Simple interest quiz with a twist

Each question comes with multiple choices—try to answer first, then check the solution after!

---

1. Emily borrowed $2,000 at a rate of 6% for 4 years. What is the total interest?

A. $360
B. $480
C. $540
D. $600

Formula:

$$\text{Interest} = 2000 \times 0.06 \times 4 = \boxed{480}$$

✅ Correct Answer: B. $480

---

2. Jack invested $1,500 for 2 years and earned $180 in interest. What was the rate?

A. 5%
B. 6%
C. 8%
D. 12%

$$\text{Rate} = \frac{180}{1500 \times 2} = 0.06 = \boxed{6\%}$$

✅ Correct Answer: B. 6%

---

3. A loan of $800 earns $96 in interest at 4% interest. How long was the loan?

A. 2 years
B. 3 years
C. 4 years
D. 5 years

$$\text{Time} = \frac{96}{800 \times 0.04} = \frac{96}{32} = \boxed{3}$$

✅ Correct Answer: B. 3 years

---

4. Sarah paid $525 in interest on a 5-year loan at 7%. What was the original principal?

A. $1,200
B. $1,400
C. $1,500
D. $1,700

$$\text{Principal} = \frac{525}{0.07 \times 5} = \frac{525}{0.35} = \boxed{1500}$$

✅ Correct Answer: C. $1,500

---

5. Tom earned $600 in interest over 3 years at 10%. How much did he invest?

A. $1,800
B. $2,000
C. $2,400
D. $2,800

$$\text{Principal} = \frac{600}{0.10 \times 3} = \frac{600}{0.30} = \boxed{2000}$$

✅ Correct Answer: B. $2,000

---

Class 8 NCERT bridge course Answers Activity W 5.3 Logic Clue Hunt on the Hundred Square

 Activity W 5.3 -  Logic Clue Hunt on the Hundred Square 

For this activity, students work in pairs or small groups. 

The students may draw a hundred square as shown below:

Procedure 

 The following clues may be written on the blackboard: 

 The number is greater than 9. 

 The number is not a multiple of 10. 

 The number is a multiple of 8. ¾ The number is even. 

The number is not a multiple of 11. 

 The number is less than 175. 

 Its ones digit is larger than its tens digit. 

 Its tens digit is odd.

Part A 

 Tell the students 

 How have a number in your mind that is on the hundred squares but you are not going to tell them what it is. 

They have to ask you for any four clues out of the given eight clues. 

With every clue they speak out, you will say just ‘YES’ or ‘NO’. 

 Try to find the set of four clues that help them to find the number in your mind. 

 Give a chance to each group to do this. 

Strategy Tip: Encourage teams to choose clues that narrow the number range quickly.

Part B 

Four of the given clues are true but they do not help in finding the number.

 Find those numbers. 

Reflection 

Consider the questions that led the students being interested and able to progress, and those you needed to clarify. 

Such reflection always helps you engage the students to find mathematics interesting and enjoyable. 

If they do not understand and do something, they are less likely to become involved.

Part A: Which clues help identify the number?

We are given 8 clues. The goal is to identify one specific number using only 4 well-chosen clues. Here’s how you can think through the process:

Let's analyze each clue for usefulness:

Clue Analysis:

1. The number is greater than 9
   ➤ Too broad — eliminates only numbers 1–9.
   Not very useful.

2. The number is not a multiple of 10
   ➤ Removes numbers ending in 0 (e.g., 10, 20, ..., 200).
   Somewhat useful.

3. The number is a multiple of 8
   ➤ Strong clue. Narrows down to numbers like 8, 16, 24, 32, etc.
   Very useful!

4. The number is even
   ➤ All multiples of 8 are even already.
   Redundant if Clue 3 is chosen.

5. The number is not a multiple of 11
   ➤ Excludes numbers like 11, 22, 33, ..., 198.
   Somewhat useful.

6. The number is less than 175
   ➤ Trims the upper end.
   Useful for narrowing down.

7. Its ones digit is larger than its tens digit
   ➤ Powerful filter (e.g., 13, 24, 57, but not 31, 43).
   Very useful!

8. Its tens digit is odd
   ➤ Limits numbers to those with tens digit as 1, 3, 5, 7, or 9.
   Very useful!

 Example: Find the Hidden Number

Let's pick a number that satisfies the following 4 helpful clues:

• Clue 3: Multiple of 8

• Clue 6: Less than 175

• Clue 7: Ones digit > Tens digit

• Clue 8: Tens digit is odd

Let’s test numbers that are:

• Multiples of 8

• Less than 175

• Have ones digit > tens digit

• Tens digit is odd

Example: 136

• Multiple of 8 

• Less than 175 

• Ones digit (6) > Tens digit (3) 

• Tens digit (3) is odd 
  → 136 is a valid hidden number!

Part B: Which clues are always true but not useful?

These clues may be true for many numbers, but don’t help narrow the list:

1. Clue 1: Greater than 9 → Always true for almost all 2- or 3-digit numbers.

2. Clue 2: Not a multiple of 10 → Excludes just a few (10, 20, ..., 200).

3. Clue 4: Even → Already covered by “multiple of 8.”

4. Clue 5: Not a multiple of 11 → Useful only if the number was close to a multiple of 11.

So, the 4 clues that don’t help much, even if true, are:

• Clue 1 (Greater than 9)

• Clue 2 (Not a multiple of 10)

• Clue 4 (Even)

• Clue 5 (Not a multiple of 11)

These clues are logically true for many numbers but don’t help you zero in on the correct number efficiently.

Class 8 NCERT bridge course Answers Activity W 5.2 Number sense

 Activity W 5.2  Number sense

Number sense involves giving meaning to numbers, that is, knowing about how they relate to each other and their relative magnitudes. 

 Having a sense of number is vital for the understanding of numerical aspects of the world. 

Here are some ideas to develop and strengthen students’ sense of numbers. 

Activity 1: Two-Digit Number Trick

Objective: Discover a pattern when reversing and adding two-digit numbers.

Procedure 

  Ask the students to choose a two-digit number. 

 Tell them to reverse the digits to get a new number. 

 Add this new number to the original number. 

 Ask students to check for their divisibility by a number.

 Check if every student gets a number by which the sum obtained in step 3 is divisible. 

 Discuss why this happens! 

Steps:

1. Ask students to choose any two-digit number (e.g., 52).

2. Reverse the digits to form a new number (e.g., 25).

3. Add the original and reversed number (e.g., 52 + 25 = 77).

4. Ask students to check the divisibility of the result.

Challenge Question:

• Is the result always divisible by a specific number?

Answer: yes, The sum is always divisible by 11.

• (Hint: Try with different numbers – 34 + 43 = 77, 61 + 16 = 77...)

✨ Discover: The sum is always divisible by 11. Why does this happen?

Why does this happen?

Answer: 

Let the two-digit number be 10a + b, where a is the tens digit and b is the units digit.

The reverse of the number is 10b + a.

Now add the two:  (10a+b)+(10b+a)=11a+11b=11(a+b)

This sum is clearly divisible by 11, because 11 is a common factor.

Activity 2: Three-Digit Number Difference

Objective: Explore divisibility through subtraction of reversed numbers.

Procedure 

 Ask students to think of a three-digit number. 

 Now, they should make a new number by putting the digits in the reverse order. 

 Subtract the smaller number from the larger one. 

 Ask the students to check by which number the difference so obtained is divisible. Which other multiple of this divisor will divide the difference? 

 Discuss how this happens! 

Steps:

1. Choose any three-digit number (e.g., 741).

2. Reverse the digits (e.g., 147).

3. Subtract the smaller from the larger (e.g., 741 - 147 = 594).

4. Ask: Which number divides this difference?

Question: What is the difference divisible by?

Answer: The difference is always divisible by 99.

Follow-Up:

• Try multiple three-digit numbers. What do you notice?

• Is there a common divisor?

✨ Discover: The difference is always divisible by 99 (or sometimes 9 and 11).

Why does it work?

Let the number be 100a + 10b + c, and the reverse is 100c + 10b + a.

Now subtract the smaller from the larger:

$$(100a+10b+c)-(100c+10b+a)=99a-99c=99(a-c)$$

So the difference is always divisible by 99.

🧠 Bonus: Since 99 = 9 × 11, it's also divisible by 9 and 11.

Activity 3: Rotating Digits of a Three-Digit Number

Objective: Explore rotational patterns in digits and their divisibility.

Procedure 

 Students may be asked to think of any 3-digit number (abc). 

 Now, using this number, students may be asked to form two more 3-digit numbers (cab, bca).

Now, add the three numbers so formed. 

 Students may explore the smallest number by which it will be divisible. 

 Discuss how this happens!

Steps:

1. Think of a three-digit number (e.g., 231).

2. Create two more numbers by rotating the digits:

   • (cab → 312)

   • (bca → 123)

3. Add the three numbers:

   • 231 + 312 + 123 = 666

4. Ask: What is the smallest number that always divides the sum?

Answer: The sum is always divisible by 37 (and also 3 and 9).

✨ Discover: The sum is divisible by 37 (and also 3 and 9)!

Why does it work?

Let the three-digit number be abc:

• Its numerical value is 100a + 10b + c

• Rotation 1 (cab): 100c + 10a + b

• Rotation 2 (bca): 100b + 10c + a

Now add all three:

$$(100a+10b+c)+(100c+10a+b)+(100b+10c+a)=111a+111b+111c=111(a+b+c)$$ $$\text{<br>}\text{<br>}$$

Class 6 NCERT bridge course Answers Activity W3.3 A Treasure Hunt

 Activity W3.3 A Treasure Hunt 

 Provide each student with a copy of the treasure map, which includes coordinates (i.e., pairs of numbers discussed in earlier activity) marking the location of the treasure. 

 Explain the objective of the activity: to use the given coordinates to locate the treasure. 

 Allow students to work individually or in pairs to navigate the map and find the treasure.

Once the treasure is found, celebrate the successful completion of the hunt and discuss the coordinates used to locate the treasure. 

 Encourage students to create their own treasure maps for future activities, incorporating coordinates and landmarks of their choice. 

Creating patterns and designs with rotational and reflection symmetry Symmetry is a property where one shape or arrangement can be transformed into another that looks the same.

Let’s imagine this map uses a simple grid system: the bottom left is (0,0) and the top right is (10,10). Here's how we can place the main points:

• Starting point (where the pirate boy is) — (9,2)

• Pirate Ship — (2,1)

• Skull Rock — (3,5)

• Crocodile Pond — (5,6)

• Lighthouse — (7,9)

• Dragon Cave — (8,6)

• X Marks the Treasure — (6,3)

Path to the Treasure:

1. Start at (9,2) — The boy.

2. Head southwest to the Pirate Ship at (2,1).

3. Move north to the Skull Rock at (3,5).

4. Go northeast to Crocodile Pond at (5,6).

5. Move southeast to reach the Treasure at (6,3).

Named Points:

• A (9,2) — Start

• B (2,1) — Pirate Ship

• C (3,5) — Skull Rock

• D (5,6) — Crocodile Pond

• E (6,3) — Treasure!

So the final treasure is at Point E (6,3).

The path starts at the boy, passes the pirate ship, skull rock, crocodile pond, and finally reaches the treasure at (6,3). 

New Route & Coordinates:

1. Start (Boy) — (8,3)

2. Dragon Cave — (7,7)

3. Snake — (5,7)

4. Water Pond — (5,6)

5. Skull Rock — (3,5)

6. Pirate Ship — (4,2)

7. Lighthouse — (6,8)

8. Crocodile Pond — (2,6)

9. Treasure — (7,4)

New Story Clues:

Clue at Start (8,3)
"Ahoy, young adventurer! Your journey begins where the sun meets the sea. Seek the ship with black sails, it waits for thee!"

"The journey starts at the shore's embrace, where adventure waits at a dragon's place."

"Set sail from the sandy shore, a bony face waits at place four."

🐉 Clue at Dragon Cave (7,7)

"The dragon sleeps but the path's not done, follow the glow of the rising sun!"

"A dragon's roar guides the way, to a slippery friend where secrets lay."

"A fiery friend guards the way, but thirst leads the next display."

Snake (5,🐍 Clue at Snake (5,7)

"Slither and hiss, the serpent's near, climb to the light, the path is clear!"

"Slither past without a sound, where water glimmers, treasure's bound."

"Slither past without delay, crocs await along the bay."

💧 Clue at Water Pond (5,6)

"Cool waters and ripples wide, but snakes nearby know where riches hide!"

"A drink for the weary, cool and clear, look for the skull that whispers fear."

"Cool water to quench your thirst, but to see the light, climb first."

💀 Clue at Skull Rock (3,5)

"A skull of stone watches the shore, to quench your thirst, seek water and more."

"The rock with eyes has seen it all, now sail to the ship that heard the call."

"The silent skull hides no lie, the ship with sails is nearby."

🦜 Clue at Pirate Ship (4,2)
"Ye found the ship, brave and bold, but the treasure's still untold. Beware the crocs — to the swamp you must go, where the water runs slow."

"Yo-ho-ho! But not yet gold — the lighthouse stands, so brave and bold."

"From deck to cave, the dragon's near, brave the beast and show no fear."

💡 Clue at Lighthouse (6,8)

"From this tower the sea is seen, but the dragon guards what's golden and green!"

"High and bright, it lights the way, where crocs are waiting to lead astray."

"From atop the world so high, the slithering snake is nearby."

🐊 Clue at Crocodile Pond (2,6)
"Snap and splash, the crocs do play, the skull-shaped rock points the way!"

"Snap and splash, the final clue! Head to (7,4) for the treasure true."

"Watch your step at the snapping jaws, the treasure waits with pirate laws."

🏴‍☠️ Final Clue at Treasure (7,4)
"X marks the spot, you’ve braved the quest, dig right here for the pirate’s chest!"

"You've solved the riddles, faced each test, dig here now and claim your chest!"

"X marks the spot — dig, and the chest is yours!"

Class 8 NCERT bridge course Answers Activity W 5.1 FRACTALS IN NATURE

 Activities for Week 5 

Activity W 5.1 FRACTALS IN NATURE

 Fractals are all around us in nature. 

They help explain the irregular, repetitive patterns in many things we see every day. 

Students may be made to explore how fractals can represent real-world data using patterns and algorithms. 

Materials Required: 

Graph paper, printouts with different fractal patterns and their corresponding data (such as trees, coastlines, plant structures or any other structure of their choice). 

Procedure 

1. Students may be asked to get a photograph of a tree. 

They may observe the branching. 

They may use this data to create a branching fractal pattern.

 The data shows how each branch divides into smaller branches at a constant angle. 

They may use a repetitive method to create fractal. 

2. This activity may be done for coastline data. The coastline is irregular and jagged, which means it’s a fractal pattern. 

Use this data to draw an irregular coastline that gets more jagged as you zoom in.

3. Another similar activity can be done for the adjoining picture. 

Extension 

Data related to clouds, or mountains, or even river systems can serve as good models for the concept of fractals.

Class 8 NCERT bridge course Answers Activity W 4.6 A fractal

 Activity W 4.6 A fractal

Objective:

 This activity will help students to: 

• Understand the concept of fractals.

• Build a simple fractal (Sierpinski Triangle   or Koch Snowflake) using repeated patterns.

Materials Required: 

Paper, markers, ruler (optional), printouts of basic geometric shapes (triangles, squares, etc.). 

A fractal is a shape or pattern that repeats itself no matter how much you zoom into it. 

Procedure

 Guide students to build fractals step-by-step. 

1. Draw a large equilateral triangle on your paper. 

2. Find the midpoints of each side and join them. Remove the middle triangle so obtained. 

It will look as shown—

 3. Again, join the midpoints of each of the sides of the three shaded triangles. 

Remove the three triangles so obtained. It will look like

4. Ask the students to continue the process for at least two more times. 

Take sufficiently large paper or cardboard for this purpose. 

Reflection

 Ask the students to describe the final structure they obtain. 

ANSWER:

The final structure looks like a big triangle made up of smaller and smaller triangles.

Does the final one looks like a big triangle made up of smaller triangles, and if we zoom in,?

ANSWER:

Yes! If we zoom in, the same pattern keeps repeating — this is the main property of a fractal.

 do we see the same pattern inside? This is a fractal. 

ANSWER:

• A fractal shows self-similarity: no matter how much you zoom in or out, the shape looks the same.

• Pattern Observation:

  • Number of triangles increases at every step:

    • Step 1: 1 large triangle.

    • Step 2: 3 shaded triangles.

    • Step 3: 9 smaller shaded triangles.

    • Step 4: 27 even smaller shaded triangles.

  The number of triangles follows this pattern:
  Number of shaded triangles = 3ⁿ (n = step number)

Extension

 1. Students may try this for different shapes, such as square, rectangle, parallelogram or any other shape of their choice. 

1. Try this fractal pattern with other shapes like:

   • Squares (creates a Sierpinski Carpet),

   • Rectangles,

   • Parallelograms,

   • Hexagons.

Squares – Sierpinski Carpet

• Step 1: Start with one large square.

• Step 2: Divide the square into 9 smaller squares (like a tic-tac-toe grid).

• Remove the middle square.

• Step 3: For each remaining square, repeat the same process: divide into 9 parts, remove the middle one.

• Keep repeating!

👉 Pattern:

• Number of remaining squares grows as 8ⁿ (n = step number).

• Creates a flat, grid-like fractal called the Sierpinski Carpet.

Rectangles

• Step 1: Start with a rectangle.

• Step 2: Divide it into equal smaller rectangles (4, 6, 8 parts or more, depending on shape).

• Remove a central part or selected parts.

• Repeat the same for the remaining rectangles.

Pattern:

• You will get a rectangular fractal where the shape keeps splitting into smaller rectangles, showing self-similarity.

Parallelograms

• Step 1: Start with a parallelogram.

• Step 2: Divide it into smaller, identical parallelograms.

• Remove one or more central parts.

• Repeat the process for each remaining parallelogram.

Pattern:

• The structure will look like a stretched or skewed version of the square fractal, but still self-similar and recursive.

Hexagons

• Step 1: Start with a regular hexagon.

• Step 2: Divide the hexagon into 7 smaller hexagons (1 central + 6 surrounding).

• Remove the central hexagon.

• Repeat the process for each remaining hexagon.

Pattern:

• A repeating honeycomb-like fractal — looks similar to snowflake patterns or beehives!

• The edges form increasingly complex, yet balanced, hexagon-based designs.

No matter the shape — square, rectangle, parallelogram, or hexagon — when you:

1. Divide,

2. Remove parts,

3. Repeat the process...

...you create a fractal: a never-ending, self-similar, and geometric pattern.

2. Students may be asked to associate numbers with each shape. 

Say, number of triangles obtained in each step or the number sides on outer edges in each step (3, 6, ...).

 They need to find a pattern.

Pattern in Outer Edges

• As you remove triangles, the number of sides or edges grows.
• For each step, the outer shape becomes more detailed but still follows a predictable mathematical pattern.

Sierpinski Triangle

• Number of Triangles at Each Step:

  Step	Triangles Formed
  0 (Start)	1
  1	3
  2	9
  3	27
  4	81

  Pattern:
  Number of triangles = 3ⁿ (where n = step number)

   Outer Edges (Visible Triangle Sides):

  Step	Outer Edges
  0	3
  1	6
  2	12
  3	24

  Pattern:
  Outer triangle sides double each step: 3 × 2ⁿ

  Sierpinski Carpet (Square Fractal)

   Number of Squares at Each Step:

  Step	Squares Remaining
  0	1
  1	8
  2	64
  3	512

  Pattern:
  Number of squares = 8ⁿ

   Outer Edges:

  Step	Squares on One Side	Total Outer Sides
  0	1	4
  1	3	12
  2	9	36
  3	27	108

  Pattern:

  • Each side is split into 3ⁿ parts.

  • Total outer sides = 4 × 3ⁿ

  Parallelogram Fractal (Custom logic – similar to square)

  If we divide into 4 parts (2×2 layout) and remove the middle one:

   Number of Parallelograms:

  Step	Shapes
  0	1
  1	3
  2	9
  3	27

  Pattern:
  Just like triangle: 3ⁿ

  Hexagon Fractal

  (Similar to honeycomb — 7 small hexagons, remove the center)

  Number of Hexagons:

  Step	Hexagons Remaining
  0	1
  1	6
  2	36
  3	216

  Pattern:
  Number of hexagons = 6ⁿ

  Summary Table:

  Shape	Shapes per Step	Pattern	Outer Sides Pattern
  Triangle	3ⁿ triangles	Exponential	3 × 2ⁿ
  Square	8ⁿ squares	Exponential	4 × 3ⁿ
  Parallelogram	3ⁿ parallelograms	Exponential	Similar to triangle logic
  Hexagon	6ⁿ hexagons	Exponential	Grows like a honeycomb

• 
  

Conclusion:

A fractal is a fascinating geometric figure where the same pattern repeats itself at every scale. 

The Sierpinski Triangle is a perfect example of this, and by exploring other shapes, we can discover many more fractals in nature and mathematics.

Class 8 NCERT bridge course Answers Activity W 4.5 Building Towers with Blocks

 Activity W 4.5: Building Towers with Blocks 

Material Required 

 Different coloured blocks. 

 Paper and pencil for recording results. 

Explain to the students that each block will represent a number, and stacking blocks will help illustrate exponents.

For example, stacking 2 blocks means 2¹ and stacking 4 blocks means 2² and so on

Procedure & Observation

Ask students to create towers representing different powers of 2: 

 For 2¹  , they stack 2 blocks. 

 For 2² , they stack 4 blocks. 

 For 2³ , they stack 8 blocks. 

Students stack blocks to represent powers of 2:

Exponent	Mathematical Form	Number of Blocks	3-D Shape Formed
2¹	2	2 blocks	Cuboid
2²	4	4 blocks	Cube (if arranged as 2x2x1)
2³	8	8 blocks	Cuboid or Cube (if arranged as 2x2x2)

1. After stacking blocks for different exponents, students may be made to explore what 3-D shape they get. 

For example, 2¹ gives a cuboid, whereas 2² gives a cube and so on. 

Discussion:

What 3-D shape do they get?

• For 2¹ = 2 blocks, students usually get a Cuboid (because two blocks form a rectangular shape).

• For 2² = 4 blocks, if arranged symmetrically (2x2), they can get a Cube.

• For 2³ = 8 blocks, if arranged as 2x2x2, it forms a Perfect Cube.

Conclusion:

• When the blocks can be arranged equally in all three dimensions (length, width, and height), the shape is a cube.

• Otherwise, it remains a cuboid.

They may see for which exponent of 2 they get a cube and for which other a cuboid. 

Can they get any other 3-D shape other than a cube or cuboid?

No, using only simple stacking of square or rectangular blocks, the shapes are usually limited to cuboids or cubes.

Other 3-D shapes like pyramids or spheres cannot be formed unless the block shape or stacking style is changed. 

2. They may observe as to how the number of blocks increases as the exponent increases. 

Is there any pattern?

Is there any pattern in the number of blocks as the exponent increases?

Yes! The number of blocks doubles every time the exponent increases by 1.

Exponent	Number of Blocks
2¹	2
2²	4
2³	8
2⁴	16

Pattern:
As the exponent increases by 1, the number of blocks is multiplied by 2.

Conclusion:
This activity helps visualize how exponents grow and how they relate to 3-D shapes like cubes and cuboids. The number of blocks follows a clear doubling pattern as the exponent increases.