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CBSE Class 6 NCERT Maths Exercise with Solution

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Chapter 1: Knowing Our Numbers

Exercise 1.1

  1. Write the place value of 5 in the number 37,215.
    • Solution: The place value of 5 in the number 37,215 is 500.
  2. Write the number 6,897 in expanded form.
    • Solution: 6,897 = 6 × 1000 + 8 × 100 + 9 × 10 + 7 × 1 = 6000 + 800 + 90 + 7 = 6,897

Exercise 1.2

  1. Write the following numbers in words:
    • 56,321
    • Solution: Fifty-six thousand, three hundred twenty-one.
  2. Write the number 89,405 in expanded form.
    • Solution: 89,405 = 8 × 10,000 + 9 × 1000 + 4 × 100 + 0 × 10 + 5 × 1 = 80,000 + 9,000 + 400 + 5 = 89,405

Exercise 1.3

  1. Compare the following numbers using greater than (>) or less than (<):
    • 46,381 __ 46,193
    • Solution: 46,381 > 46,193 (because 46,381 has 8 in the hundred’s place, whereas 46,193 has 1)
  2. Write the smallest and largest numbers from the following set:
    • 7,890; 8,908; 9,879
    • Solution: Smallest number = 7,890; Largest number = 9,879

Exercise 1.4

  1. Fill in the blanks with correct numbers:
    • (i) The greatest 5-digit number is 99,999.
    • (ii) The smallest 6-digit number is 100,000.
  2. Write a 5-digit number in which the thousands place is 6, and the hundreds place is 3.
    • Solution: One such number could be 63,000.

Exercise 1.5

  1. Round off the following numbers to the nearest hundred:
    • 6,478
    • Solution: 6,478 rounded off to the nearest hundred is 6,500.
  2. Round off the following numbers to the nearest thousand:
    • 24,752
    • Solution: 24,752 rounded off to the nearest thousand is 25,000.

Exercise 1.6

  1. Write the number 347,586 in words.
    • Solution: Three hundred forty-seven thousand, five hundred eighty-six.
  2. Write the following in expanded form:
    • 526,903
    • Solution: 526,903 = 5 × 100,000 + 2 × 10,000 + 6 × 1,000 + 9 × 100 + 0 × 10 + 3 × 1 = 500,000 + 20,000 + 6,000 + 900 + 3 = 526,903

Exercise 1.7

  1. Find the place value of 7 in 74,689.
    • Solution: The place value of 7 in 74,689 is 70,000.
  2. In a 5-digit number, the place value of 4 is 40,000. What is the number?
    • Solution: The number is 40,000 (since the place value of 4 is 40,000).

Conclusion

These solutions to the exercises in Chapter 1: Knowing Our Numbers will help you to understand key concepts like place value, expanded form, comparing numbers, rounding off, and writing numbers in words. It’s important to practice these problems to gain a solid understanding of the concepts and to develop problem-solving skills.

Make sure to regularly practice these exercises, and you’ll be confident in applying these skills to higher-level topics in mathematics.

Chapter 2: Whole Numbers

Exercise 2.1

  1. What is the successor of 12,345?
    • Solution: The successor of a number is obtained by adding 1 to the number.
    • Successor of 12,345 = 12,345 + 1 = 12,346.
  2. What is the predecessor of 78,902?
    • Solution: The predecessor of a number is obtained by subtracting 1 from the number.
    • Predecessor of 78,902 = 78,902 – 1 = 78,901.

Exercise 2.2

  1. Write the following numbers in ascending order:
    • 3,514; 3,050; 3,625; 3,400
    • Solution:
      • Ascending order: 3,050; 3,400; 3,514; 3,625.
  2. Write the following numbers in descending order:
    • 9,870; 9,000; 8,976; 9,010
    • Solution:
      • Descending order: 9,870; 9,010; 8,976; 9,000.

Exercise 2.3

  1. Which is the smallest whole number?
    • Solution: The smallest whole number is 0.
  2. Which is the largest whole number?
    • Solution: There is no largest whole number because whole numbers are infinite. They go on forever.

Exercise 2.4

  1. Find the sum of 6,215 and 3,745.
    • Solution:
      • 6,215 + 3,745 = 9,960.
  2. Find the sum of 10,021 and 2,379.
    • Solution:
      • 10,021 + 2,379 = 12,400.

Exercise 2.5

  1. Find the difference between 5,678 and 3,234.
    • Solution:
      • 5,678 – 3,234 = 2,444.
  2. Find the difference between 9,876 and 4,543.
    • Solution:
      • 9,876 – 4,543 = 5,333.

Exercise 2.6

  1. Multiply 256 by 20.
    • Solution:
      • 256 × 20 = 5,120.
  2. Multiply 453 by 12.
    • Solution:
      • 453 × 12 = 5,436.

Exercise 2.7

  1. Divide 864 by 8.
    • Solution:
      • 864 ÷ 8 = 108.
  2. Divide 1,512 by 12.
    • Solution:
      • 1,512 ÷ 12 = 126.

Summary of Key Concepts in Chapter 2: Whole Numbers

Chapter 3: Playing with Numbers

Exercise 3.1

  1. Write the following numbers in prime factorization form:
    • 36
    • Solution:
      • Prime factorization of 36: 36 = 2 × 2 × 3 × 3 = 2² × 3²
  2. Write the following numbers in prime factorization form:
    • 72
    • Solution:
      • Prime factorization of 72: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

Exercise 3.2

  1. Find the LCM (Least Common Multiple) of 6 and 8.
    • Solution:
      • The multiples of 6: 6, 12, 18, 24, 30, 36, …
      • The multiples of 8: 8, 16, 24, 32, …
      • The LCM of 6 and 8 is 24.
  2. Find the LCM of 4 and 9.
    • Solution:
      • The multiples of 4: 4, 8, 12, 16, 20, 24, …
      • The multiples of 9: 9, 18, 27, 36, …
      • The LCM of 4 and 9 is 36.

Exercise 3.3

  1. Find the HCF (Highest Common Factor) of 24 and 36.
    • Solution:
      • The factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
      • The factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
      • The common factors are 1, 2, 3, 4, 6, 12.
      • The HCF of 24 and 36 is 12.
  2. Find the HCF of 45 and 75.
    • Solution:
      • The factors of 45: 1, 3, 5, 9, 15, 45
      • The factors of 75: 1, 3, 5, 15, 25, 75
      • The common factors are 1, 3, 5, 15.
      • The HCF of 45 and 75 is 15.

Exercise 3.4

  1. Find the LCM and HCF of 15 and 20 using the prime factorization method.
    • Solution:
      • Prime factorization of 15: 15 = 3 × 5
      • Prime factorization of 20: 20 = 2² × 5
      • To find the LCM, take the highest power of each prime factor: LCM = 2² × 3 × 5 = 60
      • To find the HCF, take the lowest power of the common prime factor: HCF = 5.

Chapter 4: Basic Geometrical Ideas

Exercise 4.1

  1. Draw a rough sketch of the following shapes:
    • Square
    • Rectangle
    • Triangle
    • Circle
    • Solution: (Draw simple sketches for each shape as per the descriptions)
      • Square: A shape with 4 equal sides and 4 right angles.
      • Rectangle: A shape with 4 sides, opposite sides are equal, and 4 right angles.
      • Triangle: A 3-sided polygon.
      • Circle: A shape with all points equidistant from the center.

Exercise 4.2

  1. How many lines of symmetry does a square have?
    • Solution: A square has 4 lines of symmetry.
  2. How many lines of symmetry does a rectangle have?
    • Solution: A rectangle has 2 lines of symmetry.

Exercise 4.3

  1. Draw a quadrilateral and label its sides.
    • Solution: (Draw any quadrilateral, e.g., a parallelogram, trapezium, etc., and label the sides as ABCD).
      • Example: A quadrilateral with sides AB, BC, CD, and DA.
  2. How many edges does a cube have?
    • Solution: A cube has 12 edges.

Exercise 4.4

  1. Name the following polygons based on their number of sides:
    • 3 sides = Triangle
    • 4 sides = Quadrilateral
    • 5 sides = Pentagon
    • 6 sides = Hexagon
    • 7 sides = Heptagon
  2. Draw a triangle and a hexagon.
    • Solution: (Draw a triangle with 3 sides and a hexagon with 6 sides).

Chapter 5: Understanding Fractions

Exercise 5.1

  1. Write the following fractions in their simplest form:
    • 12/36
    • Solution:
      • To simplify 12/36, divide both the numerator and denominator by their HCF (12).
      • 12 ÷ 12 = 1 and 36 ÷ 12 = 3.
      • So, 12/36 = 1/3.
  2. Write the following fractions in their simplest form:
    • 18/24
    • Solution:
      • To simplify 18/24, divide both the numerator and denominator by their HCF (6).
      • 18 ÷ 6 = 3 and 24 ÷ 6 = 4.
      • So, 18/24 = 3/4.

Exercise 5.2

  1. Add the following fractions:
    • 1/4 + 2/4
    • Solution:
      • Since the denominators are the same, add the numerators.
      • 1/4 + 2/4 = (1 + 2)/4 = 3/4.
  2. Add the following fractions:
    • 3/8 + 1/8
    • Solution:
      • Since the denominators are the same, add the numerators.
      • 3/8 + 1/8 = (3 + 1)/8 = 4/8 = 1/2 (after simplifying).

Exercise 5.3

  1. Subtract the following fractions:
    • 5/6 – 2/6
    • Solution:
      • Since the denominators are the same, subtract the numerators.
      • 5/6 – 2/6 = (5 – 2)/6 = 3/6 = 1/2 (after simplifying).
  2. Subtract the following fractions:
    • 7/9 – 2/9
    • Solution:
      • Since the denominators are the same, subtract the numerators.
      • 7/9 – 2/9 = (7 – 2)/9 = 5/9.

Exercise 5.4

  1. Multiply the following fractions:
    • 2/5 × 3/7
    • Solution:
      • Multiply the numerators: 2 × 3 = 6.
      • Multiply the denominators: 5 × 7 = 35.
      • So, 2/5 × 3/7 = 6/35.
  2. Multiply the following fractions:
    • 4/9 × 2/3
    • Solution:
      • Multiply the numerators: 4 × 2 = 8.
      • Multiply the denominators: 9 × 3 = 27.
      • So, 4/9 × 2/3 = 8/27.

Exercise 5.5

  1. Divide the following fractions:
    • 2/3 ÷ 4/5
    • Solution:
      • To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
      • 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6 (after simplifying).
  2. Divide the following fractions:
    • 7/8 ÷ 3/4
    • Solution:
      • 7/8 ÷ 3/4 = 7/8 × 4/3 = 28/24 = 7/6 (after simplifying).

Chapter 6: Decimal Numbers

Exercise 6.1

  1. Write the following decimals in words:
    • 5.36
    • Solution: Five point three six.
  2. Write the following decimals in words:
    • 12.09
    • Solution: Twelve point zero nine.

Exercise 6.2

  1. Convert the following decimals into fractions:
    • 0.75
    • Solution:
      • 0.75 = 75/100 = 3/4 (after simplifying).
  2. Convert the following decimals into fractions:
    • 0.25
    • Solution:
      • 0.25 = 25/100 = 1/4 (after simplifying).

Exercise 6.3

  1. Add the following decimals:
    • 1.25 + 3.75
    • Solution:
      • 1.25 + 3.75 = 5.00 or simply 5.
  2. Add the following decimals:
    • 0.65 + 0.45
    • Solution:
      • 0.65 + 0.45 = 1.10.

Exercise 6.4

  1. Subtract the following decimals:
    • 4.50 – 1.25
    • Solution:
      • 4.50 – 1.25 = 3.25.
  2. Subtract the following decimals:
    • 6.75 – 2.80
    • Solution:
      • 6.75 – 2.80 = 3.95.

Exercise 6.5

  1. Multiply the following decimals:
    • 0.6 × 0.4
    • Solution:
      • 0.6 × 0.4 = 0.24.
  2. Multiply the following decimals:
    • 1.25 × 2.4
    • Solution:
      • 1.25 × 2.4 = 3.00.

Exercise 6.6

3.6 ÷ 0.9 = 4.

Divide the following decimals:

2.4 ÷ 0.6

Solution:

2.4 ÷ 0.6 = 4.

Divide the following decimals:

3.6 ÷ 0.9

Solution:

4.

Chapter 7: Ratio and Proportion

Exercise 7.1

  1. Write the ratio of 15 kg to 20 kg in the simplest form.
    • Solution:
      • To write the ratio in its simplest form, divide both terms by their greatest common divisor (GCD).
      • GCD of 15 and 20 is 5.
      • So, 15:20 = 15 ÷ 5 : 20 ÷ 5 = 3:4.
  2. Write the ratio of 8 minutes to 2 hours in the simplest form.
    • Solution:
      • First, convert 2 hours into minutes: 2 hours = 120 minutes.
      • The ratio of 8 minutes to 120 minutes is 8:120.
      • To simplify, divide both terms by their GCD, which is 8.
      • So, 8:120 = 8 ÷ 8 : 120 ÷ 8 = 1:15.

Exercise 7.2

  1. Find the ratio of the number of boys to the number of girls in a class with 24 boys and 16 girls.
    • Solution:
      • The ratio of boys to girls is 24:16.
      • To simplify, divide both terms by their GCD, which is 8.
      • So, 24:16 = 24 ÷ 8 : 16 ÷ 8 = 3:2.
  2. A recipe requires 3 parts of sugar to 4 parts of flour. If we use 12 parts of flour, how much sugar is required?
    • Solution:
      • The ratio of sugar to flour is 3:4.
      • If 4 parts of flour require 3 parts of sugar, then 12 parts of flour will require:
        • (12 × 3) ÷ 4 = 9 parts of sugar.

Exercise 7.3

  1. The ratio of the number of apples to oranges is 5:3. If there are 40 apples, how many oranges are there?
    • Solution:
      • The ratio of apples to oranges is 5:3.
      • Let the number of oranges be x.
      • Using the ratio, we set up the proportion:
        53=40x\frac{5}{3} = \frac{40}{x}35​=x40​
      • Cross-multiply to solve for x:
        5x = 3 × 40
        5x = 120
        x = 120 ÷ 5
        x = 24 oranges.
  2. The ratio of boys to girls in a class is 7:8. If there are 56 boys, how many girls are there?
    • Solution:
      • The ratio of boys to girls is 7:8.
      • Let the number of girls be y.
      • Using the ratio, we set up the proportion:
        78=56y\frac{7}{8} = \frac{56}{y}87​=y56​
      • Cross-multiply to solve for y:
        7y = 8 × 56
        7y = 448
        y = 448 ÷ 7
        y = 64 girls.

Chapter 8: Understanding Simple Equations

Exercise 8.1

  1. Solve the equation:
    • x + 5 = 12
    • Solution:
      • To find x, subtract 5 from both sides of the equation.
      • x + 5 – 5 = 12 – 5
      • x = 7
  2. Solve the equation:
    • 3x = 18
    • Solution:
      • To find x, divide both sides of the equation by 3.
      • 3x ÷ 3 = 18 ÷ 3
      • x = 6

Exercise 8.2

  1. Solve the equation:
    • x – 7 = 10
    • Solution:
      • To find x, add 7 to both sides of the equation.
      • x – 7 + 7 = 10 + 7
      • x = 17
  2. Solve the equation:
    • 5x = 35
    • Solution:
      • To find x, divide both sides of the equation by 5.
      • 5x ÷ 5 = 35 ÷ 5
      • x = 7

Exercise 8.3

  1. Solve the equation:
    • 4x – 3 = 13
    • Solution:
      • First, add 3 to both sides of the equation.
      • 4x – 3 + 3 = 13 + 3
      • 4x = 16
      • Now, divide both sides by 4.
      • 4x ÷ 4 = 16 ÷ 4
      • x = 4
  2. Solve the equation:
    • 7x + 2 = 23
    • Solution:
      • First, subtract 2 from both sides of the equation.
      • 7x + 2 – 2 = 23 – 2
      • 7x = 21
      • Now, divide both sides by 7.
      • 7x ÷ 7 = 21 ÷ 7
      • x = 3

Exercise 8.4

  1. Solve the equation:
    • x/5 = 3
    • Solution:
      • Multiply both sides of the equation by 5 to isolate x.
      • (x/5) × 5 = 3 × 5
      • x = 15
  2. Solve the equation:
    • 2x + 7 = 15
    • Solution:
      • First, subtract 7 from both sides.
      • 2x + 7 – 7 = 15 – 7
      • 2x = 8
      • Now, divide both sides by 2.
      • 2x ÷ 2 = 8 ÷ 2
      • x = 4

Chapter 9: Geometry

Exercise 9.1

  1. Name the different types of angles based on their measure.
    • Solution:
      • Acute angle: An angle less than 90°.
      • Right angle: An angle equal to 90°.
      • Obtuse angle: An angle greater than 90° but less than 180°.
      • Straight angle: An angle equal to 180°.
      • Reflex angle: An angle greater than 180° but less than 360°.
  2. Draw the following angles:
    • Right angle
    • Acute angle
    • Obtuse angle
    • Solution:
      • Draw a right angle (90°), acute angle (less than 90°), and obtuse angle (greater than 90° but less than 180°) using a protractor.

Exercise 9.2

  1. How many lines of symmetry does a rectangle have?
    • Solution:
      • A rectangle has 2 lines of symmetry. These lines are the vertical and horizontal lines passing through the center.
  2. How many lines of symmetry does an equilateral triangle have?
    • Solution:
      • An equilateral triangle has 3 lines of symmetry, which are the medians from each vertex to the opposite side.

Exercise 9.3

  1. Identify the following as types of polygons:
    • 3 sides = Triangle
    • 4 sides = Quadrilateral
    • 5 sides = Pentagon
    • 6 sides = Hexagon
    • 7 sides = Heptagon
    • 8 sides = Octagon
  2. Draw a quadrilateral and label its sides.
    • Solution:
      • A quadrilateral is any four-sided figure. You can draw a square, rectangle, rhombus, or any irregular four-sided shape and label the sides as AB, BC, CD, DA.

Exercise 9.4

  1. Draw a circle and name its parts.
    • Solution:
      • A circle consists of:
        • Centre: The middle point of the circle.
        • Radius: The distance from the center to any point on the circle.
        • Diameter: The longest chord of the circle, passing through the center.
        • Circumference: The boundary or perimeter of the circle.
  2. What is the difference between a line and a ray?
    • Solution:
      • Line: A straight path that extends infinitely in both directions.
      • Ray: A straight path that starts at a point and extends infinitely in one direction.

Chapter 10: Mensuration

Exercise 10.1

  1. Find the perimeter of a rectangle whose length is 8 cm and breadth is 5 cm.
    • Solution:
      • The perimeter of a rectangle is given by the formula:
        Perimeter = 2 × (Length + Breadth)
      • Substituting the given values:
        Perimeter = 2 × (8 + 5) = 2 × 13 = 26 cm.
  2. Find the perimeter of a square whose side is 6 cm.
    • Solution:
      • The perimeter of a square is given by the formula:
        Perimeter = 4 × Side
      • Substituting the given value:
        Perimeter = 4 × 6 = 24 cm.

Exercise 10.2

  1. Find the area of a rectangle whose length is 10 cm and breadth is 7 cm.
    • Solution:
      • The area of a rectangle is given by the formula:
        Area = Length × Breadth
      • Substituting the given values:
        Area = 10 × 7 = 70 cm².
  2. Find the area of a square whose side is 9 cm.
    • Solution:
      • The area of a square is given by the formula:
        Area = Side²
      • Substituting the given value:
        Area = 9² = 81 cm².

Exercise 10.3

  1. Find the area of a triangle whose base is 8 cm and height is 5 cm.
    • Solution:
      • The area of a triangle is given by the formula:
        Area = 1/2 × Base × Height
      • Substituting the given values:
        Area = 1/2 × 8 × 5 = 20 cm².
  2. Find the area of a right-angled triangle with base 12 cm and height 9 cm.
    • Solution:
      • The area of a triangle is given by the formula:
        Area = 1/2 × Base × Height
      • Substituting the given values:
        Area = 1/2 × 12 × 9 = 54 cm².

Exercise 10.4

  1. Find the area of a circle with radius 7 cm.
    • Solution:
      • The area of a circle is given by the formula:
        Area = π × Radius²
      • Substituting the value of radius and taking π ≈ 22/7:
        Area = (22/7) × 7² = (22/7) × 49 = 154 cm².
  2. Find the circumference of a circle with radius 6 cm.
    • Solution:
      • The circumference of a circle is given by the formula:
        Circumference = 2 × π × Radius
      • Substituting the value of radius and taking π ≈ 22/7:
        Circumference = 2 × (22/7) × 6 = 75.43 cm.

Chapter 11: Algebra

Exercise 11.1

  1. Find the value of the variable in the following expressions:
    • 3x = 12
    • Solution:
      • To find the value of x, divide both sides of the equation by 3.
      • 3x ÷ 3 = 12 ÷ 3
      • x = 4
  2. Find the value of the variable in the following expressions:
    • 5y = 35
    • Solution:
      • To find the value of y, divide both sides of the equation by 5.
      • 5y ÷ 5 = 35 ÷ 5
      • y = 7

Exercise 11.2

  1. Solve the equation:
    • x + 6 = 14
    • Solution:
      • To find x, subtract 6 from both sides of the equation.
      • x + 6 – 6 = 14 – 6
      • x = 8
  2. Solve the equation:
    • y – 9 = 15
    • Solution:
      • To find y, add 9 to both sides of the equation.
      • y – 9 + 9 = 15 + 9
      • y = 24

Exercise 11.3

  1. Solve the equation:
    • 3x + 5 = 20
    • Solution:
      • First, subtract 5 from both sides.
      • 3x + 5 – 5 = 20 – 5
      • 3x = 15
      • Now, divide both sides by 3.
      • 3x ÷ 3 = 15 ÷ 3
      • x = 5
  2. Solve the equation:
    • 4y – 3 = 21
    • Solution:
      • First, add 3 to both sides.
      • 4y – 3 + 3 = 21 + 3
      • 4y = 24
      • Now, divide both sides by 4.
      • 4y ÷ 4 = 24 ÷ 4
      • y = 6

Exercise 11.4

  1. Find the value of x:
    • 2x – 7 = 13
    • Solution:
      • First, add 7 to both sides.
      • 2x – 7 + 7 = 13 + 7
      • 2x = 20
      • Now, divide both sides by 2.
      • 2x ÷ 2 = 20 ÷ 2
      • x = 10
  2. Find the value of y:
    • 3y + 4 = 19
    • Solution:
      • First, subtract 4 from both sides.
      • 3y + 4 – 4 = 19 – 4
      • 3y = 15
      • Now, divide both sides by 3.
      • 3y ÷ 3 = 15 ÷ 3
      • y = 5

Chapter 12: Ratio and Proportion (Advanced)

Exercise 12.1

  1. In a class, the ratio of the number of girls to the number of boys is 3:4. If there are 24 girls, how many boys are there?
    • Solution:
      • The ratio of girls to boys is 3:4.
      • Let the number of boys be x.
      • Set up the proportion:
        34=24x\frac{3}{4} = \frac{24}{x}43​=x24​
      • Cross-multiply to solve for x:
        3x = 4 × 24
        3x = 96
        x = 96 ÷ 3
        x = 32 boys.
  2. The ratio of the number of apples to oranges is 5:7. If there are 35 oranges, how many apples are there?
    • Solution:
      • The ratio of apples to oranges is 5:7.
      • Let the number of apples be y.
      • Set up the proportion:
        57=y35\frac{5}{7} = \frac{y}{35}75​=35y​
      • Cross-multiply to solve for y:
        5 × 35 = 7 × y
        175 = 7y
        y = 175 ÷ 7
        y = 25 apples.

Exercise 12.2

  1. In a recipe, the ratio of sugar to flour is 2:3. If 120 grams of flour is used, how much sugar is needed?
    • Solution:
      • The ratio of sugar to flour is 2:3.
      • Let the amount of sugar be x.
      • Set up the proportion:
        23=x120\frac{2}{3} = \frac{x}{120}32​=120x​
      • Cross-multiply to solve for x:
        2 × 120 = 3 × x
        240 = 3x
        x = 240 ÷ 3
        x = 80 grams of sugar.
  2. The ratio of the cost of a pencil to an eraser is 5:2. If the cost of an eraser is ₹18, what is the cost of a pencil?
    • Solution:
      • The ratio of the cost of a pencil to an eraser is 5:2.
      • Let the cost of the pencil be y.
      • Set up the proportion:
        52=y18\frac{5}{2} = \frac{y}{18}25​=18y​
      • Cross-multiply to solve for y:
        5 × 18 = 2 × y
        90 = 2y
        y = 90 ÷ 2
        y = ₹45 (cost of a pencil).

Exercise 12.3

  1. If the ratio of the number of boys to girls in a school is 7:9, and there are 35 boys, how many girls are there?
    • Solution:
      • The ratio of boys to girls is 7:9.
      • Let the number of girls be x.
      • Set up the proportion:
        79=35x\frac{7}{9} = \frac{35}{x}97​=x35​
      • Cross-multiply to solve for x:
        7x = 9 × 35
        7x = 315
        x = 315 ÷ 7
        x = 45 girls.
  2. In a garden, the ratio of the number of roses to daisies is 5:8. If there are 40 daisies, how many roses are there?
    • Solution:
      • The ratio of roses to daisies is 5:8.
      • Let the number of roses be y.
      • Set up the proportion:
        58=y40\frac{5}{8} = \frac{y}{40}85​=40y​
      • Cross-multiply to solve for y:
        5 × 40 = 8 × y
        200 = 8y
        y = 200 ÷ 8
        y = 25 roses.

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