Class 8 NCERT bridge course Answers Activity W6.3 Digit Sum Detectives — The Mystery of 14

 Activity W 6.3 Digit Sum Detectives — The Mystery of 14

Engaging students in these puzzles will make them more observant about numbers. 

Procedure 

1. Students may be given numbers of 3 or 4 digits. 

2. They may add the digits of the number to get a two-digit number. 

3. They may then find other numbers which will give the same sum. 

Example: 

Take the number 176. 1 + 7 + 6 = 14. 

The other number is 545 5 + 4 + 5 = 14. 

Find some more such numbers that give the sum of the digits as 14. 

Extension 

 There could be variations in the puzzle. 

For example, the sum could be a one- digit number, etc.

Find different 3- or 4-digit numbers whose digit sum = 14
Examples:

• 176 → 1 + 7 + 6 = 14

• 545 → 5 + 4 + 5 = 14

• 905 → 9 + 0 + 5 = 14

• 590 → 5 + 9 + 0 = 14

• 410 → 4 + 1 + 9 = 14

• 680 → 6 + 8 + 0 = 14

• 275 → 2 + 7 + 5 = 14

• 8006 → 8 + 0 + 0 + 6 = 14

Students can discover many such combinations.

Numbers of more number of digits can also be thought of.

 Students may try to find this. 

For the sum 14, we may ask the following— 

1. What is the smallest number whose digit sum is 14?
59 → 5 + 9 = 14 (Smallest 2-digit option)

2. What is the largest 5-digit number whose digit sum is 14?
99991 → 9 + 9 + 9 + 9 + 1 = 37 (Too big!)
 Try 98000 → 9 + 8 + 0 + 0 + 0 = 17
Let’s try 99950 → 9 + 9 + 9 + 5 + 0 = 32 → still high

Highest 5-digit with sum = 14 = 95000
→ 9 + 5 + 0 + 0 + 0 = 14

3. How big a number can you form with digit sum = 14?

 No limit to digits — you could do 1400000000000000 → 1 + 4 + many 0s = 5 — too small
 Try 9992 → 9 + 9 + 9 + 2 = 29 → too large
Try numbers like 9990000000000000000005 where total digit sum = 14
→ So yes, very large numbers are possible

4. Can you make an even bigger number?
 Yes! Just keep adding zeroes after a number with digit sum 14.

Students may think and discuss about such different digit sums. 

 Find out the digit sums of all the numbers from 40 to 70. 

Number	Digit Sum
40	4
41	5
42	6
43	7
...	...
59	14
60	6
61	7
...	...
70	7

 You’ll notice that the digit sum increases until 59, then drops again at 60 (6+0=6), creating a V-shape pattern.

Calculate the digit sums of 3-digit numbers, whose digits are consecutive 

(for example, 345). 

Do you see a pattern? Will this pattern continue?

Digit Sums of 3-digit Numbers with Consecutive Digits (e.g., 345):

• 123 → 1 + 2 + 3 = 6

• 234 → 2 + 3 + 4 = 9

• 345 → 3 + 4 + 5 = 12

• 456 → 4 + 5 + 6 = 15

• 567 → 5 + 6 + 7 = 18

Pattern: Increases by 3 as digits increase
Linear increase pattern