Ratio and Proportion - Shortcuts & Tricks for Placement Tests, Job Interviews & Exams

Name: Ratio and Proportion - Shortcuts & Tricks for Placement Tests, Job Interviews & Exams
Uploaded: 2017-05-19T05:52:59.000Z
Duration: 1 h 56 min 31 s

Understanding Ratio and Proportion

Introduction to Ratios

The video introduces the concept of ratio and proportion, emphasizing its importance in quantitative aptitude for various exams including placement tests, job interviews, and banking.

It highlights that ratios are fundamental comparisons between two or more quantities.

Definition of Ratio

A ratio compares two quantities; for example, if person P is five times faster than Q, their speeds can be expressed as a ratio.

The speed of P (50 km/h) compared to Q (10 km/h) results in a simplified ratio of 5:1.

Ratios can also be expressed as fractions; thus, a ratio of A:B can be written as A/B.

Understanding Proportion

Proportion refers to the equality of two ratios. For instance, if A:B = C:D, this relationship is termed proportion.

The notation used for proportions involves four dots indicating the relationship between the ratios.

Tips for Solving Ratio Problems

Cross Multiplication Method

When comparing two ratios (A/B and C/D), if they are proportional, cross-multiplying gives AD = BC.

Determining Greater Ratios

To compare which fraction is greater without assuming equality: multiply A by D and B by C. If AD > BC, then A/B > C/D.

Component Rule (Compono)

If two ratios are equal (A/B = C/D), adding their denominators yields another valid equation: (A+B)/B = (C+D)/D.

Division Rule (Dividendo)

Similarly, subtracting the denominators leads to another valid equation: (A-B)/B = (C-D)/D.

Understanding Compono Dividendo and Proportions

Compono Dividendo Rule

The rule states that if A + B/A - B = C + D/C - D , then it can be rewritten as A/B = C/D . This is known as the compono dividendo rule for ratio and proportion.

It is crucial to remember this rule, especially for exams where values of A, B, C, and D may be provided to solve problems involving these ratios.

Inversion of Ratios

If A/B = C/D , then the inverse is also true: B/A = D/C . This concept is referred to as invertendo.

Understanding this inversion helps in manipulating ratios effectively without needing to memorize specific names.

Proportional Relationships

When dealing with three numbers in a ratio (e.g., A : B : C), A and C are termed extremes while B is the mean proportional. This structure holds true even when extending to four ratios (e.g., A : B : C : D).

The relationship can be expressed mathematically as A : B :: C : D or A/B = C/D. This notation emphasizes the equality of proportions across different sets of numbers.

Finding Actual Values from Ratios

Ratios like 5:6 represent simplified values; actual values can be derived by assuming a common factor K (e.g., 5K and 6K). For example, if actual values are 40 and 48, dividing by their GCD gives the simplified ratio of 5:6.

In problem-solving scenarios where the common factor isn't known, using K allows for flexibility in deriving actual values from given ratios. This method will become clearer through practice with sums later on.

Comparing Ratios

Determining Greater Ratios

To compare two ratios such as 17:18 and 10:11, multiply crosswise (17 * 11 vs. 18 * 10) to determine which ratio is greater; here, 17/18 proves larger than 10/11.

For multiple ratios (e.g., three or four), apply similar cross-multiplication techniques iteratively until identifying the greatest or smallest among them based on comparisons made at each step.

Finding Proportional Relationships

Third Proportional Calculation

To find the third proportional related to two numbers (e.g., 18 and 54), use the relationship defined by proportions: if A:B::B:C, then calculate using squares (B^2 = AC). Here, substituting gives us that C equals 162 when calculated correctly from given values.

Finding Proportions and Ratios in Mathematics

Mean Proportional Calculation

The value of B can be derived from the equation A/B = B/C , leading to B^2 = AC .

To find the fourth proportional of 19, 13, and 153, we set up the proportion A : B :: C : D .

The formula for finding D is given by D = BC/A . Substituting values gives D = 13 times 153/19 = 221.

Finding Mean Proportional Between Two Numbers

To find the mean proportional between 7 and 63, we need a third number (B) such that A : B :: B : C .

This leads to the equation b^2 = AC, where substituting gives us b^2 = 7 times 63.

Solving yields that the mean proportional is found to be 21.

Solving Ratios with Multiple Terms

Given ratios like 10/13 = 11/28 = K, one can simply add numerators and denominators.

Adding numerators: 10 + 11 + K + 12; adding denominators: 13 + 28 + K + K.

This results in a straightforward calculation for K as it simplifies down to a single fraction.

Income and Spending Ratios

Ramesh's income ratio compared to Sur's is given as 5:6, while their spending ratio is noted as 7:9.

Actual incomes are expressed using a common factor (K): Ramesh earns 5K, Sur earns 6K.

Spending calculations involve subtracting savings from income; thus, spending equals income minus savings.

Equating Spending Ratios

The spending ratio translates into an equation involving both individuals' incomes and savings.

By solving this equation, we derive values for both Ramesh’s and Sur’s actual incomes based on their respective ratios.

Combining Ratios into One Expression

When combining ratios like A:B (3:7) with B:C (9:5), it's essential to equalize the middle term (B).

How to Divide Numbers in Ratios

Understanding Ratios and Division

The speaker explains how to divide 3395 into the ratio of 42:32:23 by making the values of B common through multiplication.

Two methods are introduced for solving this problem, referencing a previous partnership video that covers profit sharing among three individuals with shares of 42, 32, and 23.

Method One: Partnership Approach

The first method involves calculating each part using the formula: (part/total parts) * total amount. For example, for P's share: 42/97 times 3395 = 1470 .

The second part is calculated similarly for Q's share as 32/97 times 3395 = 1120 , leading to a straightforward calculation.

Method Two: Common Factor Approach

In the second method, a common factor K is assumed. Thus, actual values become 42K, 32K, and 23K. Adding these gives 97K = 3395, leading to K = 35.

Each part can then be calculated as follows:

First part (A): 42K = 1470

Second part (B): 32K = 1120

Third part (C): Calculated by subtracting A and B from total.

Finding Unknown Numbers Using Ratios

Problem Setup

The next question involves finding three numbers where their sum equals 285. The ratios between them are given as:

First and second numbers: 3:7

Second and third numbers: 6:5

Solving the Ratios

To solve this, both ratios need to be made comparable by adjusting their values so that they align correctly.

After adjustments, the new ratios become 18:42:35, allowing for easier calculations.

Final Calculation

With all parts aligned, calculate the third number using proportions based on their sum equaling 285. This leads to determining that the third number is 105.

Determining Smaller Number from Given Conditions

Setting Up Equations

In another problem involving two numbers in a ratio of 3:8, adding five to both changes their ratio to 2:5.

Finding Actual Values

Letting actual numbers be represented as 3K and 8K, we set up an equation after adding five which results in a solvable format.

Conclusion on Smaller Number

Solving yields K's value; thus, smaller number becomes 45 when substituting back into original expressions.

Complex Ratio Relationships

Multiple Ratios Involved

A complex scenario arises where multiple ratios are provided:

A:B = 2:3

B:C = 7:9

C:D = 5:9

Addressing Different Values of B

How to Solve Complex Ratios?

Understanding the Problem

The discussion begins with the complexity of solving ratios when multiple values are involved, specifically mentioning two different values for B and C.

The speaker introduces a trick to solve these types of problems by first establishing the ratios: A is to B as 2 is to 3, B is to C as 7 is to 9, and C is to D as 5 is to 7.

Calculating Values

To find the overall ratio of A:B:C:D, the first value (A) can be calculated by multiplying the numerators: 2 times 7 times 5 = 70.

The last value (D) is found by multiplying the denominators: 3 times 9 times 7 = 189.

Finding Middle Values

The second value (B) can be determined by moving from A through B and directly connecting with D: 3 times 7 times 5 = 105.

For the third value (C), start from B again but jump across: 3 times 9 times 5 = 135.

Visualizing Ratios

The speaker emphasizes remembering a diagram that resembles a kite or rhombus for easier recall of how to calculate these ratios.

This diagram helps visualize connections between terms in a systematic way, aiding in memorization and application during problem-solving.

General Formula for Ratios

Establishing General Terms

The general formula for calculating ratios involves identifying small letters representing each term. If A:B::C:D::E:F, then:

First Value = A * C * E

Last Value = B * D * F

Simplifying Calculation Steps

To derive middle values, take combinations of terms such as BCE for one middle term and BDE for another.

Example Problem on Article Purchase

Problem Setup

An example problem presents Sur purchasing articles P, Q, and R at prices of ₹300, ₹180, and ₹120 respectively while spending a total of ₹6480 in a ratio of quantities as P:Q:R = 3:2:3.

Solving for Quantities

Let K represent a common factor; thus:

Articles bought are represented as 3K, 2K, and 3K.

Total expenditure equations are set up based on individual article costs leading back to total spent amount.

Final Calculation

After setting up equations based on expenditures per type:

Understanding Ratios and Expenditures

Ajay and Raj's Money Problem

Ajay and Raj have a combined total of 1050 rupees. If Ajay gives 150 rupees to Raj, they will have equal amounts.

Let Ajay's initial amount be A and Raj's be R. The equations formed are A + R = 1050 and A - 150 = R.

Solving these equations yields A = 600 rupees and R = 450 rupees, leading to a ratio of Ajay to Raj as 4:3.

Simplifying Ratios with Variables

For the ratio x:y = 3:4, the expression for (7x + 3y):(7x - 3y) is derived using substitution.

By substituting x = 3k and y = 4k into the expression, it simplifies to (21 + 12):(21 - 12), resulting in an answer of 11/3 .

Component Dividend Method

An alternative method involves manipulating the ratios into a component dividend format for easier calculation.

This method confirms that (7x + 3y)/(7x - 3y) equals 33/9 , which also simplifies to 11/3 .

Finding AC:BD Ratio

Given two ratios a:b = 5:7 and c:d = 2a:3b, multiplying these provides the ratio AC:BD as 50/147 .

Solving for Three Numbers in Ratio

The three numbers are represented as k/2, k2/3, and k3/4 based on their ratios of x:y:z being proportional to their respective coefficients.

The difference between the largest (k75/4 or k108 when calculated correctly) and smallest number is given as 36.

Solving this leads to finding actual values for each number; thus, they are determined as 72,96,108.

Wheat vs Patty Expenditure Ratio

The market price ratio of wheat to patty is established at a ratio of 2:3.

Understanding Expenditure Ratios

Calculating Expenditure for Wheat and Pad

The actual quantity of wheat consumed is represented as 5a and for pad as 4a. The expenditure on wheat is calculated by multiplying the price per unit (2K) with the quantity consumed, leading to a formula: [Expenditure_wheat = 2K times 5a] .

Similarly, the expenditure for pad is calculated using its rate (3K) multiplied by the quantity consumed: [Expenditure_pad = 3K times 4a]. To find the ratio of expenditures, we divide these two expressions .

After simplification, we find that the ratio of expenditure on wheat to pad simplifies to 10/12, which reduces to 5/6. This indicates that for every 5 units spent on wheat, 6 units are spent on pad .

An alternative method involves expressing expenditures in terms of their rates and quantities. The ratio can also be derived from multiplying the ratios of rates and quantities separately, confirming that both methods yield consistent results .

Dividing Shares Among A, B, C, D

Understanding Share Ratios

The total amount of Rupees 8,400 is divided among A, B, C, and D based on given ratios: A:B (2:3), B:C (4:5), and C:D (6:7). These ratios need to be combined into a single ratio for all four individuals .

To derive A:B:C:D from individual ratios requires equating them through common multipliers. For instance:

From A:B = 2x:3x

From B:C = 4y:5y

From C:D = 6z:7z

By finding a common factor across these equations we can express all shares in terms of one variable .

Calculation Steps

Using cross-multiplication techniques helps establish values for each share. For example:

Calculate intermediate products like 8 times 6 = 48 and others to derive final values.

Ultimately leads us to determine each individual's share based on their respective contributions .

Finding Number of Story Books Bought

Initial Conditions

Initially stated that the ratio of story books to non-story books was 4:3, with a total number of story books being 1,248. This allows us to calculate non-story books using proportions .

Adjusting Ratios After Purchase

When additional story books are bought (M), it’s important to note that non-story book numbers remain constant at 936. Thus new total becomes 1248 + M while maintaining a new ratio of story books at 5/3 against non-story books .

Solving for M

Setting up an equation based on this new ratio allows us to solve for M:

[3(1248 + M) = 5(936)]

This leads directly towards determining how many additional story books were purchased [].

Conclusion on Ratio Concepts

Key Takeaways

Mastery over basic concepts in ratios significantly simplifies problem-solving in quantitative aptitude.

Utilizing simple assumptions or factors can streamline calculations related to various topics including financial distributions or inventory management.