NCERT Solutions for Class 9 Maths Chapter 11 — Exercise Set 11.1#

The book’s algorithm to add two numbers:

  1. Write the numbers one below the other in the Indian base-ten place-value system so that the digits are aligned from right to left.
  2. Add the rightmost digits. If the sum is less than 10, write it below and set carry to 0; otherwise write the units digit below and set carry to 1.
  3. Move to the next column on the left. Add the two digits and the current carry. (Same rule for writing the sum and setting the carry.)
  4. Repeat Step 3 until there are no more digits on the left.
  5. If the carry is 1, write 1 to the left of the bottom row.

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? #

Solution

Take $4738 + 5964$.

Step Column Sum Write Carry
2 units: $8 + 4$ 12 2 1
3 tens: $3 + 6 + 1$ 10 0 1
3 (repeat) hundreds: $7 + 9 + 1$ 17 7 1
3 (repeat) thousands: $4 + 5 + 1$ 10 0 1
5 carry is 1 — 1 on the left —

The bottom row reads 10702, which is correct: $4738 + 5964 = 10702$. Following the steps exactly gives the right answer.

A small gap in the wording: Steps 2 and 3 say “less than 10” and “more than 10”, but do not say what to do when the sum is exactly 10 (as in the tens and thousands columns above). We treated 10 like “more than 10”: write 0 and carry 1. The step should say “10 or more”.

2. What happens if you add a 5-digit number to a 3-digit number. Do our steps handle this situation correctly? #

Solution

Take $72519 + 486$, aligned from the right:

$$\begin{array}{r} 72519 \\ +\ \ \ 486 \\ \hline \end{array}$$

The algorithm works for the units, tens and hundreds columns (giving 5, 0, 0 with carries). But in the thousands column, Step 3 says “add the two digits and the carry” — and the 3-digit number has no digit there.

Strictly, the steps do not say what to do. We must assume that a missing digit counts as 0 (as if we wrote 486 as 00486). With that assumption the answer is 73005, which is correct.

Fix: add a line to Step 1: “If one number has fewer digits, fill the empty places on the left with 0s.”

3. Why is it important to align the columns from right to left? #

Solution

In the place-value system, the rightmost digit is always the units digit, the next is tens, and so on. Lining up the numbers from the right puts units under units, tens under tens, hundreds under hundreds — so we only add digits that have the same place value.

If we lined them up from the left, a units digit could be added to a tens digit. For example, 473 + 95 lined up from the left would add 4 + 9 in the hundreds column, giving a wrong answer (1423 instead of 568).

4. In Step 3, why cannot the value of carry be more than 1? #

Solution

In any column we add two digits, each at most 9, and a carry from the column before. If that carry is at most 1, the largest possible sum is:

$$9 + 9 + 1 = 19$$

which is less than 20. So the new carry (the tens digit of the sum) is at most 1.

The very first column (Step 2) has no carry, so its largest sum is $9 + 9 = 18$, giving a carry of at most 1. Then by the argument above, every later carry is also at most 1.

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. #

Solution

Step 5 writes the final carry. Without it, the last carry is lost.

  • Works correctly when the last column produces no carry. Example: $473 + 215$: columns give 8, 8, 6 with carry 0, so the answer 688 is correct.
  • Fails when the last column produces a carry. Example: $473 + 695$: columns give 8, 6, 1 with a final carry of 1. Without Step 5 we would write 168 instead of the correct 1168.

So Step 5 is needed whenever the sum has more digits than the longer of the two numbers.

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