Class 8 NCERT bridge course Answers Activity W 5.5 exploring numbers. 

 The activity will help in developing the habit of exploring numbers. 

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Part 1: Two-Digit Number Reversal and Subtraction

Procedure 

1. Ask the students to write a two-digit number. 

2. They should reverse the digits and form a new number. 

3. Subtract the smaller of these numbers from the larger one. 

4. Using the result, repeat the process. 

5. Students may observe when the process stops. 

Discuss about it. 

Example: 

1. Choose a two-digit number (e.g., 52).

2. Reverse the digits → 25.

3. Subtract the smaller from the larger:
   → 52 - 25 = 27

4. Reverse again: 72 - 27 = 45

5. Continue the process:
   → 54 - 45 = 09
   → 90 - 09 = 81

6. Observations:
   

   • Eventually, the number 81 appears.

   • If we check 81 ÷ 9 = 9, and 8 + 1 = 9 → There's a link with the table of 9.

   • Any two-digit number (with different digits) tends to reach a multiple of 9.

   • Numbers like 81, 63, 27 may appear—all are multiples of 9.

   
   

Answer to Q1: Is there any link with the table of 9?

 Yes, the numbers generated through this process often end up being multiples of 9. This happens because reversing and subtracting two-digit numbers preserves a difference divisible by 9.

Answer to Q2: What if the two digits are the same?

• Example: 33
  → Reversed = 33
  → 33 - 33 = 0

• The process stops immediately.
  So, when digits are the same, the result is always 0, and the process ends.

 

Part 2: Three-Digit Number Digit Arrangement and Subtraction

Procedure Recap (Example: 256):

1. Descending: 652

2. Ascending: 256
   → Subtract: 652 - 256 = 396

Then:
→ 963 - 369 = 594
→ 954 - 459 = 495
→ 954 - 459 = 495 again

Observation:

 The process reaches a stable number, 495, and continues to repeat it.

This is known as a Kaprekar constant (for 3-digit numbers).

Discuss 

Answer to Q1: What if any digits are the same?

• Example: 232
  → Descending: 322
  → Ascending: 223
  → 322 - 223 = 099
  → 990 - 099 = 891
  → 981 - 189 = 792
  → 972 - 279 = 693
  → 963 - 369 = 594
  → 954 - 459 = 495

 The process still ends at 495, although it might take longer.

Answer to Q2: What if all three digits are the same?

• Example: 555
  → Descending: 555
  → Ascending: 555
  → 555 - 555 = 0

 Just like in the two-digit case, the process stops immediately.

Extension: Three-digit Number Reverse and Add

Procedure

1. Think of a 3-digit number in which the first and the last digits differ by at least 

 2. Reverse its digits and subtract the smaller number from the larger of the two. 

3. Add the resulting number and its reverse. 

Example:

1. Choose a 3-digit number with first and last digits differing by at least 2
   Example: 321

2. Reverse it: 123

3. Subtract smaller from larger:
   → 321 - 123 = 198

4. Reverse 198 = 891

5. Add both:
   → 198 + 891 = 1089

Observation:

 The final result is 1089 — a famous number trick!

Try others:

• 532 → reverse: 235
  → 532 - 235 = 297
  → 297 + 792 = 1089

This works for most such numbers.

What do you find?

Summary of Key Findings

Activity	Observation
2-digit process	Leads to multiples of 9 (e.g., 81)
2-digit same digits	Results in 0 immediately
3-digit process	Always reaches 495 (Kaprekar constant)
3-digit same digits	Results in 0
3-digit reverse/add	Final result is always 1089