Percentages

• Percentage means “per hundred” and is represented using the symbol %.
• A percentage compares a part to the whole, assuming the whole is 100.
• Percentages are used to compare quantities, calculate change, and describe growth or reduction.
• Key conversions:
  • Percentage to fraction: divide by 100
  • Percentage to decimal: divide by 100
  • Decimal to percentage: multiply by 100
• Finding a percentage of a quantity:
  Percentage of a quantity = (Percentage ÷ 100) × Quantity

Practice Questions:

• Find 15% of 240
• Convert 0.375 to a percentage
• Convert 62% into a fraction in simplest form

Percentage Increase and Decrease

• Percentage change compares how much a value has increased or decreased relative to the original value.
• Percentage Increase: final value is greater than the original value
• Percentage Decrease: final value is less than the original value
• Formula for percentage change:
  Percentage Change = (Change ÷ Original Value) × 100
• New value after increase:
  New Value = Original × (1 + percentage ÷ 100)
• New value after decrease:
  New Value = Original × (1 − percentage ÷ 100)

Practice Questions:

• A price increases from ₹800 to ₹920. Find the percentage increase.
• A population of 12,000 decreases by 8%. Find the new population.
• A jacket costing ₹2,500 is discounted by 20%. What is the sale price?

Simple Interest

• Simple interest is calculated only on the original principal.
• The interest earned each year remains the same.
• Formula:
  Simple Interest (SI) = (P × R × T) ÷ 100
  • P = Principal (initial amount)
  • R = Rate of interest (% per year)
  • T = Time (in years)
• Total amount:
  Amount = Principal + Simple Interest

Practice Questions:

• Find the simple interest on ₹4,000 at 5% per year for 3 years.
• How much interest is earned on ₹12,000 at 6% for 2 years?
• Find the total amount after 4 years if ₹7,500 is invested at 8% simple interest.

Compound Interest

• Compound interest is calculated on the principal plus accumulated interest.
• Interest grows faster than simple interest over time.
• Formula:
  Amount = P × (1 + R ÷ 100)^T
• Compound Interest:
  CI = Amount − Principal
• The exponent T represents the number of compounding periods (usually years).

Practice Questions:

• Calculate the compound interest on ₹5,000 at 10% per year for 2 years.
• Find the total amount after 3 years if ₹8,000 is invested at 5% compound interest.
• Explain why compound interest earns more than simple interest over time.

Direct and Inverse Variation

• Direct Variation: both quantities increase or decrease together.
  y ∝ x → y = kx
• Inverse Variation: one quantity increases while the other decreases.
  y ∝ 1/x → y = k/x
• k is called the constant of variation.
• Graphs:
  • Direct variation: straight line through the origin
  • Inverse variation: curved graph (hyperbola)

Practice Questions:

• If y ∝ x and y = 18 when x = 6, find y when x = 10.
• If y ∝ 1/x and y = 4 when x = 3, find y when x = 6.
• State whether the relationship between speed and time (for a fixed distance) is direct or inverse.

Time Zones, Clocks and Timetables

• The Earth is divided into time zones based on longitude.
• There are 360° of longitude and 24 hours in a day.
• Key fact:
  15° of longitude = 1 hour
• Moving east: add time
• Moving west: subtract time
• UTC (Coordinated Universal Time) is the reference time zone.

Practice Questions:

• If it is 14:00 UTC, what time is it at UTC +5:30?
• A city is 45° east of Greenwich. How many hours ahead is it?
• A flight leaves London at 09:00 and takes 8 hours to reach India (UTC +5:30). Find the arrival time.

Distance–Time Graph

• A distance–time graph shows how distance changes with time.
• Time is plotted on the horizontal axis (x-axis).
• Distance is plotted on the vertical axis (y-axis).
• Interpretation:
  • Steep line: faster speed
  • Flat line: object is stationary
  • Curved line: changing speed
• No slope calculations are required for interpretation in this unit.

Practice Questions:

• Describe the motion represented by a horizontal line on a distance–time graph.
• Which object is moving faster if one graph is steeper than another?
• Explain what a curved distance–time graph indicates about speed.

Task Overview

• This task assesses your understanding of:
  • Percentages
  • Simple Interest
  • Compound Interest
  • Averages (Mean)
  • Real-life financial decision making
• You will choose a dream college or university and plan how you manage your money as a student.
• All calculations must be shown clearly. Final answers must include correct units.

Part A: Student Banking Options

You must choose ONE of the following student banking plans:

Option 1: Student Simple Interest Savings Plan

• Interest Type: Simple Interest
• Interest Rate: 5% per year
• Interest is calculated only on the principal.
• Suitable for short-term savings or predictable income.
• Formula:
  Simple Interest (SI) = (P × R × T) ÷ 100

Option 2: Student Compound Interest Savings Plan

• Interest Type: Compound Interest (compounded annually)
• Interest Rate: 4% per year
• Interest is calculated on the principal and previously earned interest.
• Suitable for long-term savings.
• Formula:
  Amount = P × (1 + R ÷ 100)^T
• Compound Interest = Amount − Principal

Student Task:

• Assume you deposit ₹________ as your initial savings.
• Assume the time period is ________ years.
• Calculate the total amount after the given time.
• Explain which banking option is better for your situation and why.

Part B: Dormitory Cost and Averages

• You are sharing a dorm room with other students.
• The monthly dorm costs for each student are:

• Student A: ₹________ per month
• Student B: ₹________ per month
• Student C: ₹________ per month
• (Optional) Student D: ₹________ per month

• Average (Mean) Formula:
  Average = (Sum of all values) ÷ (Number of values)

Student Task:

• Calculate the average monthly dorm cost.
• State what this average represents in context.
• Explain one advantage of using the average cost.

Part C: Transportation Costs and Percentage Discounts

Cost per kilometre (before discount):

• Car: ₹12 per km
• Bus: ₹6 per km
• Bicycle: ₹1.50 per km (maintenance cost)
• Walking: ₹0 per km

Student Transport Discounts:

• Bus pass: 25% discount on total cost
• Carpool discount (car): 15% discount
• Bicycle & walking: no discount needed

Jobs:

Find an available job which you are able to do whilst attending aforementioned university

Account this into finances

Student Task:

• Estimate the distance from your home/dorm to college: ________ km.
• Calculate the daily transport cost for each option.
• Apply any potential percentage discounts.
• Decide which transport option is the most cost-effective.
• Justify your choice using calculations.
• Create a pie chart of total expenditure

Part D: Final Evaluation (Extended Response)

• Summarise your financial plan as a student.
• State:
  • Chosen banking option
  • Average dorm cost
  • Chosen transport method
• Explain how you managed percentages, averages, and interest
• Use correct mathematical reasoning

UNIT 3: ALGEBRA

Sequences

• A sequence is an ordered list of numbers that follow a rule or pattern.
• Arithmetic Sequences:
  • The difference between consecutive terms is constant.
  • This constant difference is called the common difference (d).
  • nth term formula:
    a_n = a + (n − 1)d
  • a = first term, n = term number
• Geometric Sequences:
  • Each term is obtained by multiplying the previous term by a constant.
  • This constant is called the common ratio (r).
  • nth term formula:
    a_n = a × r^{(n − 1)}
• Sequences can be increasing, decreasing, or constant.

Practice Questions:

• Find the 10th term of the arithmetic sequence: 3, 7, 11, …
• Write the nth term of the sequence: 2, 6, 18, …
• State whether the sequence −5, −2, 1, 4 is arithmetic or geometric.

Simplifying Algebraic Expressions

• Algebraic expressions consist of numbers, variables, and operations.
• Linear Expressions:
  • Highest power of the variable is 1.
  • Example: 3x + 5 − 2x
• Quadratic Expressions:
  • Highest power of the variable is 2.
  • Example: 2x² + 5x − 3
• Like terms must have the same variable raised to the same power.
• Simplification involves collecting like terms and applying correct signs.

Practice Questions:

• Simplify: 4x + 7 − 2x − 5
• Simplify: 3x² + 5x − 2x² − x
• State whether 5x and 5x² are like terms.

Factorisation

• Factorisation is the reverse process of expansion.
• Common Factor:
  • Take out the highest common factor (HCF).
  • Example: 6x + 12 = 6(x + 2)
• Algebraic Identities:
  • (a + b)² = a² + 2ab + b²
  • (a − b)² = a² − 2ab + b²
  • a² − b² = (a + b)(a − b)
• Splitting the Middle Term:
  • Used for factorising quadratics of the form ax² + bx + c.
  • The middle term is split into two terms whose product is ac and sum is b.

Practice Questions:

• Factorise: 8x − 12
• Factorise: x² + 7x + 10
• Factorise using identities: 9y² − 16

Algebraic Fractions

• Algebraic fractions contain variables in the numerator and/or denominator.
• Simplification involves factorising both numerator and denominator.
• Common factors are cancelled.
• Denominators must never be equal to zero.

Practice Questions:

• Simplify: (6x² ÷ 3x)
• Simplify: (x² − 9) ÷ (x + 3)
• State the value of x for which the expression is undefined.

Linear and Simultaneous Equations

• Linear Equations:
  • An equation with the highest power of the variable equal to 1.
  • Solved by isolating the variable.
• Changing the Subject:
  • Rearranging a formula to make a different variable the subject.
• Simultaneous Equations:
  • Solved using substitution, elimination, or graphical methods.
  • The solution satisfies both equations.

Practice Questions:

• Solve: 3x − 7 = 11
• Make x the subject: y = 4x + 3
• Solve simultaneously: x + y = 10 and x − y = 2

UNIT 4: ALGEBRA AND GRAPHS

Cartesian Plane

• The Cartesian plane consists of two perpendicular number lines.
• The horizontal axis is the x-axis and the vertical axis is the y-axis.
• The point (0, 0) is called the origin.
• Coordinates are written as (x, y).
• The plane is divided into four quadrants.

Linear Graphs

• Linear graphs form straight lines.
• They represent linear equations with two variables.
• Each point on the line satisfies the equation.

Slope–Intercept Form

• Equation: y = mx + c
• m is the gradient (rate of change).
• c is the y-intercept.
• Gradient formula:
  m = (y₂ − y₁) ÷ (x₂ − x₁)

Types of Slopes

• Positive slope: y increases as x increases.
• Negative slope: y decreases as x increases.
• Zero slope: horizontal line (y = constant).
• Infinite slope: vertical line (x = constant).

Inequalities on Graphs

• Inequalities show a range of solutions.
• Solid line for ≤ or ≥
• Dotted line for < or >
• The shaded region represents all possible solutions.

Statistics

• Mean: (Sum of values) ÷ (Number of values)
• Median: Middle value when data is ordered
• Mode: Most frequent value
• Used to describe and compare data sets.

Stem and Leaf Diagram

• Organises numerical data.
• Shows frequency while keeping original values.
• Key must always be included.

Cumulative Frequency

• Running total of frequencies.
• Used to find median, quartiles, and interquartile range.

Probability

• 0 ≤ P(E) ≤ 1
• Formula:
  P(E) = favourable outcomes ÷ total outcomes

Types of Events

• Mutually exclusive: cannot happen at the same time.
• Independent: one event does not affect the other.

Tree Diagrams

• Used for multi-stage probability.
• Probabilities along branches are multiplied.
• Total probability is found by adding outcomes.

Sets and Venn Diagrams

• Sets are collections of well-defined elements.
• Union (A ∪ B): all elements in A or B.
• Intersection (A ∩ B): common elements.
• Complement (A′): elements not in the set.