Grade 7 CBSE Mathematics Comprehensive Study Guide and Practice Paper

Structure and How to Use This Study Guide

• The material covers six key chapters of Grade 7 CBSE Mathematics:
  • 1. Large Numbers Around Us
  • 2. Arithmetic Expressions
  • 3. Peak Beyond the Point (Fractions & Decimals)
  • 4. Expressions Using Letters and Numbers
  • 5. Parallel and Intersecting Lines
  • 6. The Tale of the Three Intersecting Lines (Triangles)
• Note: Triangle construction is strictly excluded from the test paper.
• Recommended Method of Study:
  • Read and thoroughly understand the concept and definitions first.
  • Work through the step-by-step example problems.
  • Solve the "TRY IT" quick check problems independently before checking the given answers.
  • Utilize the Rapid Revision Sheet and Memory Summary prior to taking tests.
  • Always show complete working for multi-step calculations in test papers.
• Core Mathematical Rules to Master:
  • Arithmetic Hierarchy: Brackets $\rightarrow$ Multiplication / Division $\rightarrow$ Addition / Subtraction.
  • Fraction Operations: Find a common denominator for addition and subtraction; multiply by the reciprocal for division.
  • Algebraic Simplification: Combine like terms only.
  • Geometric Angle Rules: Straight line angle sum = $180^\circ$; total angles around a point = $360^\circ$.
  • Triangle Property: Interior angle sum of any triangle = $180^\circ$.

Chapter 1: Large Numbers Around Us

• Indian Place-Value System:

  • Numbers are grouped into distinct periods: Ones, Thousands, Lakhs, and Crores.
  • Commas are inserted to separate periods and assist in reading large numbers easily.
  • Place Value Structure:
  • Ones Period: Contains ones, tens, and hundreds ($1$, $10$, $100$).
  • Thousands Period: Contains thousands and ten-thousands ($1,000$; $10,000$; $1,00,000$).
  • Lakhs Period: Contains lakhs and ten-lakhs ($1,00,000$; $10,00,000$).
  • Crores Period: Contains crores and ten-crores ($1,00,00,000$ and above).
  • 

• Reading and Comparing Large Numbers:

  • Example: $7,05,43,218$ is written and read as "Seven crore five lakh forty-three thousand two hundred eighteen".
  • Comparison Rules:
  1. Compare the total number of digits first. The number with more digits is greater.
  2. If the number of digits is equal, compare digits starting from the leftmost place value.
  3. Ascending Order: Arranging numbers from smallest to greatest.
  4. Descending Order: Arranging numbers from greatest to smallest.

• Rounding Off Numbers:

  • Look at the digit immediately to the right of the place value being rounded to.
  • If that digit is $5$ or greater ($\ge 5$): Increase the rounding digit by $1$ (round up).
  • If that digit is less than $5$ ($< 5$): Keep the rounding digit unchanged (round down).
  • Example: Round $6,48,732$ to the nearest thousand. The digit in the hundreds place is $7$ ($\ge 5$), so the thousands digit $8$ becomes $9$. The rounded number is $6,49,000$.

• Try It Questions & Solutions:

  • Question 1: Round $83,649$ to the nearest thousand.
  • Question 2: Which is greater: $4,98,000$ or $5,02,000$?
  • Answers:
  • 1. $84,000$ (since the hundreds digit $6 \ge 5$).
  • 2. $5,02,000$ (comparing the leftmost digits, $5 > 4$).

Chapter 2: Arithmetic Expressions

• Order of Operations (BODMAS / PEMDAS):

  • 1. Brackets: Evaluate expressions inside brackets first.
  • 2. Multiplication and Division: Perform multiplication and division next. When both occur at the same level, work from left to right.
  • 3. Addition and Subtraction: Perform addition and subtraction last. Work from left to right.
  • Example without brackets:     $18 + 6 \times 4 = 18 + 24 = 42$
  • Example with brackets:     $(18 + 6) \times 4 = 24 \times 4 = 96$

• Fractions in Expressions:

  • For addition and subtraction, find a common denominator before combining numerators.     $\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$
  • For division, multiply the first fraction by the reciprocal of the second fraction.     $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

• Decimals in Expressions:

  • Align decimal points vertically when adding or subtracting decimal values.     $4.5 + 2.35 = 6.85$
  • For multiplication, multiply as whole numbers first and place the decimal point so that the product has as many decimal places as the sum of decimal places in the factors.

• Common Mistakes to Avoid:

  • Performing addition before multiplication (ignoring operator precedence).
  • Ignoring or misinterpreting brackets.
  • Misplacing decimal points during addition, subtraction, or multiplication.
  • Failing to simplify fractions to their simplest terms.

• Try It Questions & Solutions:

  • Question 1: Evaluate $(12 + 8) \div 4 + 3 \times 2$.
  • Question 2: Evaluate $20 - 3 \times 4 + 8 \div 2$.
  • Answers:
  • 1. $(12 + 8) \div 4 + 3 \times 2 = 20 \div 4 + 6 = 5 + 6 = 11$.
  • 2. $20 - 3 \times 4 + 8 \div 2 = 20 - 12 + 4 = 8 + 4 = 12$.

Chapter 3: Peak Beyond the Point (Fractions and Decimals)

• Fractions Fundamentals:

  • In a fraction represented as $\frac{a}{b}$, $a$ is the numerator and $b$ is the denominator ($b \neq 0$).
  • Equivalent Fractions: Fractions that represent the same value or proportion.     $\frac{3}{4} = \frac{6}{8} = \frac{9}{12}$

• Fraction Arithmetic:

  • Addition and Subtraction: Requires finding a common denominator for unlike fractions.     $\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}$
  • Multiplication: Multiply numerators together and denominators together.     $\frac{3}{5} \times \frac{10}{9} = \frac{30}{45} = \frac{2}{3}$
  • Division: Multiply the dividend by the reciprocal of the divisor.

• Decimals Fundamentals:

  • Decimal places include tenths ($\frac{1}{10}$), hundredths ($\frac{1}{100}$), and thousandths ($\frac{1}{1000}$).
  • Converting decimal to fraction:     $0.75 = \frac{75}{100} = \frac{3}{4}$
  • Trailing zeros after a decimal point do not change its value:     $0.5 = 0.50$

• Comparing Decimals and Fractions:

  • To compare decimals, compare place values from left to right starting with the whole number part, then tenths, hundredths, etc.
  • To compare fractions, convert them to equivalent fractions with a common denominator or convert them to decimal form.

• Try It Questions & Solutions:

  • Question 1: Convert $0.625$ to a fraction in simplest form.
  • Question 2: Find $\frac{3}{4} - \frac{1}{8}$.
  • Question 3: Find $2.6 \times 0.4$.
  • Answers:
  • 1. $0.625 = \frac{625}{1000} = \frac{5}{8}$.
  • 2. $\frac{3}{4} - \frac{1}{8} = \frac{6}{8} - \frac{1}{8} = \frac{5}{8}$.
  • 3. $2.6 \times 0.4 = 1.04$.

Chapter 4: Expressions Using Letters and Numbers

• Key Vocabulary:

  • Variable: A symbol (usually a letter such as $x$, $y$, $n$, $a$) representing an unknown number or quantity.
  • Constant: A fixed numerical value.
  • Coefficient: The numerical factor of a variable; in $7x$, $7$ is the coefficient.
  • Term: A part of an expression separated by addition ($+$) or subtraction (-$).\n - *Example*: In 4x + 3y - 7$, the terms are$4x$,$3y$, and$-7.\n\n- **Like and Unlike Terms**:\n - **Like Terms**: Terms with the exact same variable part. Only like terms can be combined.\n    5x + 2x = 7x\n - **Unlike Terms**: Terms with different variables or powers. They cannot be combined. 5x$and$2y are unlike terms.\n\n- **Substitution**:\n - Replace the variable with a given numerical value to compute the expression.\n - *Example*: If x = 4:\n    3x + 2 = 3(4) + 2 = 12 + 2 = 14\n\n- **Writing Expressions from Verbal Descriptions**:\n - \n - "5$more than$x$"$\rightarrow$$x + 5\n - "3$times$y$"$\rightarrow$$3y\n - "7$less than$n$"$\rightarrow$$n - 7\n - "Twice x$plus$4$"$\rightarrow$$2x + 4\n\n- **Try It Questions & Solutions**:\n - Question 1: Simplify 6a + 3 - 2a\n - Question 2: If a = 5, find the value.\n - *Answers*:\n - 1. Combine like terms: (6a - 2a) + 3 = 4a + 3.\n - 2. Substitute a = 5$:$4(5) + 3 = 20 + 3 = 23.\n\n# Chapter 5: Parallel and Intersecting Lines\n\n- **Basic Definitions**:\n - **Intersecting Lines**: Lines that cross each other and meet at a single point.\n - **Parallel Lines**: Lines that do not meet even when extended infinitely.\n - **Perpendicular Lines**: Lines that intersect at a right angle (90^\circ).\n - **Transversal**: A line that crosses two or more lines.\n\n- **Angles Formed by Intersecting Lines**:\n - **Vertically Opposite Angles**: Angles opposite each other when two lines intersect. They are always equal.\n - **Adjacent Angles on a Straight Line**: Angles lying next to each other on a straight line; their sum is 180^\circ.\n - *Example*: If one angle at an intersection is 65^\circ:\n - Its vertically opposite angle is 65^\circ.\n - Each adjacent angle on the straight line is 180^\circ - 65^\circ = 115^\circ.\n\n- **Parallel Lines Cut by a Transversal**:\n - **Corresponding Angles**: Equal in measure.\n - **Alternate Interior Angles**: Equal in measure.\n - **Interior Angles on the Same Side (Consecutive Interior)**: Add up to 180^\circ (supplementary).\n\n- **Important Angle Classification & Values**:\n - \n - Right angle = 90^\circ\n - Straight angle = 180^\circ\n - Angles around a point = 360^\circ\n - Complementary angles: Sum = 90^\circ\n - Supplementary angles: Sum = 180^\circ\n\n- **Try It Questions & Solutions**:\n - Question 1: One angle at an intersection is 48^\circ. Find the other three angles.\n - Question 2: A corresponding angle is 72^\circ. What is its corresponding partner?\n - *Answers*:\n - 1. 132^\circ$,$48^\circ$,$132^\circ\n - 2. 72^\circ\n\n# Chapter 6: The Tale of the Three Intersecting Lines (Triangles)\n\n- **Triangle Basics**:\n - A triangle is a closed 2D figure with 3 sides, 3 vertices, and 3 interior angles.\n - *Exam Notice*: Triangle construction is excluded from test evaluation.\n\n- **Classification of Triangles**:\n - *By Side Lengths*:\n - **Equilateral Triangle**: All 3 sides are equal; each interior angle equals 60^\circ.\n - **Isosceles Triangle**: 2 sides are equal; angles opposite equal sides are equal.\n - **Scalene Triangle**: All 3 sides have different lengths.\n - *By Angle Measures*:\n - **Acute-Angled Triangle**: All interior angles are less than 90^\circ.\n - **Right-Angled Triangle**: One interior angle equals 90^\circ.\n - **Obtuse-Angled Triangle**: One interior angle is greater than 90^\circ.\n\n- **Angle-Sum Property**:\n - The sum of the three interior angles of a triangle is always 180^\circ.\n - *Example*: 55^\circ + 65^\circ + x = 180^\circ \implies 120^\circ + x = 180^\circ \implies x = 60^\circ$.

• Exterior Angle Property:

  • An exterior angle of a triangle is equal to the sum of its two interior opposite angles.
  • Example: 50^\circ + 70^\circ = 120^\circ$.\n\n- **Triangle Inequality Theorem**:\n - The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.\n - *Example*: Can side lengths 4\,\text{cm}$,$5\,\text{cm}$, and$10\,\text{cm} form a triangle?\n - Check: 4 + 5 = 9\,\text{cm}$. Since$9 < 10, these side lengths cannot form a triangle.\n\n- **Try It Questions & Solutions**:\n - Question 1: Can 5\,\text{cm}$,$6\,\text{cm}$, and$8\,\text{cm} form a triangle?\n - Question 2: Angles are x^\circ$,$2x^\circ$, and$3x^\circ$. Find$x$.
  • Answers:
  • 1. Yes ($5+6=11 > 8$, $5+8=13 > 6$, $6+8=14 > 5$).
  • 2. x + 2x + 3x = 180^\circ \implies 6x = 180^\circ \implies x = 30^\circ$.\n\n# Mixed Worked Examples\n\n- **Large Numbers**:\n - Write in figures: 3 crore 7 lakh 5 thousand 40.\n - *Solution*: 3,07,05,040.\n\n- **Arithmetic**:\n - Evaluate 20 - 3 \times 4 + 8 \div 2\n - *Solution*: Perform multiplication and division first: 20 - 12 + 4 = 8 + 4 = 12.\n\n- **Fractions**:\n - Evaluate \frac{3}{4} - \frac{1}{8}\n - *Solution*: Common denominator is 8$:$\frac{6}{8} - \frac{1}{8} = \frac{5}{8}.\n\n- **Algebra**:\n - If p = 6$, evaluate$4p - 7\n - *Solution*: 4(6) - 7 = 24 - 7 = 17.\n\n- **Lines**:\n - Two adjacent angles on a straight line are (3x + 10)^\circ$and$2x^\circ$. Find$x$.
  • Solution: (3x + 10)^\circ + 2x^\circ = 180^\circ \implies 5x + 10 = 180 \implies 5x = 170 \implies x = 34$.\n\n- **Triangle**:\n - Angles are x^\circ$,$2x^\circ$, and$3x^\circ. Find the angles.\n - *Solution*: x + 2x + 3x = 180^\circ \implies 6x = 180^\circ \implies x = 30^\circ$. The angles are$30^\circ$,$60^\circ$, and$90^\circ.\n\n# Exam Checklist\n\n- Always show complete step-by-step working for multi-step questions.\n- Keep decimal points aligned vertically in addition and subtraction.\n- State the exact geometric property or theorem used in line and angle calculations.\n- Double-check positive and negative signs during algebraic operations.\n- Verify that the sum of interior angles in any triangle problem equals 180^\circ$.

Rapid Revision Summary & Memory Sheet

• 
• Key Chapter Reminders:
  • Large Numbers: Place value structure; compare from left; rounding uses the next digit.
  • Arithmetic: Brackets \rightarrow \times/\div \rightarrow +/-$.\n - **Fractions**: Common denominator for addition/subtraction; reciprocal for division.\n - **Decimals**: 0.5 = 0.50; align decimal points for addition/subtraction.\n - **Algebra**: Combine like terms only; substitute carefully.\n - **Lines**: Vertically opposite angles are equal; angles on a straight line sum to 180^\circ$.
  • Parallel Lines: Corresponding/alternate interior angles equal; same-side interior angles sum to 180^\circ$.\n - **Triangles**: Interior angles sum to 180^\circ; exterior angle equals two opposite interior angles added.\n - **Triangle Inequality**: Any two sides together must be greater than the third side.\n\n- **6-Question Rapid Self-Test**:\n - 1. Round 9,64,721 to the nearest ten thousand.\n - 2. Evaluate 7 + 3 \times 5$.
  • 3. Find \frac{2}{5} + \frac{1}{10}$.\n - 4. If x = 8$, find$5x - 6$.
  • 5. One angle at an intersection is $117^\circ$. Find its vertically opposite angle.
  • 6. A triangle has angles $45^\circ$ and $65^\circ$. Find the third angle.
  • Answers:
  • 1. $9,60,000$
  • 2. $22$
  • 3. $\frac{1}{2}$
  • 4. $34$
  • 5. $117^\circ$
  • 6. $70^\circ$

25-Mark Practice Question Paper

• Exam Specifications:

  • Grade: Grade 7 CBSE Mathematics
  • Time Allowed: 45 minutes
  • Maximum Marks: 25
  • Instructions: Answer all questions. Show working where required. Triangle construction is not tested.

• Section A (1 Mark Each = 5 Marks Total):

  • 1. Write $7,04,08,015$ in words. [1]
  • 2. Evaluate: $6 + 4 \times 3$. [1]
  • 3. Convert $0.75$ into a fraction in simplest form. [1]
  • 4. If $x = 5$, find $3x + 2$. [1]
  • 5. What is the angle between perpendicular lines? [1]

• Section B (2 Marks Each = 8 Marks Total):

  • 6. Arrange in ascending order: $5,40,000$; $5,04,000$; $5,44,000$; $5,40,400$. [2]
  • 7. Simplify: $\frac{3}{4} + \frac{1}{8} - \frac{1}{2}$. [2]
  • 8. Simplify: $7a + 4 - 3a + 2$. [2]
  • 9. Two adjacent angles on a straight line are $(3x + 10)^\circ$ and $2x^\circ$. Find $x$ and both angles. [2]

• Section C (3 Marks Each = 12 Marks Total):

  • 10. A shop has $2,50,000$ notebooks. It receives $75,000$ more and then sells $1,25,500$. How many remain? Show your calculation. [3]
  • 11. Evaluate: $(18 + 6) \div 3 + 2 \times 5$. Explain the order of operations used. [3]
  • 12. A triangle has angles $(x + 20)^\circ$, $2x^\circ$, and $(x + 40)^\circ$. Find $x$ and all three angles. [3]
  • 13. Two parallel lines are cut by a transversal. One angle is $68^\circ$. Find (a) a corresponding angle and (b) an adjacent angle on a straight line. State the properties used. [3]