Geometry: Shapes & Angles,polygons) GL ASSESSMENT QUESTIONS 11 plus exam part -1

 

Geometry: Shapes & Angles,polygons) GL ASSESSMENT QUESTIONS 11 plus exam part -1

Chapter: Geometry - Shapes, Angles & Polygons

Step 1: Understanding Angles

An angle is a measure of turn between two lines that meet at a point (the vertex).

• Types of Angles:

  • Acute Angle: Less than 90°.

  • Right Angle: Exactly 90° (often marked with a square in a corner).

  • Obtuse Angle: Greater than 90° but less than 180°.

  • Straight Line: Exactly 180°.

  • Reflex Angle: Greater than 180° but less than 360°.

  • Full Turn: Exactly 360°.

• Key Angle Facts:

  • Angles on a straight line add up to 180°.

    • e.g., If one angle is 115°, the other is 180 - 115 = 65°.

  • Angles around a point add up to 360°.

    • e.g., If three angles are 110°, 95°, and 80°, the missing angle is 360 - (110+95+80) = 75°.

  • Vertically opposite angles are equal. (Where two straight lines cross, the angles opposite each other are equal).

Step 2: Understanding Polygons

A polygon is a 2D shape with straight sides.

• Regular Polygon: All sides are the same length and all interior angles are equal (e.g., a square).

• Irregular Polygon: Sides and angles are not all equal.

• Common Polygons:

  • Triangle: 3 sides

  • Quadrilateral: 4 sides

  • Pentagon: 5 sides

  • Hexagon: 6 sides

  • Heptagon: 7 sides

  • Octagon: 8 sides

  • Nonagon: 9 sides

  • Decagon: 10 sides

• Key Polygon Facts:

  • Sum of Interior Angles:

    • Triangles always add up to 180°.

    • For any polygon, you can find the sum of interior angles by splitting it into triangles. The formula is: Sum of interior angles = (n - 2) × 180°, where n is the number of sides.

    • Example: A hexagon (6 sides): (6-2) × 180 = 4 × 180 = 720°.

  • Exterior Angles: The angles on the outside if you extend the sides. For any regular polygon, all exterior angles are equal.

    • Sum of exterior angles for ANY polygon is 360°.

    • To find one exterior angle of a regular polygon: 360° ÷ number of sides.

    • Example: A regular octagon: One exterior angle = 360 ÷ 8 = 45°.

Step 3: Special Triangles and Quadrilaterals

• Triangles:

  • Equilateral: All sides equal, all angles 60°.

  • Isosceles: Two sides equal, two angles equal.

  • Scalene: All sides and angles different.

  • Right-Angled: Has one 90° angle.

• Quadrilaterals:

  • Square: All sides equal, all angles 90°.

  • Rectangle: Opposite sides equal, all angles 90°.

  • Parallelogram: Opposite sides equal and parallel.

  • Rhombus: All sides equal, opposite angles equal (a "squashed" square).

  • Trapezium: One pair of parallel sides.

  • Kite: Two pairs of adjacent sides equal.

Step 4: Symmetry

• Line Symmetry: A shape has line symmetry if you can fold it in half and both sides match exactly. The fold line is the line of symmetry.

• Rotational Symmetry: A shape has rotational symmetry if it can be rotated (turned) about its centre and look the same in more than one position. The order of rotational symmetry is the number of times it fits into its own outline during a full 360° turn.

  • Example: A square has 4 lines of symmetry and rotational symmetry of order 4.

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Practice Questions (Modelled on GL Assessment Style)

Here are 50 questions covering all the sub-topics above.

Part A: Angles (Questions 1-15)

1. What is the size of the angle between the hands of a clock at 3 o'clock?

2. An angle is 34°. What is the size of its complement? (Complementary angles add to 90°).

3. An angle is 112°. What is the size of its supplement? (Supplementary angles add to 180°).

4. Calculate the size of angle *a* in this isosceles triangle. (Base angles are 40° each).

5. Two angles on a straight line are 3x and 2x. What is the value of x?

6. Angles around a point are 95°, 80°, 70°, and *y*. Find *y*.

7. In a right-angled triangle, one of the other angles is 25°. What is the third angle?

8. In an equilateral triangle, what is the size of each interior angle?

9. A reflex angle is 275°. What is the corresponding acute/obtuse angle?

10. In a parallelogram, one angle is 65°. What are the sizes of the other three angles?

Part B: Polygons (Questions 11-30)

11. What is the name of a polygon with 8 sides?

12. What is the sum of the interior angles of a pentagon?

13. A regular hexagon has interior angles of 120°. What is the size of one exterior angle?

14. How many sides does a regular polygon have if each interior angle is 135°?

15. How many sides does a regular polygon have if each exterior angle is 30°?

16. Is a square a regular polygon? Explain why.

17. The interior angles of a quadrilateral are 90°, 110°, and 85°. What is the fourth angle?

18. A polygon has its interior angles adding up to 900°. How many sides does it have?

19. What is the size of an exterior angle of a regular nonagon?

20. True or False: A rhombus is always a regular polygon.

21. A triangle has angles of x, x+10, and 50°. Find the value of x.

22. A hexagon can be divided into how many triangles from a single vertex?

23. What is the order of rotational symmetry of a regular pentagon?

24. How many lines of symmetry does a regular hexagon have?

25. What is the sum of the exterior angles of a heptagon?

26. An irregular octagon has seven angles of 150° each. What is the size of the eighth angle?

27. If one exterior angle of an isosceles triangle is 110°, what are the two possible sizes of the interior angles at the base?

28. A polygon has 15 sides. What is the sum of its interior angles?

29. The interior angle of a regular polygon is twice its exterior angle. How many sides does it have?

30. Three of the angles in a pentagon are 100°. The other two angles are equal. What is the size of one of these equal angles?

Part C: 2D Shapes & Properties (Questions 31-50)

31. How many pairs of parallel sides does a trapezium have?

32. What is the specific name for a quadrilateral with all sides equal and all angles 90°?

33. What type of triangle has no lines of symmetry?

34. A kite has one line of symmetry. If one of its angles is 90°, what could the other angles be? (Give one example).

35. How many right angles does a parallelogram have?

36. What is the difference between a rhombus and a square?

37. A shape has rotational symmetry of order 2 and 2 lines of symmetry. What could it be?

38. Draw a scalene triangle with one obtuse angle.

39. True or False: Every rectangle is a parallelogram.

40. What is the order of rotational symmetry of an isosceles triangle?

41. A quadrilateral has exactly two lines of symmetry and rotational symmetry of order 2. What is its name?

42. How many sides does a decagon have?

43. All rectangles are quadrilaterals. Are all quadrilaterals rectangles?

44. What is the size of one interior angle of a regular octagon?

45. A heptagon has how many diagonals? (A diagonal is a line connecting two non-adjacent vertices).

46. A shape is made by putting two equilateral triangles together. What is the name of the new quadrilateral formed?

47. What is the size of angle *b* in a right-angled isosceles triangle?

48. A polygon has an interior angle sum of 1800°. How many sides does it have?

49. True or False: A circle has infinite lines of symmetry.

50. A regular polygon has an exterior angle of 20°. What is the sum of its interior angles?

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10 Questions from Previous Year GL Assessment Style (with Solutions)

1. The diagram shows an isosceles triangle. The base angles are both 55°. What is the size of the third angle?

   • A) 55°

   • B) 60°

   • C) 70°

   • D) 80°

2. What is the name of a polygon in which the interior angles add up to 720°?

   • A) Pentagon

   • B) Hexagon

   • C) Heptagon

   • D) Octagon

3. A regular polygon has an exterior angle of 40°. How many sides does it have?

   • A) 7

   • B) 8

   • C) 9

   • D) 10

4. The sizes of three of the angles in a quadrilateral are 100°, 110°, and 85°. What is the size of the fourth angle?

   • A) 55°

   • B) 65°

   • C) 75°

   • D) 85°

5. What is the order of rotational symmetry of a rectangle?

   • A) 1

   • B) 2

   • C) 3

   • D) 4

6. The interior angle of a regular polygon is 150°. How many sides does it have?

   • A) 10

   • B) 12

   • C) 15

   • D) 18

7. The diagram shows a kite. One angle is 120° and another is 50°. What is the size of the smallest angle in the kite?

   • A) 50°

   • B) 60°

   • C) 70°

   • D) 80°

8. How many lines of symmetry does a regular pentagon have?

   • A) 3

   • B) 4

   • C) 5

   • D) 6

9. Two angles are supplementary. One angle is five times the size of the other. What is the size of the larger angle?

   • A) 30°

   • B) 120°

   • C) 150°

   • D) 160°

10. The exterior angle of an equilateral triangle is 120°. What is the sum of the exterior angles of the triangle?

    • A) 120°

    • B) 240°

    • C) 360°

    • D) 720°

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Answer Key & Solutions

Part A & B & C (Questions 1-50) -

Part A: Angles (Questions 1–10)

1. Angle between clock hands at 3 o’clock
At 3:00, hour hand at 3, minute hand at 12.
Each hour mark = 30° (360°/12).
From 12 to 3 → $3\times 30\mathrm{°}=90\mathrm{°}$.
Answer: 90°

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2. Complement of 34°
Complementary sum = 90°.
$90-34=56\mathrm{°}$.
Answer: 56°

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3. Supplement of 112°
Supplementary sum = 180°.
$180-112=68\mathrm{°}$.
Answer: 68°

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4. Isosceles triangle, base angles each 40°
Angles sum to 180°, so $a+40+40=180$ → $a+80=180$ → $a=100\mathrm{°}$.
Answer: 100°

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5. Two angles on a straight line: 3x and 2x
$3x+2x=180\mathrm{°}$ → $5x=180$ → $x=36$.
Answer: 36

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6. Angles around a point: 95°, 80°, 70°, y
Sum = 360°.
$95+80+70+y=360$ → $245+y=360$ → $y=115\mathrm{°}$.
Answer: 115°

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7. Right-angled triangle, one other angle = 25°
$90+25+\text{third}=180$ → $115+\text{third}=180$ → third = 65°.
Answer: 65°

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8. Equilateral triangle interior angle
Each angle = 60°.
Answer: 60°

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9. Reflex angle 275°, corresponding acute/obtuse angle
Reflex + other angle = 360°.
Other angle = $360-275=85\mathrm{°}$ (acute).
Answer: 85°

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10. Parallelogram with one angle 65°
Opposite angles equal, adjacent supplementary.
So angles: $65\mathrm{°},115\mathrm{°},65\mathrm{°},115\mathrm{°}$.
Answer: 65°, 115°, 65°, 115°

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Part B: Polygons (Questions 11–30)

11. Polygon with 8 sides
Octagon.
Answer: Octagon

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12. Sum interior angles of pentagon
Pentagon has 5 sides. Sum = $(5-2)\times 180=3\times 180=540\mathrm{°}$.
Answer: 540°

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13. Regular hexagon interior = 120°, exterior angle
Exterior = 180 - interior = $180-120=60\mathrm{°}$.
Answer: 60°

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14. Regular polygon, interior 135°, how many sides?
Exterior = $180-135=45\mathrm{°}$.
Number of sides $n=360\mathrm{/}45=8$.
Answer: 8

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15. Regular polygon exterior = 30°, sides?
$n=360\mathrm{/}30=12$.
Answer: 12

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16. Is a square a regular polygon?
Yes: all sides equal, all angles equal.
Answer: Yes — all sides and angles equal

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17. Quadrilateral angles 90°, 110°, 85°, fourth angle
Sum = 360°.
$90+110+85+x=360$ → $285+x=360$ → $x=75\mathrm{°}$.
Answer: 75°

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18. Interior sum = 900°, number of sides
$(n-2)\times 180=900$ → $n-2=5$ → $n=7$.
Answer: 7

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19. Exterior angle of regular nonagon (9 sides)
Exterior = $360\mathrm{/}9=40\mathrm{°}$.
Answer: 40°

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20. Rhombus always a regular polygon?
No — a rhombus has equal sides but not necessarily equal angles unless it’s a square.
Answer: False

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21. Triangle angles x, x+10, 50°, find x
$x+(x+10)+50=180$ → $2x+60=180$ → $2x=120$ → $x=60$.
Answer: 60

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22. Hexagon divided into triangles from one vertex
From one vertex: $n-3$ diagonals → $n-2$ triangles for interior sum. For hexagon (n=6), triangles = $4$ (not from one vertex — wait, actually they ask: "A hexagon can be divided into how many triangles from a single vertex?" From single vertex, you can draw diagonals to other non-adjacent vertices, giving triangles that include that vertex: number of triangles = $n-2=4$ triangles for interior sum, but in drawing from one vertex? Actually: A single vertex connects to n-2 others to form n-2 triangles? Misleading wording. Possibly they mean “dividing polygon into triangles by drawing non-intersecting diagonals from one vertex” — yes, that yields n-2 triangles.
For hexagon, $n-2=4$. But that’s from one vertex for all non-intersecting diagonals from it.
Answer: 4

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23. Rotational symmetry order of regular pentagon
Rotates into itself 5 times in 360°. Order = 5.
Answer: 5

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24. Lines of symmetry of regular hexagon
6 lines (through opposite vertices and through midpoints of opposite sides).
Answer: 6

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25. Sum exterior angles of a heptagon
Sum exterior angles of any polygon = 360°.
Answer: 360°

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26. Irregular octagon: seven angles 150° each, eighth angle
Sum interior = $(8-2)\times 180=1080\mathrm{°}$.
Seven angles sum = $7\times 150=1050$.
Eighth = $1080-1050=30\mathrm{°}$.
Answer: 30°

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27. Isosceles triangle, one exterior angle 110°, possible base interior angles
Exterior 110° → adjacent interior = 70°.
Case 1: 70° is vertex angle → base angles = $(180-70)\mathrm{/}2=55\mathrm{°}$ each.
Case 2: 70° is a base angle → vertex = 40°, other base = 70°.
So base angles possible: 55°, 55° or 70°, 70°. Wait, but the question asks for “two possible sizes of the interior angles at the base” — meaning in the two cases, the base angles are either (55°,55°) or (70°,70°)? But 70°,70° means vertex=40°, works fine. So possible base angle size = 55° or 70°.
Answer: 55° and 70° (two possible base angles depending on case)

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28. 15 sides, sum interior angles
Sum = $(15-2)\times 180=13\times 180=2340\mathrm{°}$.
Answer: 2340°

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29. Interior angle = 2 × exterior angle
Let exterior = e, interior = 2e, so $2e+e=180\mathrm{°}$ → $3e=180$ → $e=60\mathrm{°}$.
Number of sides $n=360\mathrm{/}60=6$.
Answer: 6

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30. Pentagon: three angles 100°, other two equal
Sum interior = 540°.
Three angles sum = 300°. Remainder = 240° for two equal angles → each = 120°.
Answer: 120°

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Part C: 2D Shapes & Properties (Questions 31–50)

31. Trapezium parallel sides
A trapezium (UK) has exactly 1 pair of parallel sides.
Answer: 1 pair

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32. Quadrilateral, all sides equal, all angles 90°
Square.
Answer: Square

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33. Triangle with no lines of symmetry
Scalene triangle (all sides different, no symmetry).
Answer: Scalene triangle

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34. Kite, one line symmetry, one angle 90°, example others
In kite, symmetry line through unequal angles; if 90° is at one of equal-angle pair?
Possible: 90°, 90°, x, y with x=y? No, kite has two equal angles. Let’s say 90° and 90° are the unequal ones? Not possible — unequal angles are not necessarily 90°. Better: 90°, 100°, 90°, 80° (90°,90° are adjacent around line of symmetry? Not both equal? A kite has one pair equal angles.) But with 90° as one angle, others could be e.g. 90°, 110°, 70°, 90° — but that’s two 90°. Acceptable. One example: 90°, 90°, 120°, 60° (not possible sum 360). Let’s compute:
Example: 90°, 90°, 100°, 80° sum=360. Equal angles could be 90° each (the pair of equal angles are opposite? no, adjacent). So: 90°, 90°, then the other pair equal 100°, 80°? That’s not equal pair — so must be 90° and 90° are the equal ones → other two unequal sum 180: e.g. 100°, 80°. Works.
Answer: e.g. 90°, 90°, 100°, 80°

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35. Right angles in a parallelogram
A general parallelogram has 0 right angles unless it’s a rectangle.
Answer: 0 (unless rectangle)

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36. Difference rhombus and square
Both have all sides equal, but square has all angles 90°, rhombus does not necessarily.
Answer: A square has all angles 90°, rhombus doesn’t

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37. Rotational symmetry order 2, 2 lines of symmetry
Rectangle.
Answer: Rectangle

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38. Draw scalene triangle with one obtuse angle
Cannot draw here, but description: all sides different, one angle > 90°.

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39. Every rectangle is a parallelogram
True — opposite sides parallel.
Answer: True

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40. Rotational symmetry order of isosceles triangle
Order 1 (no rotation except 360°), unless equilateral.
Answer: 1

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41. Quadrilateral, 2 lines symmetry, order 2 rotational
Rhombus (rectangle has order 2 but 2 lines symmetry for non-square? Actually square has 4 lines. Rectangle has 2 lines if not square. Rhombus if not square has 2 lines symmetry? Yes, diagonals are symmetry lines, order 2 rotation). Possible also oblong rectangle? But oblong rectangle fits: 2 lines symmetry through midpoints, rotational order 2. They might be thinking of rhombus, but rectangle also fits. But rhombus’s lines of symmetry are diagonals (if square) only if angles 90°? Wait, in a rhombus (not square), lines of symmetry are diagonals? No, a rhombus that’s not square has no lines of symmetry? Actually a rhombus has 2 lines of symmetry only if it’s a square? Correction: kite has one line. I think they intend rectangle (non-square).
Let’s check:
Rectangle: 2 lines (midpoints), order 2 rotation.
Rhombus (non-square): 2 lines of symmetry through opposite vertices, order 2 rotation.
Both fit, but a rhombus (non-square) has axes along diagonals, yes, and order 2 rotation.
Common exam answer: Rhombus.
Answer: Rhombus

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42. Decagon sides
10 sides.
Answer: 10

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43. All rectangles are quadrilaterals, are all quadrilaterals rectangles?
No — quadrilaterals include trapeziums, kites, etc.
Answer: No

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44. Interior angle of regular octagon
Each interior = $180-(360\mathrm{/}8)=180-45=135\mathrm{°}$.
Answer: 135°

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45. Heptagon diagonals
Number of diagonals = $n(n-3)\mathrm{/}2=7\times 4\mathrm{/}2=14$.
Answer: 14

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46. Two equilateral triangles together (edge sharing) → quadrilateral
Putting base-to-base → rhombus (parallelogram with 60° and 120° angles).
Answer: Rhombus

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47. Right-angled isosceles triangle angles
90°, 45°, 45°. b likely refers to one base angle = 45°.
Answer: 45°

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48. Interior sum 1800°, sides
$(n-2)\times 180=1800$ → $n-2=10$ → $n=12$.
Answer: 12

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49. Circle lines of symmetry
Infinite lines through center.
Answer: True

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50. Regular polygon exterior = 20°, sum interior angles
$n=360\mathrm{/}20=18$ sides.
Sum interior = $(18-2)\times 180=16\times 180=2880\mathrm{°}$.
Answer: 2880°

10 GL Assessment Style Questions:

1. C) 70° (180 - 55 - 55 = 70)

2. B) Hexagon ((n-2)×180=720 → n-2=4 → n=6)

3. C) 9 (Number of sides = 360 ÷ 40 = 9)

4. B) 65° (Sum of quadrilateral angles=360°. 360 - (100+110+85)=65)

5. B) 2 (It looks the same in 2 positions: upright and at 180°)

6. B) 12 (Exterior angle = 180-150=30°. Sides=360/30=12)

7. A) 50° (In a kite, two angles are equal and two are different. The 50° angle will have a matching 50° angle. Total = 120+50+50 + x = 360 → x=140. The smallest angles are the two 50° ones).

8. C) 5

9. C) 150° (Let smaller angle = x. Larger = 5x. x + 5x = 180 → 6x=180 → x=30. Larger angle=5×30=150)

10. C) 360° (The sum of exterior angles for ANY polygon is always 360°).

PROPORTIONAL REASONING-1 FIGURE IT OUT QUESTIONS & ANSWERS

PROPORTIONAL REASONING-1 

KEY POINTS & FIGURE IT OUT 

 7.2 Ratios 

In a ratio of the form a : b, we can say that for every 'a' units of the first quantity, there are 'b' units of the second quantity. 

 A more systematic way to compare whether the ratios are proportional is to reduce them to their simplest form and see if these simplest forms are the same. 

7.3 Ratios in their Simplest Form 

 We can reduce ratios to their simplest form by dividing the terms by their HCF.

When two ratios are the same in their simplest forms, we say the ratios are proportional. 

 We use the symbol :: to show proportionality.

 Example: 60 : 40 :: 30 : 20 and 60 : 40 :: 90 : 60. 

 7.4 Problem Solving with Proportional Reasoning

Example 1: 

 Are the ratios 3 : 4 and 72 : 96 proportional?

 3 : 4 is already in its simplest form. To find the simplest form of 72 : 96, we need to divide both terms by their HCF. 

 The HCF of 72 and 96 is 24. Dividing both terms by 24, we get 3 : 4. Since both ratios in their simplest form are the same, they are proportional

 Example 2: 

 Kesang wanted to make lemonade for a celebration. She made 6 glasses of lemonade in a vessel and added 10 spoons of sugar to the drink. Her father expected more people to join the celebration. So he asked her to make 18 more glasses of lemonade. To make the lemonade with the same sweetness, how many spoons of sugar should she add? 6 : 10 :: 18 : ?

The ratio of glasses of lemonade to spoons of sugar is 6 : 10. 

6 : 10 :: 18 : x.

  6 : 10 :: 18 : x \[ \frac{6}{10} = \frac{18}{x} \implies x = \frac{18 \times 10}{6} = 30 \] 

So, she should use 30 spoons of sugar to make 18 glasses of lemonade with the same sweetness as earlier

Example 3: 

 Nitin and Hari were constructing a compound wall around their house. Nitin was building the longer side, 60 ft in length, and Hari was building the shorter side, 40 ft in length. Nitin used 3 bags of cement but Hari used only 2 bags of cement. Nitin was worried that the wall Hari built would not be as strong as the wall he built because she used less cement. Is Nitin correct in his thinking?

 The ratio in Nitin’s case is 60 : 3, That is , 20 : 1 (in its simplest form). 

The ratio in Hari’s case is 40 : 2, That is , 20 : 1 (in its simplest form). 

Since both ratios are proportional, the walls are equally strong. Nitin should not worry!

Example 4: 

  In my school, there are 5 teachers and 170 students. The ratio of teachers to students in my school is 5 : 170. Count the number of teachers and students in your school. What is the ratio of teachers to students in your school? Write it below. ______ : ______ Is the teacher-to-student ratio in your school proportional to the one in my school?

Example:

My school = 5:170

Your school = 6: 240

5:170::6:240no not in proportional

  Example 5: 

Measure the width and height (to the nearest cm) of the blackboard in your classroom. What is the ratio of width to height of the blackboard? ______ : ______ Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard? Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?

 Black board Width: 120 cm Height: 90 cm HCF = 30

120:90 = 4:3

Draw rectangle with Width = 8 cm

Height = (8÷4)×3=6 cm

Yes, in terms of shape. All rectangles drawn with a 4 : 3 ratio will have the same shape. They will look like scaled versions of each other—some may be larger, some smaller, but the proportion of width to height will be identical. This is the essence of similarity. they look different in size, Precision, drawing accuracy.

Example 6: 

When Neelima was 3 years old, her mother’s age was 10 times her age. What is the ratio of Neelima’s age to her mother’s age? What would be the ratio of their ages when Neelima is 12 years old? Would it remain the same?

The ratio of Neelima’s age to her mother’s age when Neelima is 3 years old is 3 : 30 (her mother’s age is 10 times Neelima’s age). 

In the simplest form, it is 1 : 10. 

When Neelima is 12 years old (, 9 years later), the ratio of their ages will be 12 : 39 (9 years later, her mother would be 39 years old).

 In the simplest form, it is 4 : 13. 

When we add (or subtract) the same number from the terms of a ratio, the ratio changes and is not necessarily proportional to the original ratio.

Example 7: 

Fill in the missing numbers for the following ratios that are proportional to 

14 : 21.

 ______ : 42 

6 : ______

 2 : ______What factor should we multiply 14 by to get 6? Can it be an integer? Or should it be a fraction? 

14:21::28:42

14y = 6

 Factor y = \( \frac{6}{14} = \frac{3}{7} \) → \( 21 \times \frac{3}{7} = 9 \) 

 Ratio = 6 : 9. 2. ? : 42 

 Factor = \( \frac{42}{21} = 2 \) → \( 14 \times 2 = 28 \) 

So, the ratio is 6 : 9. 

 Ratio = 28 : 42

 In the third ratio, the first term is 2. 

14 : 21 divide by 7 (HCF of 14 and 21)  → 2 : 3. 

So, the ratio is 2 : 3

  7.5 Filter Coffee Example 

Filter Coffee! Filter coffee is a beverage made by mixing coffee decoction with milk. Manjunath usually mixes 15 ml of coffee decoction with 35 ml of milk to make one cup of filter coffee in his coffee shop. In this case, we can say that the ratio of coffee decoction to milk is 15 : 35. If customers want ‘stronger’ filter coffee, Manjunath mixes 20 ml of decoction with 30 ml of milk. The ratio here is 20 : 30. Why is this coffee stronger? And when they want ‘lighter’ filter coffee, he mixes 10 ml of coffee and 40 ml of milk, making the ratio 10 : 40.Why is this coffee lighter? The following table shows the different ratios in which Manjunath mixes coffee decoction with milk. Write in the last column if the coffee is stronger or lighter than the regular coffee

 Why is the 20:30 coffee stronger?

Because the coffee-to-milk ratio 2:32:3 (≈0.667≈0.667) is larger than 3:73:7 (≈0.429≈0.429), meaning more coffee per unit of milk.

Why is the 10:40 coffee lighter?

Because the coffee-to-milk ratio 1:4 (=0.25) is smaller than 3:7 (≈0.429), meaning less coffee per unit of milk.

Regular coffee: 15 ml decoction + 35 ml milk → ratio 15 : 35 → simplest 3 : 7. 

 Stronger: 20 ml decoction + 30 ml milk → ratio 20 : 30 → simplest 2 : 3.

 Lighter: 10 ml decoction + 40 ml milk → ratio 10 : 40 → simplest 1 : 4. 

 Comparison: - 

2 : 3 (≈0.667) > 3 : 7 (≈0.429) → stronger. 

 1 : 4 (0.25) < 3 : 7 (≈0.429) → lighter. 

Coffee Decoction (in mL)	Milk (in mL)	Regular/Strong/Light
300	600	Stronger
150	500	Lighter
200	400	Stronger
24	56	Regular
100	300	Lighter

Figure it Out (Page 165-167)

 1. 1. Circle the following statements of proportion that are true. (i) 4 : 7 :: 12 : 21 (iii) 7 : 12 :: 12 : 7 (v) 12 : 18 :: 28 : 12 (ii) 8 : 3 :: 24 : 6 (iv) 21 : 6 :: 35 : 10 (vi) 24 : 8 :: 9 : 3

 Solution:(1) Given statement is 4:7:: 12:21.

This is true if 4/7 = 12/21 which is true. or if 4/7 = 4/7 ... Given statement is true,

(ii) Given statement is 8:3:: 24: 6.

This is true if 8/3 = 24/6 which is false. or if 8/3 = 4 ... Given statement is not true.

(iii) Given statement is 7:12::12:7.

This is true if which is false. Given statement is not true. 7/12 = 12/7

(iv) Given statement is 21:6:: 35:10.

This is true if or if 7/2 = 7/2 which is true. .. Given statement is true.

(v) Given statement is 12:18:28:12.

This is true if 12/18 = 28/12 or if 2/3 = 7/3 or 27, which is false.

Given statement is not true,

(vi) Given statement is 24:8::9:3.

This is true if 24/8 = 9/3 or if 33, which is true.... Both simplify to 3:1 Given statement is true,

(i) 4 : 7 :: 12 : 21 ✅ 

 (ii) 8 : 3 :: 24 : 6 ❌ 

 (iii) 7 : 12 :: 12 : 7 ❌ 

 (iv) 21 : 6 :: 35 : 10 ✅ 

 (v) 12 : 18 :: 28 : 12 ❌ 

 (vi) 24 : 8 :: 9 : 3 ✅ 

2. 2. Give 3 ratios that are proportional to 4 : 9. ______ : ______ ______ : ______ ______ : ______ 

 Multiply both terms by same factor: e.g., 8:18, 12:27, 20:45

4/9 = (4 * 2)/(9 * 2) = (4 * 3)/(9 * 3) = (4 * 4)/(9 * 4) 

4/9 = 8/18 = 12/27 = 16/36

4:9::8:18, 4:9::12:27 and 4:9::16:36

  8 : 18, 12 : 27, 20 : 45  

3. Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ______ 12 : ______ 20 : ______ 27 : ______

(1) Given ratio is 18: 24, let 18:24::3:x

18/24 = 3/x

3/4 = 3/x

x = 4

let 18:24:12: x. 

18/24 = 12/x or 

3x = 48 or or 

x = 48/3 = 16 

3/4 = 12/x

Missing number in the ratio 12: is 16.

(iii) let 18:24:20: x. 

or 3x = 80 

x = 80/3 

3/4 = 20/x 

18/24 = 20/x

Missing number in the ratio 20: _is 80/3 *

(iv) let 18:24:27:x. or 3x = 108 or Missing number in the ratio 27: is 36 

x = 108/3 = 36

 3/4 = 27/x * 

18/24 = 27/x





   3 : 4, 12 : 16, 20 : \( \frac{80}{3} \), 27 : 36 

  4.  look at the following rectangles. Which rectangles are similar to each other? You can verify this by measuring the width and height using a scale and comparing their ratios.

 Solution:

Using a scale, we measure the width and height of given rectangles.

The ratio 'Width: Height' for given rectangles A, B, C, D and E are respectively 1:3, 3:2, 9:4, 7:2 and 3:1. These ratios are all distinct. The ratios of A and E are 1:3 and 3: 1 respectively

Rectangles with same width : height ratio are similar. 

5. look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio in your notebooks? Compare your rectangles with your classmates’ drawings. Are all of them the same? If they are different from yours, can you think why? Are they wrong?

Width 32 mm and height 18 mm

Ratio is 32:18.

New width = 1/2 X 32 =16mm

 new height = 1/2  x18 = 9 mm 

New similar rectangle is shown in the figure.

let 'factor of change' be 2.

 New width = 2x32= 64 mm and

new height = 2 × 18 = 36 mm

 Factor of change = 2: Smaller: 16 mm × 9 mm Larger: 64 mm × 36 mm 

6. The following figure shows a small portion of a long brick wall with patterns made using coloured bricks. Each wall continues this pattern throughout the wall. What is the ratio of grey bricks to coloured bricks? Try to give the ratios in their simplest form: Ratio of grey bricks to coloured bricks in pattern: 

 Number of grey bricks in one set of pattern = 2 + 3 + 4 = 9 

Number of coloured bricks in one set of pattern = 3 + 2 + 1 = 6 

Ratio of grey bricks to coloured bricks  9:6 = 3:2 

Ratio in the simplest form 3:2

(b) One set of pattern

Number of grey bricks in one set of pattern

= (½  + 1 + 1 + ½) + (1 + 1) + (½ + 1 + ½) + (1 + 1) + (½ + 1 + ½) + (1 + 1) + (½ + 1 + 1 + ½)

= 3 + 2 + 2 + 2 + 2 + 2 + 3 = 16

Number of coloured bricks in one set of pattern = 1 + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) +(1+1)+1  Ratio of grey bricks to coloured bricks = 16:12 = 4:3 .

 Ratio in the simplest form = 4:3 . 

(a) Grey : Coloured = 9 : 6 → simplest = 3 : 2 

 (b) Grey : Coloured = 16 : 12 → simplest = 4 : 3

7. let us draw some human figures. Measure your friend’s body — the lengths of their head, torso, arms, and legs. Write the ratios as mentioned below— head : torso ______ : ______ torso : arms ______ : ______ torso : legs ______ : ______Now, draw a figure with head, torso, arms, and legs with equivalent ratios as above. Does the drawing look more realistic if the ratios are proportional? Why? Why not? Human figure proportions: 

head : torso 25:60 (Answer: 5 : 12)

torso : arms  60:65 (Answer: 12 : 13)

torso : legs 60:80 (Answer: 3 : 4)

Now I draw: Head: 5 cm tall, Torso: 12 cm tall, Arms: 13 cm each, Legs: 16 cm each, This will be a proportional stick figure or simplified human shape.

Does it look more realistic if the ratios are proportional?

Yes, generally it does look more realistic.

Why?Human bodies have consistent proportional relationships across individuals (though they vary somewhat). If you keep the torso-to-leg ratio realistic (e.g., around 3:4 or similar), and head-to-torso ratio correct (around 1:3 to 1:4), the figure will look naturally human.

If you make the head too big compared to the torso (e.g., 1:1 ratio), it will look cartoonish or child-like. If legs are too short compared to the torso, it looks unnatural. Proportionality based on real human measurements captures the natural balance of body parts, making the drawing lifelike.

Why not (possible counterpoint)?

Sometimes in art, especially cartoons or stylized figures, artists intentionally break proportions for expressive effect (big head for cuteness, long legs for elegance). So, “realistic” isn’t always the goal in art—but if the goal is a realistic human figure, keeping ratios proportional is key.

Final thought for the student:

When you draw using the actual ratios from your friend, you are drawing a figure that matches their specific body proportions. This will look like a recognizable representation of that person, which is what “realistic” means in this activity.

 Example: head : torso = 25 : 60 → 5 : 12 Proportional drawing looks realistic because body parts maintain natural ratios. 

7.6 Rule of Three (Trairasika)

Two ratios are proportional if their terms are equal when cross multiplied. The fourth unknown quantity can be found through such cross multiplication. a : b :: c : d. 

In ancient India, Āryabhaṭa (199 CE) and others called such problems of proportionality Rule of Three problems. 

There were 3 numbers given — the pramāṇa (measure — ‘a’ in our case), the phala (fruit — ‘b’ in our case), and the ichchhā (requisition — ‘c’ in our case). To find the ichchhāphala (yield — ‘d’ in our case),

 Āryabhaṭa says, “Multiply the phala by the ichchhā and divide the resulting product by the pramāṇa.”

 In other words, Āryabhaṭa says, “pramāṇa : phala :: ichchhā : ichchhāphala,”

 therefore, pramāṇa × ichchhāphala = phala × ichchhā. 

ichchhāphala = "phala × ichchhā" /"pramāṇa" . Using the cross multiplication method proposed by Āryabhaṭa, ancient Indians solved complex problems that involved proportionality

If a : b :: c : d, then \[ d = \frac{b \times c}{a} \] Ancient Indian method: - pramāṇa (a) - phala (b) - ichchhā (c) - ichchhāphala (d) = \( \frac{b \times c}{a} \) 

  Example 8: 

 For the mid-day meal in a school with 120 students, the cook usually makes 15 kg of rice. On a rainy day, only 80 students came to school. How many kilograms of rice should the cook make so that the food is not wasted? The ratio of the number of students to the amount of rice needs to be proportional. So, 120 : 15 :: 80 : ? What is the factor of change in the first term? 

For 120 students, required rice is 15 kg

Let x kg of rice be required for 80 students

Ratios 120:15 and 80 : x  are in proportion

120 : 15 : : 80 : x

by dividing the terms 80 :120 = 2 : 3. 

The number of students is reduced by a factor of 2 : 3 .

 On multiplying the weight of rice by the same factor, we get, 15 × 2 /3 = 10. 

 School: 120 students → 15 kg rice. 80 students → ? kg rice. 120 : 15 :: 80 : x \[ x = \frac{15 \times 80}{120} = 10 \ \text{kg} \] 

So, the cook should make 10 kg of rice on that day

  Example 9: 

(i) A car travels 90 km in 150 minutes. If it continues at the same speed, what distance will it cover in 4 hours? If it continues at the same speed, the ratio of the time taken should be proportional to the ratio of the distance covered. 

(ii) 150 : 90 :: 4 : ? Is this the right way to formulate the question? 

(iii) How can you find the distance covered in 240 minutes?

 (i) 4 hours = 4 x 60 = 240 minutes

In 150 minutes, distance covered 90 km

Let x km be covered in 4 hours in 240 minutes

The ratios 150:90 and 240:x are in proportion.

150:90::240:x

(ii) unit must be same in comparing ratios 
so 150:90::4:x is meaningless.

and150:90::240:x is correct.

(iii) 150 : 90 :: 240 : x. 

By cross multiplication, we get 150 × x = 240 × 90 

Therefore, x = 144. 

 Car: 90 km in 150 min. Distance in 4 hours (240 min)? 150 : 90 :: 240 : x \[ x = \frac{90 \times 240}{150} = 144 \ \text{km} \] 

The distance covered by the car in 4 hours is 144 km

  Example 10: 

A small farmer in Himachal Pradesh sells each 200 g packet of tea for ₹200. A large estate in Meghalaya sells each 1 kg packet of tea for ₹800. Are the weight-to-price ratios in both places proportional? Which tea is more expensive? Which tea is more expensive? Why?

 The ratio of weight to price of the Himachal tea is 200 : 200. 

So, the weight to price ratio is 1000 : 800 in Meghalaya after we convert the weight to grams.

The Himachal tea ratio in its simplest form is 1 : 1. 

The Meghalaya tea ratio in its simplest form is 5 : 4. 

So, the ratios are not proportional. 

The price of 1 kg of tea is x rupees. 

200 g is 1/5  of 1 kg. So, 1/5 x = 200. 

1/5 x × 5 = 200 × 5 

x = 1000. So, the cost of 1 kg of tea is ₹800 in Meghalaya and ₹1,000 in Himachal Pradesh. Therefore, the tea from Himachal Pradesh is more expensive.

Himachal tea: 200 g → ₹200 → ratio 1 : 1. Meghalaya tea: 1 kg (1000 g) → ₹800 → ratio 5 : 4. Not proportional. Cost per kg: Himachal = ₹1000, Meghalaya = ₹800 → Himachal more expensive. --- ### 

Figure it Out (page 170 - 171)

1. The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?

Solution:

1 million  = 10 lakh = 10,00,000 

 1 year = \( \frac{365}{7} \) weeks

940 million kilometres = 940 X 10 ,00,000 kilometres are travelled by the Earth in 1 year.

 let x kilometres be travelled by the Earth in 1 week 

 The ratios 940 × 10,00,000: \( \frac{365}{7} \)  and x: 1  are in proportion.

\[ x = \frac{940 \times 1000000}{365/7} \] = $ \frac{x}{1} $ 

\[ x = \frac{188 \times 7000000}{73} \]

In 1 week, Earth travels nearly 1,80,27,397 kilometres around the Sun.

2. A mason is building a house in the shape shown in the diagram. He needs to construct both the outer walls and the inner wall that Proportional Reasoning-1 separates two rooms. To build a wall of 10-feet, he requires approximately 1450 bricks. How many bricks would he need to build the house? Assume all walls are of the same height and thickness. 

Total length of walls = 

= AI + CH + DE + FG + IG + AF + CD 

=12 + (9 + 12) + 9 + 12 + (9 + 15) + (9 + 15) + 6 = 108ft 

let x bricks be required for 108 ft long wall.

Ratio of length of wall to number of bricks = 108 : x 

These ratios are in proportion. 10:1450::108: x 

 \( \frac{10}{1450} \) = \( \frac{108}{x} \) 

X = 145 x 108 = 15660

Number of required bricks = 15660

Puneeth’s father went from lucknow to Kanpur in 2 hours by riding his motorcycle at a speed of 50 km/h. If he drives at 75 km/h, how long will it take him to reach Kanpur? Can we form this problem as a proportion —  50 : 2 :: 75 : __ Would it take Puneeth’s father more time or less time to reach Kanpur? Think about it. Even though this problem looks similar to the previous problems, it cannot be solved using the Rule of Three! The time of travel would actually decrease when the speed increases. So this problem cannot be modelled as 50 : 2 :: 75 : __

Solution:

Time taken at the speed of 50 km/h = 2 hours .. 

Distance = Speed × Time = 50 × 2 = 100 km 

At speed of 75km / h time taken = \( \frac{100}{75} \)  = $ \frac{4}{3} $  hours

The ratios 50: 2 and proportion. 75 : \( \frac{4}{3} \)  are not in proportion.

We cannot write: 50:2 :: 75 : $ \frac{4}{3} $

At speed of 75km / h Puneeth's father will take $ \frac{4}{3} $   hours to reach Kanpur, which is less than  2 hours.

if we want to divide a quantity x in the ratio of m: n, 

then the parts will be 

Example 11: 

Prashanti and Bhuvan started a food cart business near their school. Prashanti invested ₹75,000 and Bhuvan invested ₹25,000. At the end of the first month, they gained a profit of ₹4,000. They decided that they would share the profit in the same ratio as that of their investment. What is each person’s share of the profit?

The ratio of their investment is 75000 : 25000. 

Reducing this ratio to its simplest form, we get 3 : 1. 

3 + 1 is 4 and dividing the profit of 4000 by 4, we get 1000. 

So, Prashanti’s share is 3 × 1000 and Bhuvan’s share is 1 × 1000. 

So, Prashanti would get ₹3,000 and Bhuvan would get ₹1,000 of the profit.

Example 12: 

A mixture of 40 kg contains sand and cement in the ratio of 3 : 1. How much cement should be added to the mixture to make the ratio of sand to cement 5 : 2?

let us find the quantity of sand and cement in the original mixture.

 The ratio is 3 : 1 and the total weight is 40 kg. 

So, the weight of sand is 3/4   x 40 = 30 kg. 

The weight of cement is 1/4   x 40 = 10 kg.

 The weight of sand is the same in the new mixture. It remains 30. 

But the new ratio of sand to cement is 5 : 2. 

So the question is, 5 : 2 :: 30 : ? 

If the ratio is 5 : 2, then the second term is 2 5 times the first term. 

Since the new ratio is equivalent to 5 : 2, the second term in the new ratio should also be 2 /5 times of 30. 

2/5  × 30 = 12. 

The new mixture should have 12 kg of cement if the ratio of sand to cement is to be 5 : 2. 

There is 10 kg of cement already. So, we need to add 2 kg of cement to the original mixture

 Mixture: 40 kg, sand : cement = 3 : 1 → sand = 30 kg, cement = 10 kg. New ratio = 5 : 2, sand unchanged (30 kg). Cement needed = \( \frac{2}{5} \times 30 = 12 \) kg → add 2 kg. 

Figure it Out (Page 175) 

1. Divide ₹4,500 into two parts in the ratio 2 : 3

Given ratio = 2 : 3 

Amount to be divided = ₹4,500 . 

First part = 2/5 × 4,500 = 2 x 900 = ₹1,800 

Second part = 3/5 × 4,500 = 3 x 900 = ₹2,700 

 Two parts are ₹1,800 and ₹2,700, 

 2. In a science lab, acid and water are mixed in the ratio of 1 : 5 to make a solution. In a bottle that has 240 ml of the solution, how much acid and water does the solution contain?

Solution:

Ratio of acid and water = 1 : 5 

Quantity of solution = 240 ml. 

Quantity of acid = 1/6 x 240 = 40 ml

Quantity of water = 5/6 x 240 = 200 ml 

Quantity of acid and water in the solution are 40 ml and 200 ml.

 Acid = \( \frac{1}{6} \times 240 = 40 \) ml Water = \( \frac{5}{6} \times 240 = 200 \) ml.

3. Blue and yellow paints are mixed in the ratio of 3 : 5 to produce green paint. To produce 40 ml of green paint, how much of these two colours are needed? To make the paint a lighter shade of green, I added 20 ml of yellow to the mixture. What is the new ratio of blue and yellow in the paint? 

Solution:

Ratio of blue and yellow paints = 3 : 5 

Quantity of green paint = 40 ml. 

Quantity of blue paint = 3/8 × 40 = 15mI

Quantity of yellow paint = 5/8 x 40 = 25ml

Addition of yellow paint to the mixture = 20 ml

New quantity of blue paint = 15 ml

New quantity of yellow paint = 25ml + 20ml = 45ml

New ratio of blue and yellow paints =15 : 45 = 1 :3

 Blue = \( \frac{3}{8} \times 40 = 15 \) ml Yellow = \( \frac{5}{8} \times 40 = 25 \) ml. Add 20 ml yellow → new ratio = 15 : 45 = 1 : 3. 

4. To make soft idlis, you need to mix rice and urad dal in the ratio of 2 : 1. If you need 6 cups of this mixture to make idlis tomorrow morning, how many cups of rice and urad dal will you need?

 Solution:

Ratio of rice and urad dal 2 :1

Total number of cups of mixture = 6

Number of cups of rice = 2/3 x 6 = 4

Number of cups of urad dal = 1/3   x 6 = 2

4 cups of rice and 2 cups of urad dal are to be mixed.

Rice = \( \frac{2}{3} \times 6 = 4 \) cups Urad dal = \( \frac{1}{3} \times 6 = 2 \) cups. 

5. I have one bucket of orange paint that I made by mixing red and yellow paints in the ratio of 3 : 5. I added another bucket of yellow paint to this mixture. What is the ratio of red paint to yellow paint in the new mixture?

Solution:

let capacity of one bucket be x litre.

Ratio of red paint and yellow paint = 3:5 

Quantity of red paint in the bucket = 3/8 x 𝑥 = 3𝑥/8

Quantity of yellow paint in the bucket = 5/8 x 𝑥 = 5𝑥/8  

One bucket of yellow paint is added to the mixture.

New quantity of red paint in the mixture = 3𝑥/8

 New quantity of yellow paint in the mixture = = 5𝑥/8 + 𝑥 = 13𝑥/8

New ratio of red paint and yellow paint in the mixture = 3𝑥/8 : 13𝑥/8 = 3 : 13

 New ratio = 3 : 13.

Unit Conversions (Reference) 

 Length: 

1 m = 3.281 ft 

 Area: 

1 m² = 10.764 ft², 

1 acre = 43,560 ft², 

1 hectare = 2.471 acres 

 Volume: 

1 L = 1000 mL = 1000 cc 

 Temperature: \[ °F = \frac{9}{5} \times °C + 32, \quad °C = \frac{5}{9} \times (°F - 32) \] 

Figure it Out Page 176-177

1. Anagh mixes 600 ml of orange juice with 900 ml of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form

Solution:

Quantity of orange juice = 600ml 

Quantity of apple juice = 900ml 

Ratio of orange juice to apple juice = 600: 900

Ratio in the simplest form = 600/900=  2/3=2:3

 Orange juice : apple juice = 600 : 900 → simplest = 2 : 3. 

2. last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full?

Solution:

Buses: 162 people → 3 buses (54 each).

Number of buses for 162 students and teachers =3

The buses were full, the capacity of 1 bus = 162/3 = 54

Ratio of number of seats to the number of buses is 54: 1.

54:1 = 2(54): 2(1) = 108 : 2 

54:1 = 3(54): 3(1)= 162: 3 

54:1 = 4(54): 4(1)=216:4 

Capacity of 4 buses = 216 

For 204 students, we shall need 4 buses. 

Since 216 – 204 = 12, 

therefore 12 vacant seats in the buses.

   For 204 people → 4 buses needed (12 vacant seats). 

3. The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so?

 Delhi vs Mumbai crowding: 

 Solution:

Ratio of area to population for Delhi = 1,484:30

let density of Delhi and Mumbai be same and there be x people in Mumbai.

 The ratio 1,484 : 30 and 550 : x are in proportion.

 1484/30=  550/𝑥

 1,484x= 30 x 550 = 16,500  

X = 16500/1484

x=1,484 = 11.118 

There should be 11.118 million people in Mumbai. But population of Mumbai is 20 million. Mumbai is more crowded than Delhi.

Another Method

Area of Delhi  = 1,484 sq. km

Population of Delhi  = 30 million

Area of Mumbai = 550 sq. km

Population of Mumbai = 20 million

Ratio of area to population for Delhi = 1,484:30

Ratio of area to population for Mumbai = 550: 20

Factor of change of area = 550/1484 0.371 (approx) 

Factor of change of population = 20/30=0.667 (approx) 

Since 0.667 > 0.371, Mumbai is more crowded than Delhi.

Mumbai more crowded because population density higher.

4. A crane of height 155 cm has its neck and the rest of its body in the ratio 4 : 6. For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be?

 Crane neck : body = 4 : 6. 

Solution: 

Ratio of height of neck and height of rest of body of a crane is 4:6

My height is 65 inches =  165 cm. ( 1 inch = 2.54cm) (65 x 2.54 = 165.1 cm)

let the ratio of height of my neck and height of rest of my body be also 4:6.

Height of my neck  = ( 4/(4+6)  x 165 ) cm

= 660/10 =  66cm 

 If my height = 165 cm, neck height = \( \frac{4}{10} \times 165 = 66 \) cm. 

5. let us try an ancient problem from lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. “If 2 𝟏/𝟐 palas of saffron costs 𝟑/𝟕 niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?”

Ancient problem: 

 Solution:

unit of weight in palas and unit of money is niskas.Cost of 2 𝟏/𝟐  palas of saffron = 𝟑/𝟕  niskas 

 Ratio of weight to price is 2 𝟏/𝟐 :  𝟑/𝟕 or  5/2 x 14 :  𝟑/𝟕   x 14 = 35:6

let x palas of saffron be bought for 9 niskas. 

Ratio of weight to price is x : 9. 

These ratios are in proportion. 

35 : 6:: x : 9 

 35/6=  (𝑥 )/9

6x = 35 x 9

x = 315/6  = 52.5

52.5 palas of saffron can be bought for 9 niskas

\( 2 \frac{1}{2} \) palas saffron → \( \frac{3}{7} \) niskas. 9 niskas → \( \frac{35}{6} \times 9 = 52.5 \) palas saffron. 

6. Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain’s age when the ratio of her age to her brother’s age is 1 : 2?

 Harmain age 1, brother 5. When ratio 1 : 2? 

 Solution:

Age of Harmain and her brother are 1 year and 5 years.

let after x years, the ratio of their ages be 1:2

Age of Harmain after x years = (1 + x) years.

Age of her brother after x years = (5 + x) years After x years, ratio of their ages = 1 + x : 5 + x 

The ratios are in proportion. ⇒ 1:2:: 1 + x : 5 + x 

 1/2=(1+𝑥)/(5+𝑥)

5 + x = 2(1 + x) 

5+x = 2+2x 

 2x - x = 5 - 2 

x = 3 

After 3 years, age of Harmain = 1+3 = 4 years. 

After 3 years → ages 4 and 8 → ratio 1 : 2. 

7. The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold?

 Gold : water mass ratio = 37 : 2. 

 Solution:

The ratio of masses of gold and water, when their volumes are same, is 37: 2. 

Mass of 1 litre of water = 1kg 

let mass of 1 litre of gold = x kg 

With equal volumes, ratio of masses of gold and water is x :1. 

These ratios are in proportion. 

37: 2 :: 1: x 

 37/2=  𝑥/1

Thus, the mass of 1 litre of gold is  37/2 kg

1 litre water = 1 kg → 1 litre gold = \( \frac{37}{2} = 18.5 \) kg.

8. It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy? (Please refer to the section on Unit Conversions earlier in this chapter).

 Manure for farming: 

Solution: 1 ton = 1000kg

10 tonnes 10×1,000 = 10,000 kg 

1 acre = 43,560 sq. ft. 

Ratio of cow manure to area of land in kg and sq. ft =10,000 : 43,560 Size of plot = 200 ft. by 500 ft. 

Area of plot = 200 x 500 = 1,00,000 sq. ft. 

let cow manure required be x kg. 

Ratio of cow manure to area of plot = x : 100000

These ratios are in proportion.

 10,000 : 43,560 :: x : 1,00,000

 10000/43560=  𝑥/100000

43, 560x =10,000 x 1,00,000 = 1,00,00,00,000

x = 1,00,00,00,000/43560

Required cow manure = 22956.84 kg = 22.95684 tonnes

 Plot = 200 ft × 500 ft = 100,000 sq ft. Manure needed ≈ 22.96 tonnes. 

9. A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 ml. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?

 Tap fills 500 ml mug in 15 seconds. 

 Solution:

Time taken by tap for 500 ml water  = 15 seconds 

Ratio of volume to time = 500:15  (1 litre = 1,000 ml)

 10 litre = 10 x 1,000= 10,000 ml 

let time taken to fill a bucket of 10,000 ml be x seconds. 

Ratio of volume to time = 10,000 : x 

These ratios are proportional. 

500 :15 ::10,000 : x

 500/15=  10000/𝑥  

500x = 1,50,000 

X = 1500000/500  = 300 

Time to fill bucket = 300 seconds = 300/60 = 5 minutes

10 L bucket → 300 seconds = 5 minutes. 

10. One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?

 Land cost: 1 acre = 43,560 sq ft → ₹15,00,000.

Solution: 1 acre = 43,560 square feet. 

Cost of 43,560 sq. ft. land = ₹15,00,000 

Ratio of area of land to cost =  43,560: 15,00,000 

let cost of 2,400 sq. ft. land be ₹x. 

 Ratio of area of land to cost = 2,400 : x 

These ratios are proportional. 

43,560 :15,00,000 :: 2,400 : x 

43560/1500000=  2400/𝑥 

43,560x = 2,400 × 15,00,000 

x= (2400 𝑥 1500000)/43560    = 82,664.63 

Cost of land = ₹82,664.63.

2,400 sq ft → ₹82,664.63. 

  11. A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead?

Tractor vs oxen ploughing: 

 Solution : 

Ratio of efficiency of a tractor to a pair of oxen = 4:1

Time taken by a pair of oxen to plough 1 acre field = 6 hours 

Time taken by a tractor to plough 

 1 acre field = 6/4 =  1.5 hours 

Time taken by a pair of oxen to plough 20 acre field = 20×6=120 hours 

Time taken by a tractor to plough 20 acre field = 20×1.5 = 30 hours

Oxen: 20 acres × 6 hours = 120 hours. Tractor (4× faster) → 30 hours. 

12. The ₹10 coin is an alloy of copper and nickel called ‘cupro-nickel’. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?

 ₹10 coin: copper : nickel = 3 : 1. 

Solution: 

Ratio of copper and nickel in ₹10 coin = 3:1

Mass of one 10 coin = 7.74 grams

Mass of copper in one ₹10 coin 

=3/(3+1) x 7.74 = 3/4 x7.74 = 5.805 grams

Mass of nickel in one ₹10 coin

= 1/(3+1) x 7.74 = 1/4 x7.74 = = 1.935 grams 

Cost of 1 kg copper = ₹906 

Cost of 1000 grams copper = ₹906 

Cost of 5.805 grams copper = 906/1000 x 5.805 = ₹5.26

Cost of 1 kg nickel =₹1341 

Cost of 1000 grams nickel =₹1341 

Cost of 1.935 grams nickel = 1341/1000 x 1.935 = ₹2.59

 In one ₹10 coin, cost of copper and cost of nickel are respectively ₹5.26 and ₹2.59.

 Mass = 7.74 g → copper = 5.805 g, nickel = 1.935 g. Cost: copper = ₹5.26, nickel = ₹2.59.