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Unit 1: Numbers, Quantification and Numerical Applications

Introduction

In this unit, we explore advanced concepts of modulo arithmetic, congruence modulo, basic mathematical applications like alligation, mixtures, boats and streams, pipelines, races, and numerical inequalities. Understanding these concepts is fundamental to applied mathematics and various competitive examinations.

Modulo Arithmetic and Congruence

Modulo operator refers to the remainder of a division operation. If we have two integers \(a\) and \(n\) (with \(n > 0\)), then \(a \pmod n\) is the remainder when \(a\) is divided by \(n\).

\[ a \equiv b \pmod n \implies n \mid (a - b) \]

This means that \(a\) and \(b\) leave the same remainder when divided by \(n\).

Example of Congruence Modulo

Let’s check if \( 38 \equiv 14 \pmod{12} \). Since \( 38 - 14 = 24 \), and \( 24 \) is divisible by \( 12 \) (\( 24 = 12 \times 2 \)), the congruence holds true.

Alligation and Mixture

Alligation is a rule that enables us to find the ratio in which two or more ingredients at a given price must be mixed to produce a mixture of a desired price.

Cheaper Quantity (C) Dearer Quantity (D) Mean Price (M) D - M M - C

Numerical Problems

Boats and Streams

When a boat moves in the direction of the stream, it is called downstream, and when it moves against the stream, it is called upstream.

Let the speed of the boat in still water be \(u\) km/hr and the speed of the stream be \(v\) km/hr. Then:

  • Speed Downstream = \( (u + v) \) km/hr
  • Speed Upstream = \( (u - v) \) km/hr

Competency-Based Questions

Q1. The time shown on a 12-hour clock modulo 12 can be thought of as a congruence class. If it is currently 8:00 AM, what time will it be in 137 hours?

  1. Calculate \( 137 \pmod{12} \).
  2. Add the remainder to the current time.

Q2. Two types of rice costing ₹ 45/kg and ₹ 60/kg are mixed together. At what ratio must they be mixed to have a mixture worth ₹ 54/kg?

  • Apply the rule of alligation where \( C = 45 \), \( D = 60 \), and \( M = 54 \).
  • Analyze the ratio \( \frac{D - M}{M - C} \).

Q3. A pipe can fill a cistern in 10 hours, and another pipe can empty it in 15 hours. If both pipes are opened simultaneously, when will the cistern be full?

  • Determine the net work done in an hour.
  • Formulate the equation mathematically to find the total time.

Q4. Compare the performance: Athlete A completes a 400m race in 45 seconds, while Athlete B completes it in 48 seconds. If they both maintain their speeds, what would be the distance of Athlete B from the finish line when Athlete A completes the race?

Answers

Ans 1. 137 mod 12 = 5. Currently 8:00 AM + 5 hours = 1:00 PM.

Ans 2. Using alligation: \( \frac{D - M}{M - C} = \frac{60 - 54}{54 - 45} = \frac{6}{9} = 2:3 \).

Ans 3. Net work in an hour = \( \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30} \). Therefore, it takes 30 hours to fill the cistern.

Ans 4. Speed of A = \( \frac{400}{45} \) m/s. Speed of B = \( \frac{400}{48} \) m/s. When A completes 400m (45s), distance of B = \( \frac{400}{48} \times 45 = 375 \)m. So B is 25m from the finish line.