Ratio and Proportion for the Assam Police Constable Exam
Ratio and proportion is the chapter that turns comparison into a calculation. In the SLPRB Assam Police Constable (AB & UB) paper it appears again and again, sometimes directly as a question on sharing an amount, and sometimes hidden inside questions on partnership, mixture, ages, maps and scale drawings. The reason is simple: the examination frequently describes a situation as a relation between two quantities, and a candidate who can translate that relation into a ratio can find every unknown in the problem. Ratio is a way of comparing two quantities of the same kind by division, and proportion is a statement that two ratios are equal. Once these two ideas are secure, questions that look complicated reduce to writing a proportion and solving one equation. The treatment here follows the Master Book: Maths Masti (D. Hurina), keeping the unit and the order of the terms always in view, because the commonest mistakes in this chapter come from reversing the order or from comparing quantities that are not in the same unit.
Why Ratio and Proportion Fit the Assam Police Paper So Well
The Assam Police written test favours questions that can be settled with a small amount of clean reasoning, and ratio and proportion are exactly that. The chapter also links many other topics together: percentage is a ratio with a denominator of one hundred, speed is a ratio of distance to time, and a partnership share is a ratio of money and time. Because of this the topic is what the examiners call high-yield, meaning a small investment of study returns marks in several sections at once.
Terms You Should Know
- Ratio: a comparison of two quantities of the same kind by division, written with a colon.
- Antecedent and consequent: the first and the second terms of a ratio.
- Proportion: an equality of two ratios.
- Extremes and means: the first and last terms and the two middle terms of a proportion.
- Continued proportion: a set of three or more quantities in which each ratio equals the next.
- Direct proportion: a relation in which an increase in one quantity increases the other.
- Inverse proportion: a relation in which an increase in one quantity decreases the other.
A ratio has no unit, but the two quantities being compared must be in the same unit before the ratio is written. Converting both to a common unit is therefore the first action in almost every question.
Writing a Ratio and Reducing It to the Lowest Terms
To write a ratio, express both quantities in the same unit and then divide each by their highest common factor. The ratio of forty-five minutes to one and a half hours is written by first changing the hours into ninety minutes, giving the ratio forty-five to ninety, which reduces to one to two. Ratios should be written in whole numbers whenever possible, and a ratio of fractions can be cleared by multiplying all the terms by the lowest common multiple of the denominators.
Compound Ratio and the Different Types of Ratio
The compound ratio of two ratios is the ratio of the products of their corresponding terms. A ratio is a ratio of greater inequality if the first term is larger, of lesser inequality if it is smaller, and a ratio of equality if the terms are equal. Two ratios can be compared by cross-multiplication: the ratio with the larger cross product is the larger ratio. These small facts settle many objective questions on which ratio is greater.
Dividing a Quantity in a Given Ratio
To divide a quantity in a given ratio, add the terms of the ratio to obtain the total number of parts, divide the quantity by this total to find the value of one part, and multiply by each term. If an amount is to be divided in the ratio of two to three, there are five equal parts, and the two shares are two-fifths and three-fifths of the amount. This single method answers a large family of questions on profit sharing, inheritance and the division of a sum among workers.
Proportion and the Product of Extremes and Means
In a proportion, the product of the extremes equals the product of the means. This one rule gives the missing term immediately: if three numbers of a proportion are known, the fourth is found by multiplying the two known terms that are diagonally opposite and dividing by the remaining term. A proportion can also be formed by inverting both ratios, or by alternating the terms, without changing its truth.
Continued Proportion and Mean Proportional
When three quantities are in continued proportion, the square of the middle term equals the product of the first and the third, so the middle term is the square root of that product and is called the mean proportional. The third term, found from the first and the middle, is called the third proportional. These two terms appear often in questions on geometric growth and on similar figures.
Direct and Inverse Proportion
Two quantities are in direct proportion when their ratio stays constant, so that doubling one doubles the other, and the cross product of the two pairs is equal. They are in inverse proportion when their product stays constant, so that doubling one halves the other. Deciding which relation the question describes is the key step: more workers finish a job in less time, so men and days are in inverse proportion, while more goods cost more money, so quantity and cost are in direct proportion.
Ratio in Partnership, Mixture and Age Problems
In partnership the profit is divided in the ratio of the capital multiplied by the period for which it was invested. In mixture questions the ratio of two ingredients is adjusted by finding the total and the parts, and alligation is used when two mixtures of different strengths are combined. In age problems the ratio between ages changes with time, so a common approach is to take the present ages as a multiple of the ratio and add the same number of years to both.
Shortcut Techniques for the Exam Hall
Reduce every ratio to its lowest terms before doing anything else. When two ratios must be combined, make the common term equal by multiplying. For a proportion, use the product of extremes equals the product of means instead of cross-multiplying from scratch. For a partnership, multiply capital by time first and then form the ratio. Keep the unknown as a single symbol multiplied by the ratio, which usually turns the question into a simple linear equation.
Common Traps and Errors
- Writing a ratio without first converting the two quantities to the same unit.
- Reversing the order of the terms in a ratio or a proportion.
- Forgetting to add all the terms of a ratio before dividing a quantity.
- Treating an inverse proportion as a direct proportion.
- In partnership questions, ignoring the time for which the capital was invested.
- Reducing a ratio of fractions without clearing the denominators first.
A Study Plan for This Chapter
- Learn to write and reduce a ratio, always converting to a common unit first.
- Memorise the rule that the product of the extremes equals the product of the means.
- Practise dividing a quantity in a given ratio and finding a missing proportion term.
- Work through direct and inverse proportion, then partnership and mixture.
- Finish with age and mixture problems and revise the direction of the comparison each week.
Frequently Asked Questions
Does a ratio have a unit?
No. A ratio compares two quantities of the same kind, so the units cancel and the ratio is a pure number. The two quantities must, however, be in the same unit before the ratio is written.
How do I find the missing term in a proportion?
Use the rule that the product of the extremes equals the product of the means. Multiply the two terms that are diagonally opposite the unknown and divide by the remaining known term.
What is the difference between direct and inverse proportion?
In direct proportion the ratio of the two quantities is constant, so both increase together. In inverse proportion the product is constant, so when one increases the other decreases.
How is profit shared in a partnership?
The profit is shared in the ratio of each partner's capital multiplied by the time for which it was invested. Money and time must both be taken into account, because a small capital kept for a long time can equal a large capital kept for a short time.
How is alligation used in mixture problems?
Alligation finds the ratio in which two mixtures of different prices or strengths must be combined to obtain a mixture of a given middle value. The required quantities are in the inverse ratio of the differences between each value and the mean value.