Understanding Ratios: The Logical Foundation
In competitive examinations like the UPSC CSAT, SSC CGL, and CGPSC, questions on ratios rarely demand pure algebraic calculation. Instead, they test your logical understanding of relative values. A ratio, expressed as a:b or a/b, does not represent absolute quantities; rather, it indicates the relative strength of the quantities. For instance, if the ratio of savings of two individuals is 3:4, their actual savings could be ₹30 and ₹40, or ₹3,000 and ₹4,000. Here, the constant multiplier (often denoted as k or x) is the factor that scales the ratio to its real-world value.
One of the most frequent operations required in exam questions is the compounding of ratios. If you are given the ratio of $A:B$ and $B:C$, you cannot directly compare $A$ and $C$ until you establish a common platform for $B$. This process is known as balancing the ratio. To combine $A:B = 2:3$ and $B:C = 4:5$, you must make the term representing $B$ equal in both ratios by finding the Least Common Multiple (LCM) of 3 and 4, which is 12. Multiplying the first ratio by 4 and the second by 3 yields the combined ratio $A:B:C = 8:12:15$.
The Unitary Method Concept: Instead of using complex algebraic variables like $x$ and $y$, think of ratios as “parts” or “units”. If a sum of ₹150 is divided between $P$ and $Q$ in the ratio $2:3$, the total units are $2 + 3 = 5$ units. Therefore, 5 units correspond to ₹150, meaning 1 unit equals ₹30. Consequently, $P$ gets 2 units (₹60) and $Q$ gets 3 units (₹90). This logical approach eliminates calculation errors and saves crucial time during the exam.
Proportion and Continued Proportion
When two ratios are equal, they are said to be in proportion. Mathematically, if $a:b = c:d$, then the four quantities $a$, $b$, $c$, and $d$ are in proportion. This is represented as $a:b :: c:d$. In this expression, $a$ and $d$ are called the extremes, while $b$ and $c$ are called the means. A fundamental rule of proportion is that the product of the extremes is always equal to the product of the means ($a \times d = b \times c$). This rule is highly useful for finding missing terms in data interpretation tables and logical puzzles.
When three quantities $a$, $b$, and $c$ are such that $a:b = b:c$, they are said to be in continued proportion. In this scenario, $b$ is known as the mean proportional between $a$ and $c$, while $c$ is referred to as the third proportional to $a$ and $b$. Solving these relations yields the following standard formulas:
- Mean Proportional: $b = \sqrt{a \times c}$
- Third Proportional: $c = \frac{b^2}{a}$
- Fourth Proportional: In $a:b = c:d$, the fourth proportional is $d = \frac{b \times c}{a}$
Distribution of Quantities and Partnership Ratios
An essential application of ratio and proportion is the distribution of resources, profits, or assets. In Partnership problems, which are a staple of the SSC and state PCS exams, profits are distributed based on two factors: the amount of capital invested and the time period of the investment. The logical formula governing this distribution is:
Profit Ratio = (Capital_1 × Time_1) : (Capital_2 × Time_2)
Another classic application is the Coin-based logical puzzle. Candidates are often given a total monetary value and the ratio of different coin denominations (e.g., 1-rupee, 50-paise, and 25-paise coins). To solve these, you must convert the ratio of coin counts into a ratio of monetary values by multiplying the count of each coin by its respective face value. This prevents the common mistake of equating the physical count of coins directly to the total value of the money.
Important Formulas & Relations
| Concept / Term | Mathematical Representation | Exam Application / Shortcut |
|---|---|---|
| Compounded Ratio | Compounded ratio of $(a:b)$ and $(c:d)$ is $ac:bd$ | Used in successive percentage changes and population growth. |
| Duplicate & Triplicate Ratios | Duplicate: $a^2:b^2$ Triplicate: $a^3 Share: You May Also Like
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