AIOU 9424 Solved Assignment Autumn 2025

---

ALLAMA IQBAL OPEN UNIVERSITY

(Department of Mathematics)

---

WARNING

1. Plagiarism or hiring of ghost writer(s) for solving the assignment(s) will debar the student from award of degree/certificate if found at any stage.

2. Submitting assignment(s) borrowed or stolen from other(s) as one's own will be penalized as defined in the "Aiou Plagiarism Policy".

Assignment Submission Schedule
6 Credit Hours	Due Date	3 Credit Hours	Due Date
Assignment 1	15-12-2025	Assignment 1	08-01-2026
Assignment 2	08-01-2026
Assignment 3	30-01-2026	Assignment 2	20-02-2026
Assignment 4	20-02-2026

Course: Quantitative Reasoning (9424)	Semester: Autumn-2025
Level: BS

Total Marks: 100	Pass Marks: 50

---

ASSIGNMENT No. 1

---

Q1(a). Which fundamental concepts and methods in mathematics did the Babylonians introduce and develop? ▶

---

Fundamental Concepts and Methods Introduced by the Babylonians

The Babylonians were among the earliest civilizations to develop systematic mathematical practices. Their contributions were practical yet advanced for their time, and many of their ideas continue to influence modern mathematics. Their work was primarily recorded on clay tablets using cuneiform script, which has given historians and mathematicians valuable insight into their techniques and problem-solving approaches.

---

Sexagesimal Number System

One of the most significant achievements of Babylonian mathematics was their adoption of the sexagesimal, or base-60, number system. Unlike the decimal system, base-60 offered great flexibility because 60 is divisible by many numbers such as 2, 3, 4, 5, and 6. This made fractional calculations easier. The legacy of this system still exists today in our division of time into 60 seconds per minute and 60 minutes per hour, and in the measurement of angles with 360 degrees in a circle.

---

Place Value System

The Babylonians developed an early place value system where the value of a symbol depended on its position. This was a revolutionary step compared to other ancient systems like Egyptian hieroglyphics. Though initially they lacked a symbol for zero, their positional system laid the groundwork for later numerical systems. Eventually, they introduced a placeholder symbol, which helped avoid ambiguities in large numbers.

---

Mathematical Tables

The Babylonians created extensive mathematical tables on clay tablets to simplify calculations. These included multiplication tables, tables of squares, cubes, reciprocals, and square roots. These resources functioned much like early calculators, enabling scribes and scholars to solve complex problems more efficiently.

---

Quadratic Equations

Babylonian mathematicians could solve quadratic equations, often applying methods similar to what we now recognize as the quadratic formula. They used these techniques for practical problems, such as calculating the dimensions of fields or the sides of geometric figures. Their ability to solve such equations shows a highly developed form of algebra.

---

Pythagorean Triples

Tablets such as Plimpton 322 demonstrate that the Babylonians were familiar with Pythagorean triples long before Pythagoras. These sets of integers satisfied the relation a² + b² = c², suggesting that they had significant knowledge of right triangles and geometric properties, which were applied in surveying and construction.

---

Practical Geometry

Geometry played a practical role in Babylonian society. They used formulas to approximate areas and volumes, such as calculating the area of a trapezoid or the volume of a cylinder. They also made approximations for π, sometimes taking it as 3 or as the fraction 25/8. These methods were essential for architecture, land measurement, and irrigation projects.

---

Algebraic Problem Solving

Babylonian texts reveal strong algebraic thinking. They worked with equations involving one or two variables and often solved simultaneous equations. These were not abstract exercises but connected to real-life problems, such as distributing land, calculating shares, or trade-related computations.

---

Astronomical Applications

Babylonian mathematics was closely tied to astronomy. They used their base-60 system to make accurate astronomical calculations, including predictions of lunar eclipses and planetary movements. This precise use of mathematics for celestial observations heavily influenced later Greek and Islamic astronomy.

---

Fractions and Reciprocals

The Babylonians made extensive use of fractions and reciprocals. Unlike Egyptians, who mainly used unit fractions, Babylonians had a more versatile approach due to their base-60 system, which allowed simple representation of many fractions. This ability was particularly important in trade, measurements, and dividing resources.

---

Legacy of Babylonian Mathematics

The mathematical innovations of the Babylonians formed the basis of later mathematical traditions. Their systematic methods, use of place value, and integration of mathematics into practical and astronomical contexts made their work foundational for Greek, Indian, and later Islamic mathematics. Even today, remnants of their sexagesimal system survive in our timekeeping and angle measurement.

---

Conclusion

The Babylonians were pioneers in mathematics, introducing fundamental concepts such as the sexagesimal system, place value, quadratic solutions, practical geometry, and astronomical mathematics. Their ability to merge practical needs with systematic problem-solving reflects their ingenuity. The impact of their mathematical system is still visible in modern life, making their contributions timeless and universal.

---

---

Q1(b). From A to B, Aryan travels 1 km. He travels back at a pace of three kilometers per hour faster than he did in the beginning. The travel takes 0.5 hours in total. Determine the pace for the initial and subsequent portions of the trip. ▶

---

Problem Restatement

Aryan travels 1 km from A to B at some initial speed \( v \) km/h. On the return trip, he travels 1 km back from B to A at a faster speed of \( v + 3 \) km/h. The total travel time is 0.5 hours. We need to determine both speeds.

---

Step 1: Express the travel times

The time taken to travel from A to B is:

\[ t_1 = \frac{1}{v} \]

The time taken to return from B to A is:

\[ t_2 = \frac{1}{v+3} \]

---

Step 2: Total time equation

The total time for the trip is 0.5 hours, therefore:

\[ t_1 + t_2 = 0.5 \]

\[ \frac{1}{v} + \frac{1}{v+3} = 0.5 \]

---

Step 3: Simplify the equation

Multiply through by \( v(v+3) \) to remove denominators:

\[ (v+3) + v = 0.5 \cdot v(v+3) \]

\[ 2v + 3 = 0.5v^2 + 1.5v \]

Multiply both sides by 2 to remove the decimal:

\[ 4v + 6 = v^2 + 3v \]

Rearranging gives:

\[ v^2 - v - 6 = 0 \]

---

Step 4: Solve the quadratic equation

The quadratic can be factorized as:

\[ (v - 3)(v + 2) = 0 \]

So the solutions are:

\[ v = 3 \quad \text{or} \quad v = -2 \]

Since speed cannot be negative, we take:

\[ v = 3 \, \text{km/h} \]

---

Step 5: Find the return speed

The return speed is:

\[ v + 3 = 3 + 3 = 6 \, \text{km/h} \]

---

The speed from A to B was \( 3 \, \text{km/h} \). The speed from B to A was \( 6 \, \text{km/h} \).

---

---

Q2(a). Consider two candidates who appeared for the selection of the post of senior scientist. Both persons were thoroughly examined by a panel of examiners. Person A got 80 points out of 100, while Person B got 70 points out of 100. Scientist A got the position. Explain how numbers helped the panel of examiners to decide for selection. ▶

---

Introduction

In any evaluation process, numbers play a crucial role in ensuring fairness and objectivity. When two candidates appear for the same position, their performance is often assessed through a scoring system that provides a measurable basis for comparison. This makes the selection process transparent and evidence-based.

---

Performance Evaluation Through Scores

In the given situation, both Person A and Person B appeared for the selection of the post of senior scientist. Their performances were thoroughly examined by a panel of experts. After evaluation, Person A secured 80 points out of 100, while Person B scored 70 points out of 100. These numbers represent the overall assessment of their skills, knowledge, and suitability for the role.

---

Numerical Comparison of Candidates

The scores provide a direct comparison between the two candidates. Person A’s score of 80 shows a higher level of achievement compared to Person B’s score of 70. The difference of 10 points reflects that Person A demonstrated greater competence, accuracy, and command over the subject matter. Without numbers, the judgment could have been subjective, but numerical values made the comparison clear and straightforward.

In mathematical form, we can write:

\[ \text{Person A's Score} = \frac{80}{100} = 0.8 \, (80\%) \]

\[ \text{Person B's Score} = \frac{70}{100} = 0.7 \, (70\%) \]

Thus, Person A outperformed Person B by: \[ 0.8 - 0.7 = 0.1 = 10\% \]

---

Role of Objectivity in Decision-Making

Numbers remove personal bias from the selection process. The panel could rely on the numerical scores instead of personal impressions or preferences. This ensured that the decision to select Person A was based purely on merit, as reflected by the higher score. The use of a scoring system guarantees transparency and fairness.

---

Why Person A Was Selected

Since Person A obtained 80 points, which is significantly higher than Person B’s 70 points, the numbers clearly indicated that Person A performed better overall. The panel used these results to select the candidate who demonstrated superior knowledge and skill. The numerical evidence provided a reliable justification for choosing Person A as the senior scientist.

---

Conclusion

The use of numbers in evaluation ensures that decisions are logical, fair, and evidence-based. In this case, the numerical difference between the scores of Person A and Person B guided the panel to select Person A. Thus, numbers acted as a decisive tool, ensuring meritocracy and fairness in the selection process.

---

---

Q2(b). In this case, the person who sold the pen for Rs. 96 made the same amount of money as the watch's cost price. If he sells it for twice as much as the percentage profit he made on it before, what will the new price be? ▶

---

Problem Restatement

A person sold a pen for Rs. 96. The money he received from selling the pen is equal to the cost price of a watch. He made a certain profit percentage on the pen. If he sells the pen again but this time at twice the earlier percentage profit, what will be the new selling price?

---

Step 1: Let the cost price of the pen be x

The selling price of the pen is 96, so the profit made is:

\[ \text{Profit} = 96 - x \]

The profit percentage is:

\[ \text{Profit \%} = \frac{96 - x}{x} \times 100 \]

---

Step 2: New profit percentage condition

If the pen is sold at twice the earlier percentage profit, then the new selling price will be:

\[ \text{New SP} = x \Bigg(1 + \frac{2(96 - x)}{x}\Bigg) \]

Simplifying,

\[ \text{New SP} = x + 2(96 - x) = 192 - x \]

---

Step 3: Interpretation

The new selling price depends on the cost price of the pen (x). Without knowing x exactly, the formula remains:

\[ \text{New SP} = 192 - x \]

---

Step 4: Numerical Example

Suppose the cost price of the pen was Rs. 80.

Then the original profit is:

\[ 96 - 80 = 16 \]

The profit percentage is:

\[ \frac{16}{80} \times 100 = 20\% \]

If the profit percentage is doubled, it becomes 40%. Therefore, the new selling price is:

\[ 80 + \frac{40}{100} \times 80 = 80 + 32 = 112 \]

---

In general, the new selling price of the pen is \(192 - x\), where \(x\) is the cost price of the pen. For example, if the cost price is Rs. 80, then the new selling price will be Rs. 112.

---

---

Q3(a). Four shirts, four sets of pants, and two hats cost $560 in a store. Nine shirts, nine pairs of pants, and six hats total $1,290 in cost. How much does one shirt, one pair of pants, and one item cost altogether? ▶

---

Problem Restatement

We are given that four shirts, four sets of pants, and two hats cost $560 in a store. Nine shirts, nine pairs of pants, and six hats cost $1,290. We need to find the cost of one shirt, one pair of pants, and one hat altogether.

---

Step 1: Define the variables

Let the price of one shirt be \( S \), one pair of pants be \( P \), and one hat be \( H \).

From the given information we can write:

\[ 4S + 4P + 2H = 560 \]

\[ 9S + 9P + 6H = 1290 \]

---

Step 2: Simplify the first equation

Divide the first equation by 2:

\[ 2S + 2P + H = 280 \quad (1) \]

---

Step 3: Eliminate H

Subtract 6 times equation (1) from the second equation:

\[ 9S + 9P + 6H - 6(2S + 2P + H) = 1290 - 1680 \]

\[ 9S + 9P + 6H - (12S + 12P + 6H) = -390 \]

\[ -3S - 3P = -390 \]

\[ S + P = 130 \]

---

Step 4: Solve for H

Substitute \( S + P = 130 \) into equation (1):

\[ 2(130) + H = 280 \]

\[ 260 + H = 280 \]

\[ H = 20 \]

---

Step 5: Final Calculation

Now the total cost of one shirt, one pair of pants, and one hat is:

\[ S + P + H = 130 + 20 = 150 \]

---

The cost of one shirt, one pair of pants, and one hat altogether is $150.

---

---

Q3(b). If the Golden Ratio is 1.618 and you have a piece of cardboard that is 7/8 ft wide. How long should you cut, so it follows the Golden Ratio? ▶

---

Problem Restatement

The Golden Ratio is approximately \( \phi = 1.618 \). We are given a piece of cardboard that is \( \tfrac{7}{8} \) ft wide. We need to find the length so that the dimensions follow the Golden Ratio.

---

Step 1: Recall the Golden Ratio Formula

For an object to follow the Golden Ratio:

\[ \frac{L}{W} = \phi \]

where \( L \) is the length and \( W \) is the width.

---

Step 2: Substitute the Width

The width is given as:

\[ W = \tfrac{7}{8} \, \text{ft} = 0.875 \, \text{ft} \]

---

Step 3: Calculate the Length

Using the Golden Ratio:

\[ L = \phi \times W \]

\[ L = 1.618 \times 0.875 \]

\[ L = 1.41575 \, \text{ft} \]

---

The required length of the cardboard should be approximately 1.416 ft in order to follow the Golden Ratio with a width of 7/8 ft.

---

---

Q4(a). ▶

---

---

---

Q4(b). ▶

---

---

---

Q5(a). ▶

---

---

---

Q5(b). ▶

---

---

---

ASSIGNMENT No. 2

---

Q1(a). ▶

---

---

---

Q1(b). ▶

---

---

---

Q2(a). ▶

---

---

---

Q2(b). ▶

---

---

---

Q3(a). ▶

---

---

---

Q3(b). ▶

---

---

---

Q4(a). ▶

---

---

---

Q4(b). ▶

---

---

---

Q5(a). ▶

---

---

---

Q5(b). ▶

---

---