NCERT Solutions Class 9 Mathematics Ganita Manjari 2 Chapter 11 The World of Algorithms Exercise 11.1

Exercise Set 11.1

Question 1. Add two 4-digit numbers using the steps we have written down. Make sure you follow the steps precisely; do not perform any action that is not explicitly mentioned. Are you able to obtain the correct result?
Answer:
Take 4756 + 3869 and apply the steps word for word.
Step 1: Write 3869 below 4756 so that the digits line up from the right.
Step 2: Units: 6 + 9 = 15. This is more than 10, so write 5 and set carry = 1.
Step 3: Tens: 5 + 6 + 1 = 12, more than 10, so write 2 and set carry = 1.
Step 4 (repeat Step 3): Hundreds: 7 + 8 + 1 = 16, write 6, carry = 1. Thousands: 4 + 3 + 1 = 8, less than 10, write 8, carry = 0. No digits remain on the left.
Step 5: carry is 0, so nothing more is written.

The steps give 4756 + 3869 = 8625, which is correct (8625 − 3869 = 4756).
But look carefully at the wording. The steps only say what to do when a column total is less than 10 or more than 10. Try 2345 + 1665:

In the units column 5 + 5 = 10, which is neither less than 10 nor more than 10. Followed literally, the steps give no instruction here, and we get stuck.
With the rule "write the units digit and carry 1 when the total is 10 or more": 5 + 5 = 10 → write 0, carry 1; 4 + 6 + 1 = 11 → write 1, carry 1; 3 + 6 + 1 = 10 → write 0, carry 1; 2 + 1 + 1 = 4 → write 4. So 2345 + 1665 = 4010.
Conclusion: for most pairs (such as 4756 + 3869 = 8625) the steps give the right sum, but they are incomplete: nothing is said when a column total is exactly 10. "More than 10" must be changed to "equal to or more than 10" in Steps 2 and 3.

Teacher's Note:
1. The point of the question is precision: an algorithm must cover every case, including the boundary case of exactly 10.
2. Ask students to invent their own pair of numbers with a column total of 10 and try the literal steps.

Question 2. What happens if you add a 5-digit number to a 3-digit number. Do our steps handle this situation correctly?
Answer:
Try 45678 + 789.
Units: 8 + 9 = 17 → write 7, carry = 1.
Tens: 7 + 8 + 1 = 16 → write 6, carry = 1.
Hundreds: 6 + 7 + 1 = 14 → write 4, carry = 1.
Thousands: Step 4 asks us to repeat Step 3, which says "add the two digits and the carry". But this column has only one digit (5); 789 has no thousands digit. The steps do not say what to do, nor whether "no more digits on the left" refers to one number or both.
Repair: treat every missing digit as 0, that is, write 789 as 00789.
Thousands: 5 + 0 + 1 = 6 → write 6, carry = 0. Ten-thousands: 4 + 0 + 0 = 4 → write 4, carry = 0. Step 5: carry is 0, nothing more to write.

So 45678 + 789 = 46467 (check: 46467 − 789 = 45678).
Answer: the steps, as written, do not fully handle this case, because Step 3 cannot be carried out once the shorter number runs out of digits. Adding the instruction "if a number has no digit in a column, take that digit as 0" fixes the algorithm, and then it gives the correct sum.

Teacher's Note:
1. Filling with leading zeros does not change the value of a number, so it is always safe.
2. This is a good example of a hidden assumption that a human reader supplies automatically but an algorithm must state.

Question 3. Why is it important to align the columns from right to left?
Answer:
In our place-value system the rightmost digit of every whole number is the units digit, the next is the tens digit, and so on. Numbers may have different lengths, so place values match only at the right end.
The algorithm adds column by column, which is correct only when each column holds digits of the same place value.
Example: 473 + 25.

Aligned from the left, the 2 tens of 25 sit under the 4 hundreds and the 5 units under the 7 tens. We are really adding 473 + 250 and get 723, which is wrong.
Aligned from the right, units are under units and tens under tens, giving 473 + 25 = 498, which is correct.
Also, a carry is a group of ten of one place value, which becomes one of the next place value to the left. This works only if the column on the left really is the next higher place value.
Right alignment makes sure that digits of the same place value are added together and that each carry goes to the next higher place value; otherwise the sum is wrong.

Question 4. In Step 3, why cannot the value of carry be more than 1?
Answer:
Every digit is at most 9.
In Step 2 only two digits are added, so the largest total is 9 + 9 = 18 < 20. This total contains at most one ten, so the first carry is 0 or 1.
In Step 3 we add two digits and the carry. If the incoming carry is at most 1, the largest total is 9 + 9 + 1 = 19 < 20. Again at most one ten is formed, so the new carry is 0 or 1.
The first carry is at most 1, and each use of Step 3 keeps it at most 1, so however many times Step 3 is repeated, the carry never reaches 2.
The biggest possible column total is 9 + 9 + 1 = 19, which is less than 20, so a column can never produce two tens. Hence carry is always 0 or 1.

Teacher's Note:
1. This is really a small proof by induction: the carry starts at most 1 and each step preserves that.
2. Contrast with adding three numbers, where a column can total 9 + 9 + 9 + 2 = 29 and the carry can be 2.

Question 5. What happens if we do not include the fifth step in the algorithm above? Give examples where the algorithm will work correctly and where it will fail to work.
Answer:
Step 5 writes the last carry (a 1) at the left end of the answer. If it is dropped, any carry produced by the leftmost column is lost, and the answer comes out short by 1 followed by as many zeros as there are columns.
Examples where it still works (the leftmost column gives no carry):
(a) 234 + 512: 4 + 2 = 6, 3 + 1 = 4, 2 + 5 = 7 → 746. Correct.
(b) 4756 + 3869: the last column gives 4 + 3 + 1 = 8 with carry 0 → 8625. Correct.
Examples where it fails (the leftmost column gives a carry):
(c) 873 + 456: 3 + 6 = 9 (carry 0); 7 + 5 = 12 → write 2, carry 1; 8 + 4 + 1 = 13 → write 3, carry 1. Without Step 5 we stop at 329, but the true sum is 1329.

(d) 473 + 695 gives 168 instead of 1168.
(e) 5 + 7 gives 2 instead of 12.
Without Step 5 the algorithm is correct only when the sum has the same number of digits as the longer number (no carry out of the leftmost column). It fails whenever the sum has one extra digit, e.g. 873 + 456 would give 329 instead of 1329.