Ratio and Proportion: Solving Comparison and Distribution Problems

1. The Building Blocks: Ratio and Proportion

At its core, a ratio is simply a way to compare two quantities. A proportion is a statement that two ratios are equal. These simple ideas are the foundation for a vast range of quantitative aptitude questions.

This infographic provides a concise overview of the key concepts.

A crucial rule for proportions is the relationship between the terms. When two ratios are in proportion, say $a:b=c:d$, we can write this as $\frac{a}{b}=\frac{c}{d}$. If you cross-multiply, you get $a\times d=b\times c$.

The terms $a$ and $d$ are called the extremes (the outer terms), and $b$ and $c$ are called the means (the inner terms).

This principle is not just a theoretical rule; it's a practical tool for solving problems where one term is unknown. Let's see how Gagan Pratap Sir applies this to a typical SSC exam question.

2. Core Techniques for Comparison and Distribution

Many SSC CGL problems don't just give you a single ratio; they give you several related ratios that you must first combine.

Combining Ratios

A very common task is to find a combined ratio like $A:B:C$ when you are given $A:B$ and $B:C$. The key is to make the common term (in this case, B) equal in both ratios.

Let's watch Gagan Pratap Sir demonstrate the standard method for this.

Distribution Problems

Distribution problems involve dividing a total amount (money, items, etc.) among individuals according to a specific ratio. The trick is to think of the ratio parts as "shares" or "units".

Example: Divide ₹5625 among A, B, and C such that A receives 1/2 as much as B and C together, and B receives 1/4 as much as A and C together. (This is Question 4 from the Testbook resource).

Here's how to approach this:

1. From the first condition, $A=\frac{1}{2}(B+C)$, which means $\frac{A}{B+C}=\frac{1}{2}$. The ratio of A's share to the combined share of B and C is $1:2$. This implies that out of a total of $1+2=3$ units, A gets 1 unit. So, $A=\frac{1}{3}$ of the total.

   • A's share = $\frac{1}{3}\times 5625=\text{Rs. }1875$.

2. From the second condition, $B=\frac{1}{4}(A+C)$, which means $\frac{B}{A+C}=\frac{1}{4}$. The ratio of B's share to the combined share of A and C is $1:4$. This implies that out of a total of $1+4=5$ units, B gets 1 unit. So, $B=\frac{1}{5}$ of the total.

   • B's share = $\frac{1}{5}\times 5625=\text{Rs. }1125$.

The question might then ask for C's share, or the difference between shares, which you can now easily calculate.

Income, Expenditure, and Savings Problems

This is a classic and very important application of ratios for your exam. The basic relationship is:
$\text{Income}-\text{Expenditure}=\text{Savings}$
You'll typically be given the ratio of incomes for two people (e.g., $I_A:I_B$) and the ratio of their expenditures ($E_A:E_B$). You cannot simply subtract the expenditure ratio from the income ratio because they are on different scales. You must first link them using a piece of information given in the problem.

This is a critical technique, and Gagan Pratap Sir provides an outstanding explanation.

3. The SSC CGL Edge: Solving with "Ultracalc"

As you requested, let's focus on the tricks that save precious seconds in the exam. Abhinay Sharma's "Ultracalc" approach is all about using the options and number properties to find the answer without heavy calculation.

The core idea is simple: quantities like money, people, or items must be integers. If a total is divided in a certain ratio, the final numbers must be multiples of the parts of that ratio.

This way of thinking is a game-changer for speed. Always glance at the options and check if a simple ratio multiple can give you the answer before diving into full calculations.

4. Let's Practice: Applying the Concepts

Let's look at some solved problems from a dedicated SSC CGL resource to see these methods in action.

Conclusion

Today, we've built a solid foundation in ratio and proportion, moving from basic definitions to advanced, exam-oriented strategies. You are now equipped with a versatile toolkit to handle comparison and distribution problems effectively.

Key Takeaways:

• Proportion Rule: The product of the means equals the product of the extremes ($a\times d=b\times c$).
• Combining Ratios: Always equalize the common term before creating a continuous ratio ($A:B:C$).
• Distribution: Treat ratio parts as "units" or "shares" to divide a total quantity.
• Income-Expenditure: Link the separate income and expenditure ratios using a given condition. A powerful method is to make the relevant terms equal first, then apply the required ratio.
• Speed Tricks: Always check the options. Look for multiples of the ratio parts or the sum of the ratio. This can often lead you to the answer in seconds.

In our next lesson, we will continue exploring this topic by focusing on direct and inverse proportion. You'll see how these specific relationships are used to solve a wide array of word problems, particularly those involving time and work, and speed, distance, and time.