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Algorithms · 2 marks

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

Answer: The biggest possible column sum is 9 + 9 + 1 = 19, which is less than 20. So a column never makes two tens, and the carry is always 0 or 1.

Step-by-step solution

Idea: Find the largest number a column can ever add up to. Each digit is at most 9 and the carry coming in is at most 1, so the total is at most 19.

  1. In Step 2 (the rightmost column) we add two digits only. Each digit is at most 9, so the sum is at most 9 + 9 = 18 < 20. So the first carry is 0 or 1.½ mark
  2. In Step 3 we add two digits and the carry. If the carry coming in is at most 1, the sum is at most 9 + 9 + 1 = 19.½ mark
  3. A sum of at most 19 is less than 20, so it has at most one ten. The carry passed to the next column is therefore 0 or 1.½ mark
  4. The same argument then works for the next column, and the next, all the way to the left. So the carry can never be more than 1.½ mark
Because the largest possible column sum is 9 + 9 + 1 = 19 < 20, a column produces at most one ten, so the carry is always 0 or 1.

Answer to write in the exam

Each digit ≤ 9.

Step 2: sum ≤ 9 + 9 = 18 < 20, so carry ≤ 1.

Step 3: sum ≤ 9 + 9 + 1 = 19 < 20, so carry passed on ≤ 1.

Repeating for every column, carry is never more than 1.

∴ The carry is always 0 or 1.

Common mistakes that cost marks

  • Writing the largest sum as 9 + 9 = 18 and forgetting the carry in Step 3. The correct largest sum is 19.
  • Saying “because we only carry 1” without explaining why the sum can never reach 20.
  • Bringing in the three-number case. With three numbers a carry of 2 is possible, but this algorithm adds only two numbers, so the largest column sum is 9 + 9 + 1 = 19.

How this can come in the exam

MCQ (1 mark)

When two numbers are added column by column, the largest possible sum in a single column (two digits plus the carry) is

  1. 18
  2. 19
  3. 20
  4. 27
Show answer

(B) 19
9 + 9 + 1 = 19.

Assertion–Reason (1 mark)

Assertion (A): When three numbers are added column by column, the carry can be 2.
Reason (R): Three digits and a carry can add up to more than 19.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
With three numbers a column can reach 9 + 9 + 9 + 2 = 29 (for example 9 + 9 + 9 = 27 gives carry 2), so the carry can be 2. R is true and explains A.

Try one yourself

Add 9999 + 9999 by the algorithm. What carry does each column produce, and what is the largest column sum?

Show answer

Units 9 + 9 = 18: write 8, carry 1. Tens, hundreds, thousands: 9 + 9 + 1 = 19: write 9, carry 1 each time. Step 5 writes the last 1. Answer 19998; the largest column sum is 19, so every carry is 1.

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