Mixture and Alligation

Mixture and Alligation is a crucial topic in quantitative aptitude, testing your ability to solve problems involving the mixing of different quantities of substances or items to achieve a desired proportion or price. It's fundamentally about ratios and proportions applied to practical scenarios.

What is Mixture?

A mixture is formed when two or more than two ingredients (or quantities) are mixed together. These ingredients can be of different types, such as liquids (milk, water, acid) or solids (rice, wheat, pulses), or even different qualities of the same item.

What is Alligation?

Alligation is a method used to find the ratio in which two or more ingredients of different prices/qualities are mixed to produce a mixture of a desired mean price/quality. It's a rule derived from the rule of proportion.

The Rule of Alligation

When two ingredients of quantities Q1 and Q2 and prices P1 and P2 respectively are mixed to obtain a mixture with mean price Pm, the ratio of their quantities is inversely proportional to the difference of their prices from the mean price.

Mathematically:

(Q1 / Q2) = (P2 - Pm) / (Pm - P1)

This can be visualized using a diagram:

Ingredient 1 (Quantity Q1, Price P1) ------> (P2 - Pm)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               &l Ingredient 2 (Quantity Q2, Price P2) ------> (Pm - P1)

The ratio of quantities Q1 : Q2 is given by (P2 - Pm) : (Pm - P1).

Key Concepts and Formulas

1. Ratio of Quantities: The ratio in which two ingredients are mixed is inversely proportional to the difference of their prices from the mean price.
2. Mean Price: The average price of the mixture. It is calculated as:

   Mean Price (Pm) = (Total Cost of Ingredients) / (Total Quantity of Mixture)

   Pm = (Q1 * P1 + Q2 * P2) / (Q1 + Q2)

3. Problems involving Milk and Water: These often involve removing a certain quantity of the mixture and replacing it with another ingredient. The formula for the remaining quantity of the original ingredient is:

   Final Quantity of Original Ingredient = Initial Quantity * (1 - (Quantity Removed / Initial Quantity))ⁿ

   Where 'n' is the number of times the operation (removal and replacement) is performed.

Types of Problems

1. Type 1: Finding the ratio of mixing. Given the prices of two ingredients and the mean price of the mixture, find the ratio in which they are mixed.
2. Type 2: Finding the mean price. Given the prices of two ingredients and the ratio in which they are mixed, find the mean price of the mixture.
3. Type 3: Finding the quantity of one ingredient. Given the price of one ingredient, the ratio of mixing, and the mean price, find the quantity of the other ingredient.
4. Type 4: Problems involving removal and replacement. A certain quantity of a mixture is removed and replaced by another ingredient. These are common and require careful application of the formula.

Example 1 (Type 1)

In what ratio should rice costing Rs. 10 per kg be mixed with rice costing Rs. 15 per kg so that the mixture costs Rs. 12 per kg?

Here, P1 = 10, P2 = 15, Pm = 12.

Ratio Q1 : Q2 = (P2 - Pm) : (Pm - P1)

Ratio Q1 : Q2 = (15 - 12) : (12 - 10)

Ratio Q1 : Q2 = 3 : 2

So, the rice should be mixed in the ratio 3:2.

Example 2 (Type 4)

A vessel contains 100 litres of pure milk. 10 litres of milk is drawn out and replaced by water. Then, 10 litres of the mixture is drawn out and replaced by water. Find the quantity of milk left in the final mixture.

Initial Quantity of Milk = 100 litres.

First operation: 10 litres removed, replaced by water.

Quantity of milk left = 100 * (1 - 10/100) = 100 * (90/100) = 90 litres.

Second operation: 10 litres of mixture removed, replaced by water.

Quantity of milk left = 90 * (1 - 10/100) = 90 * (90/100) = 81 litres.

Alternatively, using the formula directly for two operations (n=2):

Final Quantity of Milk = 100 * (1 - 10/100)² = 100 * (90/100)² = 100 * (0.9)² = 100 * 0.81 = 81 litres.

Mensuration

Mensuration deals with the study of geometrical shapes, their areas, volumes, and other related properties. It's a vital part of geometry and is frequently tested in competitive exams.

Basic Concepts

Mensuration involves calculating measurements for 2D (plane figures) and 3D (solid figures) shapes.

2D Shapes (Plane Figures)

These figures lie on a flat surface. Key measurements are Perimeter (or Circumference) and Area.

1. Square

• Definition: A quadrilateral with all four sides equal and all four angles right angles (90 degrees).
• Side length: Let the side be 'a'.
• Perimeter: P = 4a
• Area: A = a²
• Diagonal: d = a√2

2. Rectangle

• Definition: A quadrilateral with opposite sides equal and parallel, and all four angles right angles.
• Length: l, Width: b
• Perimeter: P = 2(l + b)
• Area: A = l * b
• Diagonal: d = √(l² + b²)

3. Triangle

• Definition: A polygon with three sides.
• Types: Equilateral, Isosceles, Scalene, Right-angled, Acute-angled, Obtuse-angled.
• Area (General): A = (1/2) * base * height
• Area (Heron's Formula): If sides are a, b, c, and semi-perimeter s = (a+b+c)/2, then A = √[s(s-a)(s-b)(s-c)]
• Area (Equilateral Triangle): A = (√3 / 4) * a², where 'a' is the side.
• Area (Isosceles Triangle): If equal sides are 'a' and base is 'b', A = (b/4) * √(4a² - b²)

4. Circle

• Definition: A set of points equidistant from a central point.
• Radius: r, Diameter: d = 2r
• Circumference (Perimeter): C = 2πr = πd
• Area: A = πr²
• Area of a Semicircle: A = (1/2)πr²
• Perimeter of a Semicircle: P = πr + 2r = r(π + 2)

5. Parallelogram

• Definition: A quadrilateral with opposite sides parallel and equal.
• Let base be 'b' and height be 'h'.
• Area: A = b * h
• Perimeter: P = 2(a + b), where a and b are adjacent sides.

6. Rhombus

• Definition: A parallelogram with all four sides equal. Diagonals bisect each other at right angles.
• Let diagonals be d1 and d2.
• Area: A = (1/2) * d1 * d2
• Side: a = (1/2)√(d1² + d2²)
• Perimeter: P = 4a

7. Trapezium (Trapezoid)

• Definition: A quadrilateral with one pair of opposite sides parallel.
• Let parallel sides be 'a' and 'b', and height be 'h'.
• Area: A = (1/2) * (a + b) * h

3D Shapes (Solid Figures)

These figures occupy space. Key measurements are Surface Area (Total and Lateral/Curved) and Volume.

1. Cuboid

• Definition: A rectangular prism with six rectangular faces.
• Length: l, Width: b, Height: h
• Volume: V = l * b * h
• Total Surface Area (TSA): TSA = 2(lb + bh + hl)
• Lateral Surface Area (LSA): LSA = 2(lh + bh) = 2h(l + b)
• Length of Diagonal: d = √(l² + b² + h²)

2. Cube

• Definition: A cuboid with all six faces as squares (all edges equal).
• Let the edge length be 'a'.
• Volume: V = a³
• Total Surface Area (TSA): TSA = 6a²
• Lateral Surface Area (LSA): LSA = 4a²
• Length of Diagonal: d = a√3

3. Cylinder

• Definition: A solid with two parallel circular bases connected by a curved surface.
• Radius: r, Height: h
• Volume: V = πr²h
• Curved Surface Area (CSA): CSA = 2πrh
• Total Surface Area (TSA): TSA = CSA + Area of two bases = 2πrh + 2πr² = 2πr(h + r)

4. Cone

• Definition: A solid with a circular base and a vertex connected by a curved surface.
• Radius of base: r, Height: h, Slant height: l
• Note: l = √(r² + h²)
• Volume: V = (1/3)πr²h
• Curved Surface Area (CSA): CSA = πrl
• Total Surface Area (TSA): TSA = CSA + Area of base = πrl + πr² = πr(l + r)

5. Sphere

• Definition: A perfectly round geometrical object in three-dimensional space.
• Radius: r
• Volume: V = (4/3)πr³
• Surface Area (SA): SA = 4πr²

6. Hemisphere

• Definition: Half of a sphere.
• Radius: r
• Volume: V = (2/3)πr³
• Curved Surface Area (CSA): CSA = 2πr²
• Total Surface Area (TSA): TSA = CSA + Area of base = 2πr² + πr² = 3πr²

Combined Shapes

Problems often involve combining two or more basic shapes (e.g., a cylinder topped with a cone, a cube with a hemispherical depression). To solve these, calculate the volume/surface area of each component shape and then add or subtract them as required.

Example (Combined Shape)

A toy is in the form of a cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the total height of the toy is 15.5 cm. Find the total surface area of the toy.

Radius of hemisphere (r) = 3.5 cm.

Height of hemisphere = radius = 3.5 cm.

Total height of toy = 15.5 cm.

Height of cone (h) = Total height - Height of hemisphere = 15.5 - 3.5 = 12 cm.

Radius of cone (r) = Radius of hemisphere = 3.5 cm.

First, find the slant height (l) of the cone: l = √(r² + h²) = √(3.5² + 12²) = √(12.25 + 144) = √156.25 = 12.5 cm.

TSA of the toy = CSA of cone + CSA of hemisphere

CSA of cone = πrl = (22/7) * 3.5 * 12.5 = 11 * 12.5 = 137.5 cm².

CSA of hemisphere = 2πr² = 2 * (22/7) * (3.5)² = 2 * (22/7) * 12.25 = 2 * 22 * 1.75 = 77 cm².

Total Surface Area = 137.5 + 77 = 214.5 cm².

Data Interpretation (DI)

Data Interpretation is a critical section that assesses your ability to analyze and interpret data presented in various formats like tables, charts, and graphs. It requires logical reasoning and calculation skills.

Types of Data Presentation

1. Tables: Data organized in rows and columns.
2. Bar Graphs: Data represented by rectangular bars, where the length or height of the bar is proportional to the value it represents. Can be simple, multiple, or stacked.
3. Line Graphs: Data points plotted on a graph and connected by line segments. Excellent for showing trends over time.
4. Pie Charts: Data represented as sectors of a circle, where each sector's angle (and thus area) is proportional to the quantity it represents. Usually shows distribution or percentages.
5. Histograms: Similar to bar graphs but used for continuous data, with bars touching each other.
6. Tabulation: A more complex form of table, often involving multiple categories and sub-categories.

Key Skills Required

• Reading and Understanding Data: Accurately extracting information from the given source.
• Calculations: Performing basic arithmetic operations (addition, subtraction, multiplication, division) quickly and accurately.
• Percentage Calculations: Finding percentages, calculating percentage increase/decrease, comparing values using percentages.
• Ratio and Proportion: Calculating ratios between different data points, understanding proportional relationships.
• Averages: Calculating mean, weighted average.
• Approximation: Estimating values when exact calculations are time-consuming or unnecessary.
• Logical Reasoning: Identifying patterns, trends, and making logical deductions based on the data.

Common DI Questions

• What is the total value of X over Y years?
• What is the average value of X in a given period?
• What is the ratio of X to Y?
• By what percentage is X greater/less than Y?
• What is the percentage increase/decrease from X to Y?
• In which year/category was the value of X maximum/minimum?
• What is the difference between the maximum and minimum values?
• What is the sum of values for categories X and Y?
• What percentage of the total does category X represent?

Strategies for Solving DI Problems

1. Understand the Context: Read the title, labels, units, and any accompanying notes carefully.
2. Identify the Question Type: Is it asking for a total, average, ratio, percentage, difference, or comparison?
3. Locate Relevant Data: Pinpoint the exact numbers needed for the calculation.
4. Perform Calculations Accurately: Use shortcuts and approximations where appropriate, but ensure accuracy.
5. Check Your Answer: Does the answer make sense in the context of the data? Is it within a reasonable range?

Example: Table DI

Consider the following table showing the number of students (in thousands) enrolled in five different courses (A, B, C, D, E) in a college over two years (2022 and 2023).

Course	2022 (Thousands)	2023 (Thousands)
A	120	150
B	90	110
C	150	160
D	110	130
E	80	95

Question 1: What is the total number of students enrolled in Course A in both years?

Answer: 120 + 150 = 270 thousand students.

Question 2: What is the percentage increase in enrollment for Course B from 2022 to 2023?

Increase = 110 - 90 = 20 thousand.

Percentage Increase = (Increase / Original Value) * 100 = (20 / 90) * 100 = (2/9) * 100 ≈ 22.22%.

Question 3: What is the ratio of students enrolled in Course D in 2023 to Course E in 2022?

Ratio = (Students in D in 2023) : (Students in E in 2022) = 130 : 80 = 13 : 8.

Question 4: What is the average number of students enrolled across all courses in 2022?

Total students in 2022 = 120 + 90 + 150 + 110 + 80 = 550 thousand.

Average = Total / Number of courses = 550 / 5 = 110 thousand students.

Probability

Probability is the branch of mathematics concerned with the likelihood of an event occurring. It's expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Basic Definitions

• Experiment: An action or process that leads to a set of outcomes. (e.g., tossing a coin, rolling a die).
• Outcome: A single possible result of an experiment. (e.g., getting 'Heads' when tossing a coin).
• Sample Space (S): The set of all possible outcomes of an experiment. (e.g., For a coin toss, S = {H, T}). The number of outcomes in the sample space is denoted by n(S).
• Event (E): A subset of the sample space, representing a specific outcome or a set of outcomes we are interested in. (e.g., Getting an even number when rolling a die). The number of outcomes in an event is denoted by n(E).

Formula for Probability

The probability of an event E occurring is given by:

P(E) = Number of favorable outcomes / Total number of possible outcomes

P(E) = n(E) / n(S)

Key Properties of Probability

• 0 ≤ P(E) ≤ 1 (Probability is always between 0 and 1, inclusive).
• If an event is impossible, its probability is 0.
• If an event is certain, its probability is 1.
• The sum of probabilities of all possible outcomes in a sample space is 1.

Types of Events

• Mutually Exclusive Events: Events that cannot occur at the same time. (e.g., Getting 'Heads' and 'Tails' on a single coin toss).
• Independent Events: The occurrence of one event does not affect the probability of another event occurring. (e.g., Tossing a coin twice; the outcome of the first toss doesn't affect the second).
• Dependent Events: The occurrence of one event affects the probability of another event. (e.g., Drawing two cards from a deck without replacement).

Formulas for Combined Events

• Probability of A or B (Mutually Exclusive): P(A or B) = P(A) + P(B)
• Probability of A or B (Not Mutually Exclusive): P(A or B) = P(A) + P(B) - P(A and B)
• Probability of A and B (Independent): P(A and B) = P(A) * P(B)
• Probability of A and B (Dependent): P(A and B) = P(A) * P(B|A) (where P(B|A) is the conditional probability of B given A).
• Complementary Event: The probability of an event NOT happening is 1 minus the probability of it happening. P(not E) = 1 - P(E).

Common Scenarios

1. Coin Tosses

• One coin: S = {H, T}, n(S) = 2. P(H) = 1/2, P(T) = 1/2.
• Two coins: S = {HH, HT, TH, TT}, n(S) = 4. P(2 Heads) = 1/4, P(1 Head, 1 Tail) = 2/4 = 1/2, P(2 Tails) = 1/4.
• 'n' coins: Total outcomes = 2ⁿ.

2. Rolling Dice

• One die: S = {1, 2, 3, 4, 5, 6}, n(S) = 6. P(any specific number) = 1/6. P(even number) = P(2, 4, 6) = 3/6 = 1/2.
• Two dice: Total outcomes = 6 * 6 = 36. (e.g., Sum of 7: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} - 6 outcomes. P(Sum=7) = 6/36 = 1/6).
• 'n' dice: Total outcomes = 6ⁿ.

3. Playing Cards (Standard Deck of 52 cards)

• Total cards = 52.
• Suits: Hearts (Red), Diamonds (Red), Clubs (Black), Spades (Black) - 13 cards each.
• Ranks: 2, 3, ..., 10, J, Q, K, A (13 ranks).
• Face Cards: J, Q, K (3 per suit, total 12).
• Aces: 4.
• Red cards = 26, Black cards = 26.
• P(drawing an Ace) = 4/52 = 1/13.
• P(drawing a Heart) = 13/52 = 1/4.
• P(drawing a face card) = 12/52 = 3/13.

4. Balls in a Bag/Urn

Problems involve drawing balls of different colors. If drawing multiple balls, consider whether it's "with replacement" (independent events) or "without replacement" (dependent events).

Example 1 (Dice)

Two dice are rolled simultaneously. What is the probability of getting a sum of 5?

Total possible outcomes n(S) = 6 * 6 = 36.

Favorable outcomes (sum of 5): {(1, 4), (2, 3), (3, 2), (4, 1)}. Number of favorable outcomes n(E) = 4.

Probability P(E) = n(E) / n(S) = 4 / 36 = 1/9.

Example 2 (Cards)

A card is drawn from a pack of 52 cards. What is the probability that the card drawn is a face card or a black card?

Total outcomes n(S) = 52.

Event A: Drawing a face card. n(A) = 12. P(A) = 12/52.

Event B: Drawing a black card. n(B) = 26. P(B) = 26/52.

Event (A and B): Drawing a black face card. There are 3 face cards in Clubs and 3 in Spades. n(A and B) = 6. P(A and B) = 6/52.

Since the events are not mutually exclusive (some cards are both face cards and black), we use:

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) = (12/52) + (26/52) - (6/52) = (12 + 26 - 6) / 52 = 32 / 52 = 8 / 13.

Permutation and Combination

Permutation and Combination are topics that deal with counting the number of ways to arrange or select items from a set. They are fundamental in probability and combinatorics.

Permutation (Arrangement)

A permutation is an arrangement of objects in a specific order. The order matters.

• Formula for Permutation of 'n' distinct objects taken 'r' at a time:

ⁿP_r = n! / (n-r)!

Where '!' denotes factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1).

• Formula for Permutation of 'n' distinct objects taken all at a time:

ⁿP_n = n! / (n-n)! = n! / 0! = n! (since 0! = 1).

• Permutations with Repetitions: If there are 'n' objects where there are n1 identical objects of type 1, n2 identical objects of type 2, ..., nk identical objects of type k, the number of distinct permutations is:

n! / (n1! * n2! * ... * nk!)

Combination (Selection)

A combination is a selection of objects where the order does not matter.

• Formula for Combination of 'n' distinct objects taken 'r' at a time:

ⁿC_r = n! / [r! * (n-r)!]

Note the relationship: ⁿC_r = ⁿP_r / r!

• Key Property: ⁿC_r = ⁿC_n-r. This means selecting 'r' items is the same as choosing 'n-r' items to leave behind.

Distinguishing Permutation and Combination

Ask yourself: Does the order of selection/arrangement matter?

• Permutation: Used for problems like arranging letters in a word, forming committees with specific roles (President, Secretary), assigning tasks.
• Combination: Used for problems like selecting a team, choosing a subset of items, forming groups without specific roles.

Example 1 (Permutation)

How many ways can the letters of the word 'APPLE' be arranged?

Total letters (n) = 5.

Repetitions: 'P' appears 2 times (n1 = 2). 'A', 'L', 'E' appear once.

Number of arrangements = n! / n1! = 5! / 2! = (5 * 4 * 3 * 2 * 1) / (2 * 1) = 120 / 2 = 60 ways.

Example 2 (Combination)

From a group of 5 students, how many ways can a committee of 3 students be selected?

Here, n = 5, r = 3. The order of selection doesn't matter for a committee.

Number of ways = ⁵C₃ = 5! / [3! * (5-3)!] = 5! / (3! * 2!) = (5 * 4 * 3 * 2 * 1) / [(3 * 2 * 1) * (2 * 1)] = (5 * 4) / (2 * 1) = 20 / 2 = 10 ways.

Example 3 (Permutation vs Combination)

In how many ways can the first and second prizes be awarded to 5 contestants?

Here, order matters (Contestant A getting 1st and B getting 2nd is different from B getting 1st and A getting 2nd). This is a permutation.

n = 5, r = 2.

Number of ways = ⁵P₂ = 5! / (5-2)! = 5! / 3! = (5 * 4 * 3 * 2 * 1) / (3 * 2 * 1) = 5 * 4 = 20 ways.

In how many ways can a group of 2 winners be selected from 5 contestants?

Here, order does not matter. This is a combination.

n = 5, r = 2.

Number of ways = ⁵C₂ = 5! / [2! * (5-2)!] = 5! / (2! * 3!) = (5 * 4) / (2 * 1) = 10 ways.