Yes. Ratio thinking helps in percentages, profit and loss, partnership, time and work, interest and index numbers. A firm grip here makes those chapters easier.