Meet the professionals who contribute specialised experience in life skills, leadership, digital business, technology, communication and career development.
RealPath Academy is an education, training and consultancy organization working with academic institutions, professionals and organizations to create meaningful opportunities for learning, development and growth.
We bring together knowledge, skills and real-world application to help individuals and organizations navigate change, develop their capabilities and move forward with greater confidence and purpose.
Our work extends from academic and career development to professional learning, leadership development and organizational growth, through programmes, training initiatives and customized solutions.
At RealPath Academy, we believe meaningful learning should go beyond acquiring knowledge. It should create the understanding, capability and confidence to turn learning into action and purposeful progress.
Direction and Purpose
Our Vision & Mission
01
A Future of Purposeful Learning and Growth
To contribute to a world where individuals and organizations can discover their potential, develop meaningful capabilities and find purposeful paths for growth and contribution.
02
Turning Learning into Capability and Action
To create meaningful learning and development experiences that help individuals and organizations discover potential, develop capability and turn knowledge into purposeful action.
03
Learning, Application and Reflection
Our programmes support people and organizations as they learn, adapt, perform and grow through application, collaboration, reflection and continuous development.
Our Values
Purpose • Trust • Integrity
Our values guide how we design our programmes, work with people and build relationships.
Purpose
Meaningful work begins with a clear sense of purpose. We connect learning and development with meaning, direction and contribution.
Trust
Meaningful relationships are built through respect, reliability, openness and mutual confidence.
Integrity
We remain honest, responsible and principled—even when doing so is difficult, inconvenient or personally disadvantageous.
Our Philosophy
Discover. Develop. Direct.
We believe meaningful development involves three interconnected stages.
Discover
Discover potential, strengths, interests, opportunities and possibilities.
Develop
Develop knowledge, skills, capabilities, confidence and character through meaningful learning and experience.
Direct
Direct those capabilities towards purposeful goals, meaningful action and positive contribution.
Our Story
From a Belief to a Growing Platform
RealPath Academy was established with a simple belief: Learning becomes meaningful when it connects with life.
Our journey has been shaped by experience in education, teaching, training, leadership development and institutional engagement. These experiences reinforced our belief that effective learning is not simply about delivering information. It is about creating opportunities for people to understand, experience, reflect and apply.
As we grow, our aim is to create programmes and initiatives that help people discover possibilities, develop meaningful capabilities and move forward with clarity and purpose.
Why RealPath Exists
Connecting Learning with the Real World
Learners today need more than academic knowledge. They need to think critically, communicate effectively, solve unfamiliar problems, collaborate, adapt and make informed decisions.
Knowledge with ApplicationLearning with ExperienceSkills with PurposeEducation with Real LifeDevelopment with Growth
Contact Us
Let’s Create Meaningful Learning Experiences
Connect with RealPath Academy for academic programmes, training initiatives, professional development and customized consultancy solutions.
Location: Humayun Nagar, Mehdipatnam, Hyderabad, Telangana, India
Website: realpathacademy.com
Area of Work
Career Planning and Guidance
Discover Your Strengths. Build a Purposeful Future.
Career planning is a structured process of understanding interests, strengths, values, abilities and aspirations, and connecting them with suitable educational and professional opportunities. RealPath Academy helps students and professionals make informed decisions and develop practical action plans.
Programme Areas
Self-awareness, interests, values and strengths
Career awareness, exploration and decision-making
Educational pathway guidance and skills mapping
Goal setting, action planning and career readiness
Communication, teamwork, interview preparation and employability
Applications and Future Scope
The programme supports school subject selection, higher-education choices, graduate employability, career transitions and institutional career-guidance initiatives. It also prepares learners for changing opportunities influenced by technology, entrepreneurship, sustainability and emerging industries.
Expected Outcomes
Participants gain stronger self-awareness, clearer career direction, realistic goals, better decision-making ability and an individual career action plan. Programmes may be offered as workshops, counselling sessions, bootcamps or customised institutional programmes.
RealPath Academy helps learners move beyond passive technology use and understand how digital tools, data and intelligent systems work. The programmes combine clear instruction, practical exercises and project-based learning.
Technology Areas
Python Programming, Artificial Intelligence and Machine Learning
Data Science, Data Analytics and Data Visualisation
Generative AI and Prompt Engineering
Microsoft Azure and Cloud Fundamentals
Digital literacy, responsible technology use and project development
Why It Matters
Technology education develops curiosity, logical thinking, creativity, experimentation, problem-solving, data awareness and ethical responsibility. It enables learners to become creators, innovators and responsible contributors rather than only consumers of technology.
Applications, Outcomes and Future Scope
Participants can apply these capabilities in education, business, research, healthcare, automation, entrepreneurship and social-impact projects. They build technical confidence, computational thinking and a foundation for future careers in AI, software, analytics, cloud computing and other emerging fields.
Qualifications and technical knowledge become more valuable when individuals can communicate clearly, understand others, manage responsibilities and work effectively in teams. RealPath Academy uses practical activities, examples and guided practice to develop these capabilities.
Programme Areas
Language proficiency and communication skills
Presentation skills and public speaking
Teamwork, collaboration and empathy
Time management and personal responsibility
Workplace etiquette, grooming and professional behaviour
Customer management, conflict handling and leadership readiness
Applications, Outcomes and Future Scope
These skills strengthen classroom participation, interviews, teaching, customer interaction, teamwork, leadership and workplace performance. Participants develop confidence, active listening, professional conduct, better time management and the ability to build productive relationships—capabilities required across every career and industry.
This programme helps students, educators and professionals examine information objectively, identify underlying causes and develop practical, well-reasoned solutions. Structured methods, discussions and activities help participants move beyond memorisation and assumption-based decisions.
Programme Areas
Critical, creative, logical and analytical thinking
Problem identification and root-cause analysis
Decision-making and solution development
Evaluation of alternatives and consequences
Practical problem-solving and clear communication of reasoning
Applications
These skills support academic learning, research, career planning, leadership, engineering, technology, entrepreneurship, project management, conflict resolution and everyday decision-making.
Outcomes and Future Scope
Participants learn to distinguish facts from assumptions, analyse patterns, consider multiple perspectives and make informed decisions. As AI, automation and rapidly changing information reshape society, independent thinking and the ability to solve unfamiliar problems will become increasingly valuable.
Abid Mohiuddin is an educator and learning-and-development professional with an M.Tech. in Mechanical Engineering. He conducts practical workshops for students, educators and professionals across schools, colleges, universities and organisations.
Areas of Expertise
Career Planning and Guidance
Emotional Intelligence, Problem-Solving and Conflict Resolution
Time Management, Goal Setting and Habit Building
Gap Analysis and Quality Improvement
Distinctive Approach and Achievements
His approach combines simplicity, visionary thinking, originality and humility. He received the Excellent Teaching Practices Award at King Khalid University, Saudi Arabia, and authored Life Skills for Children. His institutional engagements include IIIT Basar, IIIT Nuzvid, IIIT Ongole, Government Degree College Nampally and several other educational institutions.
Data Analytics • AI and Python • Microsoft Azure Trainer
Sanjeevani Shere is a technology trainer from Navi Mumbai with more than ten years of experience in technical training, curriculum development, learner mentoring and practical project delivery.
Technical Expertise
Python, SQL, Power BI, Excel and Data Analytics
Artificial Intelligence, Machine Learning and Generative AI
Prompt Engineering, Data Visualisation and Cloud Computing
Microsoft Azure Fundamentals, Administration and Architecture
Additional exposure to R, Java, C, C++, Tableau, Zoho Analytics and Flourish
Experience and Approach
She has trained learners through schools, academies and professional organisations, combining clear explanations with hands-on exercises, assessments and projects. Her professional training includes Data Science, Power BI and SQL, and Microsoft Azure courses AZ-900, AZ-104 and AZ-305.
Salma Khan is a soft-skills trainer and leadership coach with a multidisciplinary background in communication, professional grooming, Montessori education and behavioural development.
Qualifications and Expertise
B.A., Bangalore University
Certified Master Trainer and Certified Montessori Trainer
Communication, public speaking and leadership development
Sales skills, customer management and teacher training
Grooming, personal hygiene and empathy
Distinctive Approach and Engagements
Her training is simple, practical and activity-oriented, helping participants apply complex ideas confidently in real situations. Her professional engagements include Joy Alukkas, Conrad Hilton, Hilton Garden Inn, The Ritz-Carlton Bangalore, Sheraton Grand, Forum Mall, Treamis International School and Suffah Public School.
Mechanical Engineering Educator • Industry Expert • Consultant
Mohammed Shafiuddin holds a Master of Engineering degree in Mechanical Engineering and has more than 25 years of combined experience in higher education and industry in India and Saudi Arabia. He received the Excellence in Academic Education Award from King Khalid University.
Professional Expertise
Problem-Solving and Critical Thinking
Workplace Communication, Leadership and Team Building
Performance Analysis and Improvement
Engineering Project Management
Finite Element Analysis and ANSYS
Programme Design, E-Learning Management, Mathematics and Science
Teaching and Leadership
He has taught and mentored engineering students, supervised projects, contributed to curriculum development and supported career planning. His academic leadership responsibilities included coordinating examinations and helping plan institutional events, while his industrial experience enables him to connect theory with professional practice.
Business Coach, E-commerce Consultant and Digital Marketing Trainer
More than two decades of experience in website development, e-commerce and digital marketing.
Professional Expertise
Website design and development
Digital-marketing strategy
SEO and social-media marketing
Branding and online-reputation management
Google Ads, WooCommerce and e-commerce platforms
Training Approach
A certified HubSpot Inbound Marketing and Google Ads professional, he has helped businesses grow online since 2002. His training combines practical industry insights, digital content and real-world strategies to help individuals and businesses succeed in the digital economy.
Python Trainer, Data Science Enthusiast and Emerging Educator
A dedicated trainer with an academic foundation in data science and an emphasis on accessible, practical learning.
Core Skills
Python
Machine learning
Artificial intelligence
Content creation
Communication
Training Approach
Her teaching focuses on clarity, simplicity and real-world application. Experience in web development, content creation, social-media management, team leadership, hackathons and innovation programmes supports her goal of helping beginners develop practical knowledge and confidence.
More than eight years of experience helping learners build and manage online ventures.
Areas of Expertise
Launching and scaling online businesses
eBay, Amazon and Etsy
E-commerce strategy and implementation
Independent income generation
Entrepreneurship mentoring
Training Approach
His engineering and legal background contributes analytical and strategic depth. He uses hands-on learning, live case studies and practical implementation to help aspiring entrepreneurs move from theory towards sustainable online business.
Life and Mindset Coach, English Language Coach and YouTuber
Certified TEFL Tutor and Master Trainer with more than seven years of international teaching and coaching experience.
Areas of Expertise
Self-confidence, self-esteem and self-love
Communication and interpersonal skills
Stress and anxiety management
Career guidance and finance
Overcoming self-limiting beliefs
Distinctive Approach and Achievements
Author of Prayers of the Dying Undead and creator of the YouTube channels Figuring It Out Alive and Language Is Fun. Her humble, authentic and compassionate approach helps learners understand how emotions, beliefs and attitudes influence success and happiness.
Data Scientist, AI Consultant and Corporate Trainer
An IT professional and trainer with more than 15 years of combined experience in industry and engineering education. He is pursuing a Ph.D. in Information Technology with Artificial Intelligence as his research area.
Core Competencies
Generative AI, Large Language Models and Retrieval-Augmented Generation
AI agents, prompt engineering and vector databases
Advanced Python, SQL, Pandas, NumPy and Scikit-learn
Machine learning, deep learning, TensorFlow and Keras
Power BI, Tableau, Excel and data visualisation
Professional and Academic Experience
His experience includes four years as a Data Scientist and AI Consultant at Tata Consultancy Services and ten years in engineering academia. He has served as a Head of Department, Assistant Placement Officer, Project Coordinator and Online Examination Coordinator.
He has conducted more than 50 guest lectures, Faculty Development Programmes and technical workshops, and has trained and mentored engineering students, faculty members and working professionals.
Training Areas
Artificial Intelligence and Generative AI
Python programming and the Python ecosystem
Machine learning and data science
SQL, Power BI and business analytics
Industry-oriented technical bootcamps and career mentoring
Leadership and Distinctions
Member, Board of Studies, JNTU Hyderabad
IIT Bombay-ISTE certified workshops in DBMS and Design and Analysis of Algorithms
External Examiner, Observer, Chief Examiner, Paper Setter and Evaluation Expert
Jury member for INFINITY-2K15 at Osmania University
The Career Planning Summit 2027 will feature keynote addresses, featured presentations, open forums, audience-participation activities and opportunities for professional interaction.
Please note: The programme is tentative and may be revised according to speaker availability, venue requirements and organisational considerations.
Time
Programme
8:30–9:30 a.m.
Registration, welcome refreshments and arrival of guests
9:30–9:45 a.m.
Inaugural Ceremony
9:45–10:00 a.m.
Chief Guest's Address: Education, Ethical Journalism and Nation Building
10:00–10:30 a.m.
Opening Keynote: Journalism and the Cost of Lies: Ethics, Accountability and Public Responsibility
10:30–10:50 a.m.
Audience Participation Block 1: Digital Quiz Round 1 and three student video messages
10:50–11:30 a.m.
One-to-One Conversation: Ethical Journalism: Principles, Courage and the Pursuit of Justice
11:30–11:45 a.m.
Tea and networking break
11:45 a.m.–12:05 p.m.
Featured Address: From Potential to Profession
12:05–12:25 p.m.
Featured Address: Innovation and Entrepreneurship in Journalism and Digital Media
12:25–12:40 p.m.
Audience Participation Block 2: Digital Quiz Round 2 and three student video messages
12:40–1:20 p.m.
Open Forum: From Potential to Impact
1:20–2:15 p.m.
Lunch, prayer and networking break
2:15–2:35 p.m.
Featured Address: Faith, Sacred Figures and Journalism
2:35–3:15 p.m.
Moderated Open Forum: Faith, Sacred Personalities, Freedom of Expression, Media Narratives and Responsible Journalism
3:15–4:05 p.m.
Thematic Speaker Series: Journalism and the Power of Narratives
4:05–4:25 p.m.
Open Forum: Journalism and the Power of Narratives
4:25–4:40 p.m.
Tea and networking break
4:40–4:55 p.m.
Audience Participation Block 3: Digital Quiz Final Round and three student video messages
4:55–5:20 p.m.
Featured Address: Journalism and Nation-Building
5:20–5:40 p.m.
Closing Keynote: Young Voices in Digital Journalism: Vision, Transformation and the Global Media Landscape
5:40–6:05 p.m.
Addresses by three Guests of Honour
6:05–6:30 p.m.
Felicitation Ceremony
6:30–6:45 p.m.
Summit Declaration, Ethical-Journalism Commitment, Future Initiatives and Recognition of the Digital Knowledge Champion
6:45–7:00 p.m.
Vote of Thanks and Closing Remarks
From 7:00 p.m.
Official photographs, media interactions, refreshments and informal networking
Career Planning Summit 2027
Speakers and Panellists
The Summit will bring together experienced journalists, educators, academics, media professionals, career-development experts and other distinguished voices to share their knowledge and perspectives.
Profiles are being published progressively. A “Confirmed” badge is displayed only after formal confirmation of participation.
A respected voice promoting interfaith dialogue, communal harmony and compassionate human values.
About
Maulana Syed Taqi Raza Abidi is a theologian, philosopher, writer and speaker who has worked to advance interfaith understanding, social harmony and constructive engagement among communities. His academic background includes advanced study and research in Islamic sciences and philosophy.
Proposed Summit Contribution
His perspective can enrich the Summit’s discussion on faith, sacred personalities and responsible journalism, with emphasis on factual accuracy, contextual understanding, sensitivity and social harmony.
Journalist, Editor, Author and Journalism Educator
An experienced media professional whose work engages with journalism, ethics, accountability and the public sphere.
About
A. S. Panneerselvan served for nearly a decade as the Readers’ Editor of The Hindu, acting as an independent internal news ombudsman. He has worked across print and television journalism, taught journalism and written extensively on media and public affairs. He is also the author of Karunanidhi: A Life.
Proposed Summit Contribution
His experience can contribute meaningfully to discussions on editorial independence, ethical decision-making, media accountability, public trust and the responsibilities of journalists in a democratic society.
The Career Planning Summit 2027 will be honoured by the presence of distinguished personalities from education, journalism, media, public service, social development and other professional fields.
Guest Announcements
The names and profiles of confirmed Chief Guests, Guests of Honour, Special Guests and VIP Guests will be published after receiving their formal consent.
Career Planning Summit 2027
Advisory Committee
The Advisory Committee brings together expertise in education, journalism, youth development and responsible public communication. Select a member to view basic details.
Abid Mohiuddin is an educator and learning-and-development professional with an M.Tech. in Mechanical Engineering. He works in career planning, life skills, problem-solving, emotional intelligence and institutional learning programmes.
Summit Role
As the organiser of the Career Planning Summit 2027, he supports the programme’s educational direction, coordination and commitment to responsible communication.
Professor and Head, Department of Mass Communication and Journalism, MANUU
Journalism educator and academic associated with the Career Planning Summit 2027.
Basic Details
Prof. Ehtesham Ahmad Khan contributes academic insight relating to journalism education, media studies and the professional development of aspiring communicators.
Summit Contribution
He is associated with the Summit as an Advisory Committee Member, Featured Speaker and Open Forum Panellist, supporting its academic relevance and engagement with ethical journalism.
The names and responsibilities of confirmed committee members will be published progressively.
Career Planning Summit 2027
Sponsors
We welcome responsible institutional support for the Summit’s educational and community-focused objectives.
Current Status
No sponsor has been confirmed at present. Confirmed sponsors, if any, will be listed here with their permission.
Career Planning Summit 2027
Volunteers
Volunteers can support participant guidance, registration, guest assistance, communications, media coordination and other event-day activities.
Volunteer Information
Roles, eligibility, responsibilities and the application process will be announced after the volunteer plan is finalised.
Career Planning Summit 2027
Partners
RealPath Academy welcomes suitable partnerships that extend educational opportunity and responsible community engagement.
Educational Institutions
Schools, colleges, universities and training institutions may participate through student nominations, academic collaboration, outreach and knowledge-sharing.
Non-Governmental Organisations
NGOs working in education, youth development, media literacy or community welfare may collaborate through outreach and programme support.
Confirmed partner organisations will be displayed here after formal approval.
RealPath Academy Events
Career Planning Summit 2027
A Career in Journalism
Opportunities • Ethics • Nation Building
A focused one-day summit created to help young people understand journalism as a career, recognise its ethical responsibilities and explore its contribution to an informed and responsible society.
17 January 2027HyderabadOrganised by RealPath Academy
The Summit Focus
Career Clarity with Social Purpose
The Summit will bring students, aspiring journalists, educators, media professionals and distinguished voices together for meaningful conversations about the changing world of journalism.
01
Opportunities
Explore career pathways across reporting, broadcasting, digital media, independent journalism, multimedia storytelling and communication.
02
Ethics
Understand truth, accuracy, fairness, credibility, professional courage and responsible decision-making in journalism.
03
Nation Building
Examine how credible journalism strengthens public awareness, accountability, informed citizenship and social understanding.
What Participants Can Expect
Career guidance from experienced voices
Discussions on contemporary journalism
Insights into professional skills and challenges
Conversations on misinformation and credibility
Opportunities for interaction and learning
Who Should Attend?
School, junior-college and university students
Aspiring and early-career journalists
Educators and career counsellors
Media and communication professionals
People interested in ethical public communication
Programme, speakers and registration details will be announced progressively.
RealPath Academy
Gallery
A glimpse into our learning, development and engagement initiatives.
RealPath Academy Academics
School Education
Explore subject-based learning, practical activities and resources for school students.
Chapter-wise words in Telugu, Roman Telugu and English.
Telangana Class X Telugu First Language — సింగిడి–2
447 vocabulary entries across all 12 chapters. This edition covers selected introductory and thematic words; it is not a complete word-by-word glossary of every lesson or exercise.
Roman Telugu helps with reading aloud. English meanings explain the words in their chapter context. Some entries use the base form of an inflected word.
Read the corrected reference directly on this page: number concepts, algebra, geometry, trigonometry and essential methods.
Number Classification
Complex numbers have the form a + bi, with real a and b; i² = −1. Real numbers have b = 0 and are rational or irrational. Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real ⊂ Complex.
Natural numbers (here): 1, 2, 3, …
Whole numbers start from 0: 0, 1, 2, 3, …
Integers: …, −2, −1, 0, 1, 2, …
Rational numbers: p/q, where p and q are integers and q ≠ 0; includes terminating and recurring decimals.
Irrational numbers cannot be expressed as p/q: √2, √3, π, e.
Pure imaginary numbers: bi, with real b ≠ 0.
Prime, Composite, Even and Odd Numbers
A prime has exactly two positive factors: 1 and itself.
sin(90° − θ) = cos θ; cos(90° − θ) = sin θ; tan(90° − θ) = cot θ.
Ratios and identities apply where their denominators are nonzero.
Standard Trigonometric Values
θ
0°
30°
45°
60°
90°
sin θ
0
1/2
1/√2
√3/2
1
cos θ
1
√3/2
1/√2
1/2
0
tan θ
0
1/√3
1
√3
Undefined
Mathematical Essentials — Concepts & Steps
1. Addition of Numbers
i. If both numbers are positive: Add them and put a positive sign.
ii. If both numbers are negative: Add them and put a negative sign.
iii. If one number is positive and the other is negative: Subtract the smaller absolute value
from the larger absolute value and put the sign of the number with the larger absolute value.
3. Multiplication of Integers with Signs
First, multiply the signs, then multiply the numbers.
4. Division of Integers with Signs
i. If the numerator and denominator have positive signs: Divide the numbers; the answer is
positive.
ii. If the numerator and denominator have negative signs: Cancel the signs and divide the
numbers; the answer is positive.
iii. If the numerator and denominator have opposite signs: Divide the numbers; the answer is
negative.
5. Addition of Fractions
Make the denominators the same, add the numerators, keep the common denominator, and
simplify the fraction.
6. Multiplication of Fractions
i. Cancel common factors between the numerator and denominator, if possible.
ii. Multiply the numerators together and multiply the denominators together.
iii. Simplify the answer.
7. Division of Fractions
Keep the first fraction as it is, change division into multiplication, take the reciprocal of the
second fraction, and multiply.
8. Converting a Fraction into a Decimal
Divide the numerator by the denominator to get the decimal.
9. Converting a Decimal into a Fraction
Remove the decimal point and write the number over the corresponding power of 10 in the
denominator, then simplify.
10. Cross-Multiplication
For an equation involving two fractions, multiply the numerator of one fraction by the
denominator of the other fraction and set the two cross-products equal.
11. Divisibility Criteria
i. Divisibility by 2: A number is divisible by 2 if its last digit is even.
ii. Divisibility by 3: Add the digits. If the sum is divisible by 3, the number is divisible by 3.
iii. Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
12. Finding Square Roots by Prime Factorisation
Write the number as a product of prime factors and take one factor out of the square root for
every pair of identical factors.
13. Finding Cube Roots by Prime Factorisation
Write the number as a product of prime factors and take one factor out of the cube root for
every group of three identical factors.
14. Laws of Exponents
i. If the bases are the same and multiplication is between them, add the powers.
ii. If the bases are the same and division is between them, subtract the powers.
iii. If a power is raised to another power, multiply the powers.
15. Simplifying Algebraic Expressions
Remove brackets, combine like terms, and arrange the expression in standard form.
16. Solving a Linear Equation in One Variable
i. Bring the variable terms to one side and the constants to the other side. When a term is
moved to the other side, its sign changes.
ii. Finally, take the coefficient of the variable to the other side. If it is multiplied by the variable,
divide by it. If it is dividing the variable, multiply by it.
17. Solving Two Linear Equations by the Elimination Method
i. Make the coefficient of one variable the same in both equations.
ii. Check the signs of the same variable. If the signs are the same, change the signs of one
equation and add them. If the signs are opposite, simply add the equations.
iii. Eliminate that variable.
iv. Find the value of the remaining variable.
v. Substitute the value in one of the equations and find the value of the other variable.
18. Solving a Factorable Quadratic Equation by Splitting the Middle Term
i. Bring all the terms to one side and write the equation in standard form.
ii. Multiply the coefficient of the x² term by the constant term and call the result M.
iii. Choose two numbers whose product is M and whose sum is equal to the coefficient of x.
iv. Rewrite the x term using these two numbers.
v. Take common factors from the first two terms and the last two terms.
vi. Take the common factor from the two groups. Now the equation is in factorised form.
vii. Equate each factor to zero and find the two values of the variable.
Build your foundation for Class X Mathematics. These guided problems help Class VIII and IX students practise solving equations formed from word problems. Read each question, use the given equations, and work on paper before opening the answer. The final five tasks are Challenge Practice. Exercise labels follow the supplied worksheet.
4.1 Q.3 — Pants and skirts
Guided Practice
Question
Neha went to a sale to purchase some pants and skirts. When her friend asked her how many of each she had bought, she answered, “The number of skirts is two less than twice the number of pants purchased, and the number of skirts is four less than four times the number of pants purchased.” Help her friend find how many pants and skirts Neha bought.
Variables and Given Equations
x = pants; y = skirts
y = 2x − 2
y = 4x − 4
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 1; y = 0.
The supplied conditions give one pair of pants and zero skirts.
4.1 Q.4 — Quiz participants
Guided Practice
Question
Ten students of Class X took part in a mathematics quiz. The number of girls was four more than the number of boys. Find the number of boys and girls who took part in the quiz.
Variables and Given Equations
x = boys; y = girls
x + y = 10
y = x + 4
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 3; y = 7.
4.1 Q.5 — Pencils and pens
Guided Practice
Question
Five pencils and seven pens together cost ₹50, whereas seven pencils and five pens together cost ₹46. Find the cost of one pencil and one pen.
Variables and Given Equations
x = cost of a pencil (₹); y = cost of a pen (₹)
5x + 7y = 50
7x + 5y = 46
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 3; y = 5.
4.1 Q.6 — Garden dimensions
Guided Practice
Question
Half the perimeter of a rectangular garden is 36 m. If its length is 4 m more than its width, find the dimensions of the garden.
Variables and Given Equations
x = length (m); y = width (m)
x + y = 36
x = y + 4
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 20; y = 16.
4.1 Q.7 — Rectangle area
Guided Practice
Question
The area of a rectangle is reduced by 80 square units when its length is reduced by 5 units and its breadth is increased by 2 units. If the length is increased by 10 units and the breadth is decreased by 5 units, the area increases by 50 square units. Find the length and breadth of the rectangle.
Variables and Given Equations
x = length; y = breadth (units)
2x − 5y = −70
−5x + 10y = 100
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 40; y = 30.
These come from (x − 5)(y + 2) = xy − 80 and (x + 10)(y − 5) = xy + 50.
4.1 Q.9 — Students and benches
Guided Practice
Question
In a class, if three students sit on each bench, one student will be left without a seat. If four students sit on each bench, one bench will be left unoccupied. Find the number of students and benches in the class.
Variables and Given Equations
x = students; y = benches
x − 3y = 1
x − 4y = −4
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 16; y = 5.
4.2 Q.1 — Income and expenditure
Guided Practice
Question
The ratio of the incomes of two persons is 9 : 7, and the ratio of their expenditures is 4 : 3. If each of them saves ₹2,000 per month, find their monthly incomes.
Variables and Given Equations
Incomes are ₹9x and ₹7x; expenditures are ₹4y and ₹3y.
9x − 4y = 2000
7x − 3y = 2000
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 2000; y = 4000.
Monthly incomes: ₹18,000 and ₹14,000.
4.2 Q.2 — Reversed digits
Guided Practice
Question
The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?
Variables and Given Equations
x = tens digit; y = units digit. Solve both cases.
x + y = 6
Case 1: x − y = 2; Case 2: y − x = 2
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 4; y = 2.
Case 1: 42. Case 2: x = 2, y = 4, giving 24. Two possible numbers.
4.2 Q.3 — Supplementary angles
Guided Practice
Question
The larger of two supplementary angles exceeds the smaller angle by 18°. Find the two angles.
Variables and Given Equations
x = larger angle; y = smaller angle (degrees)
x + y = 180
x − y = 18
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 99; y = 81.
4.2 Q.4 — Taxi charges
Guided Practice
Question
Taxi charges in Hyderabad consist of a fixed charge and an additional charge based on the distance travelled. A minimum fixed amount is charged for the first 3 km. After that, an additional amount is charged for every kilometre travelled. For a journey of 10 km, the total charge is ₹166. For a journey of 15 km, the total charge is ₹256. i. Find the fixed charge and the charge per kilometre after the first 3 km. ii. How much will a person have to pay for travelling a distance of 25 km?
Variables and Given Equations
x = fixed charge for first 3 km (₹); y = charge per additional km (₹). Hypothetical exercise.
x + 7y = 166
x + 12y = 256
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 40; y = 18.
For 25 km: 40 + (25 − 3) × 18 = ₹436.
4.2 Q.5 — Fraction
Guided Practice
Question
A fraction becomes 4/5 when 1 is added to both its numerator and denominator. If 5 is subtracted from both its numerator and denominator, the fraction becomes 1/2. Find the original fraction.
Variables and Given Equations
x = numerator; y = denominator
5x − 4y = −1
2x − y = 5
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 7; y = 9.
Original fraction: 7/9. Derived from (x + 1)/(y + 1) = 4/5 and (x − 5)/(y − 5) = 1/2.
4.2 Q.6 — Car speeds
Guided Practice
Question
Places A and B are 100 km apart on a highway. One car starts from A and another car starts from B at the same time, travelling at different speeds. If the cars travel in the same direction, they meet after 5 hours. If they travel towards each other, they meet after 1 hour. Find the speeds of the two cars.
Variables and Given Equations
x = faster speed; y = slower speed (km/h)
x − y = 20
x + y = 100
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 60; y = 40.
4.2 Q.7 — Complementary angles
Guided Practice
Question
Two angles are complementary. The larger angle is 3° less than twice the measure of the smaller angle. Find the measure of each angle.
Variables and Given Equations
x = smaller angle; y = larger angle (degrees)
x + y = 90
y = 2x − 3
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 31; y = 59.
4.2 Q.8 — Dictionary pages
Guided Practice
Question
A dictionary has a total of 1,382 pages. It is divided into two parts. The second part has 64 pages more than the first part. How many pages are there in each part of the book?
Variables and Given Equations
x = pages in first part; y = pages in second part
x + y = 1382
y = x + 64
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 659; y = 723.
4.2 Q.9 — Solution mixture
Guided Practice
Question
A chemist has two solutions of hydrochloric acid in stock. One is a 50% solution and the other is an 80% solution. How much of each solution should be used to obtain 100 ml of a 68% solution?
Variables and Given Equations
x = volume of 50% solution; y = volume of 80% solution (ml). Mathematics exercise.
x + y = 100
0.50x + 0.80y = 68
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 40; y = 60.
4.2 Q.10 — Investment
Guided Practice
Question
You have savings of ₹12,000 and want to invest the amount in two schemes offering annual interest rates of 10% and 15%. How much should be invested in each scheme to earn an overall annual interest rate of 12%?
Variables and Given Equations
x = amount at 10%; y = amount at 15% (₹). Hypothetical annual simple interest.
x + y = 12000
0.10x + 0.15y = 1440
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
x = 7200; y = 4800.
4.3 Q.1(i) — Reciprocal substitution
Challenge Practice
Question
Solve the pair of equations by reducing them to a pair of linear equations: 5/(x − 1) + 1/(y − 2) = 2 and 6/(x − 1) − 3/(y − 2) = 1.
Variables and Given Equations
Let A = 1/(x − 1), B = 1/(y − 2). x ≠ 1; y ≠ 2.
5A + B = 2
6A − 3B = 1
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
A = 1/3; B = 1/3.
A = 1/3, B = 1/3. Therefore original x = 4, y = 5.
4.3 Q.1(iv) — Product equations
Challenge Practice
Question
Solve the equations 6x + 3y = 6xy and 2x + 4y = 5xy. Check whether any solutions are excluded when dividing by xy.
Variables and Given Equations
Original: 6x + 3y = 6xy; 2x + 4y = 5xy. For nonzero x, y, let A = 1/x, B = 1/y.
3A + 6B = 6
4A + 2B = 5
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
A = 1; B = 1/2.
A = 1, B = 1/2, giving original x = 1, y = 2. Also (0, 0) satisfies the original equations and is excluded by division by xy.
4.3 Q.2(i) — Boat and stream
Challenge Practice
Question
A boat travels 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can travel 40 km upstream and 55 km downstream. Determine the speed of the stream and the speed of the boat in still water.
Variables and Given Equations
Let A = 1/(x − y), B = 1/(x + y); x = boat speed, y = stream speed. x > y ≥ 0.
30A + 44B = 10
40A + 55B = 13
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
A = 1/5; B = 1/11.
Upstream speed 5 km/h; downstream speed 11 km/h. Boat speed 8 km/h; stream speed 3 km/h.
4.3 Q.2(ii) — Train and car
Challenge Practice
Question
Rahim travels 600 km to his home, partly by train and partly by car. He takes 8 hours when he travels 120 km by train and the remaining distance by car. He takes 20 minutes longer when he travels 200 km by train and the remaining distance by car. Find the speeds of the train and the car.
Variables and Given Equations
Let A = 1/x, B = 1/y; x = train speed, y = car speed (km/h).
120A + 480B = 8
200A + 400B = 25/3
Solve using substitution or elimination. Verify your answer in both equations.
Check Answer
A = 1/60; B = 1/80.
Train speed 60 km/h; car speed 80 km/h.
4.3 Q.2(iii) — Work rates
Challenge Practice
Question
Two women and five men can together complete an embroidery work in 4 days, while three women and six men can complete it in 3 days. Find the time taken by one woman working alone and one man working alone to complete the work.
Variables and Given Equations
x = work done by one woman per day; y = work done by one man per day
2x + 5y = 1/4
3x + 6y = 1/3
Solve using substitution or elimination. Verify your answer in both equations.
