Ratio and Proportion
Ratio and proportion form the bridge between arithmetic and algebra in the ADRE Grade IV syllabus, and they appear in more places than the candidate first expects. A ratio compares two quantities of the same kind, and a proportion states that two ratios are equal. The chapter is used directly in problems on partnership, mixture and alligation, time and work, and in the scaling of maps and diagrams. The method followed in the Master Book: Maths Masti (D. Hurina) is to write every comparison as a ratio in its lowest terms and then to convert the given information into an equation of proportions. Once the candidate is comfortable with the idea that a ratio has no units of its own and is only a comparison, most of the difficulty disappears. These questions deserve daily practice because they build the habit of setting up equations neatly, and that habit is worth many marks across the paper.
Basic Terms
- Ratio: the comparison of two quantities of the same kind by division.
- Antecedent: the first term of a ratio, written before the colon.
- Consequent: the second term of a ratio, written after the colon.
- Proportion: the equality of two ratios.
- Continued proportion: three quantities in which the ratio of the first to the second equals the ratio of the second to the third.
- Mean proportional: the middle term of a continued proportion.
A ratio has no unit because both quantities must be of the same kind and in the same unit. If the two quantities are not in the same unit, they must first be converted so that the units agree, and only then may the ratio be written.
Writing a Ratio and Reducing It
A ratio is written in the form a is to b and may also be read as the fraction a over b. To reduce a ratio to its lowest terms, divide both terms by their highest common factor. The ratio of twelve to eighteen reduces to the ratio of two to three after dividing both by six. A ratio may also be scaled up by multiplying both terms by the same number without changing its value, and this property is the main working tool of the chapter.
Types of Ratio
- Compound ratio: found by multiplying the corresponding terms of two or more ratios.
- Inverse or reciprocal ratio: obtained by swapping the antecedent and the consequent.
- Duplicate ratio: the ratio of the squares of two quantities.
- Sub-duplicate ratio: the ratio of their square roots.
- Triplicate ratio: the ratio of their cubes.
- Sub-triplicate ratio: the ratio of their cube roots.
Dividing a Quantity in a Given Ratio
To divide a quantity in a given ratio, first add the terms of the ratio to obtain the number of parts. The value of one part is the total quantity divided by the number of parts. Then multiply the value of one part by each term in turn to obtain the shares. If two thousand rupees is to be divided in the ratio of two is to three, the number of parts is five, the value of one part is four hundred rupees, and the two shares are eight hundred rupees and one thousand two hundred rupees.
Proportion and Its Properties
A proportion is written as a is to b equals c is to d, and here the outer terms are called the extremes while the inner terms are called the means. The fundamental property is that the product of the means equals the product of the extremes, so b multiplied by c equals a multiplied by d. This single relation solves most proportion problems. The first and fourth terms are the extremes, the second and third are the means, and in a continued proportion the middle term is the mean proportional between the other two.
Direct and Inverse Proportion
Two quantities are in direct proportion when they increase or decrease together, so that doubling one doubles the other. They are in inverse proportion when an increase in one causes a proportional decrease in the other, so that the product of the two remains constant. Wages and the number of days worked are directly proportional, while the number of workers and the time taken to finish a job are inversely proportional. Identifying which of the two applies is the first and most important step in word problems on this chapter.
Ratio in Partnership and Mixture Problems
In a partnership the profit is divided in the ratio of the capital contributed, and when the capitals are invested for different periods the profit is divided in the ratio of capital multiplied by time. In mixture and alligation the quantities of two ingredients are in the inverse ratio of the differences of their values from the mean value. These two standard applications show how the ratio chapter quietly supports several other chapters, and they are the reason a weak grasp of ratios costs marks in unexpected places.
Shortcut Methods for Rapid Calculation
A few shortcuts make the questions quick. To compare two ratios, cross-multiply and compare the products. To find the number of parts, add the terms of the ratio and use that single figure throughout. When the same quantity is added to both terms of a ratio, write the new ratio as a fraction and set it equal to the given new value; the equation is short and needs no guessing. In problems of the form a is to b equals c is to d, always write the product of the means equal to the product of the extremes first, because that single step turns the word problem into simple arithmetic.
How This Chapter Appears in the Examination
Ratio and proportion questions in ADRE Grade IV, APSC and Assam Police papers usually ask the candidate to simplify a ratio, to divide an amount in a given ratio, to find the mean proportional or a missing term in a proportion, or to apply direct and inverse proportion to a word problem. Partnership, mixture and alligation, and several questions on time and work are built on these same ideas. Because the equations are short and the arithmetic is simple, a steady practice routine makes this one of the surest scoring sections.
Common Mistakes to Avoid
- Writing a ratio between quantities that are in different units without converting them first.
- Adding the same number to both terms of a ratio, which changes its value.
- Confusing direct proportion with inverse proportion in word problems.
- Forgetting the mean proportional while solving a continued proportion.
- Leaving the final ratio in a form that is not in its lowest terms.
How to Prepare This Topic
- Learn the meaning of antecedent, consequent, extremes and means.
- Practise reducing ratios and scaling them up and down.
- Master dividing a quantity in a given ratio with different numbers of parts.
- Work on proportions using the product of the means equals the product of the extremes.
- Finish with mixed word problems on direct and inverse proportion.
Frequently Asked Questions
Does a ratio have a unit?
No. A ratio compares two quantities of the same kind, so the units cancel out and the ratio is a pure number. If the two quantities are in different units, they must be converted to the same unit before the ratio is written.
What is the difference between a ratio and a proportion?
A ratio compares two quantities by division, while a proportion states that two ratios are equal. A ratio is a single comparison; a proportion is an equation between two such comparisons.
How do I divide a sum in a given ratio?
Add the terms of the ratio to get the number of parts, divide the total by this number to find the value of one part, and then multiply that value by each term. For the ratio two to three on two thousand rupees, one part is four hundred rupees and the shares are eight hundred and one thousand two hundred rupees.
How do I identify inverse proportion in a word problem?
Look for a fixed total. If the product of the two quantities stays constant as one changes, the relation is inverse. The number of workers and the days taken, or the speed and the time for a fixed distance, are the standard examples.
What is the mean proportional?
In a continued proportion of the form a is to b equals b is to c, the middle term b is the mean proportional between a and c, and its square equals the product of a and c. The mean proportional is always the square root of that product.