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

Step-by-step solution

Idea: Step 3 always expects two digits in a column. When one number is shorter, the left-hand columns have only one digit. Writing zeros in front of the shorter number (which does not change its value) fixes this.

1. Take 52746 + 389 and line them up from the right: 9 under 6, 8 under 4 and 3 under 7.
2. Units: 6 + 9 = 15, write 5, carry 1. Tens: 4 + 8 + 1 = 13, write 3, carry 1. Hundreds: 7 + 3 + 1 = 11, write 1, carry 1.1 mark
3. Now the thousands column has only one digit (2); the 3-digit number has run out. Step 3 says “Add the two digits and the current value of carry”, so taken precisely the step cannot be carried out. Step 4 says repeat “until there are no more digits on the left”, but it is not clear whether that means in both numbers or in one. So the steps do not handle this case as written.1 mark
4. Fix: write zeros in front of the shorter number so both have the same number of digits: 389 = 00389. Zeros in front do not change the value. Now every column has two digits: thousands 2 + 0 + 1 = 3, carry 0; ten-thousands 5 + 0 + 0 = 5. Result 53135.1 mark
5. Check: 52746 + 389 = 53135 ✓. (Another fix is to add one line to Step 3: “if a column has only one digit, add that digit and the carry”.)

Check: 53135 − 389 = 52746 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Lining up the 3-digit number under the first three digits from the left, which adds hundreds to ten-thousands.
• Saying “yes, it works” without noticing that Step 3 always talks about two digits.
• Dropping the carry when moving into the columns that have only one digit (writing 2 instead of 3 in the thousands place here).

How this can come in the exam

Try one yourself

Add 41858 + 77 by the column method. Which columns have only one digit before you pad with zeros?

More questions like this

• Why is it important to align the columns from right to left?
• In Step 3, why cannot the value of carry be more than 1?
• 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.
• 1. See if you can complete the argument about grouping by units, tens, hundreds, … to justify why the addition algorithm works.
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• Algorithm to find the divisors of n: 1. Start with an empty list-of-divisors. 2. For each number j in the sequence 1, 2, 3, …, n – if j divides n, add j to the list-of-divisors. Let us execute this algorithm for a small number, say 18.

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