Verified step-by-step solutions for your class notes.
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: Let's add 4567 and 8924 using the algorithm:
1. Align right to left:
4567
+ 8924
2. Rightmost digits: 7 + 4 = 11. More than 10. Write 1. Carry = 1.
3. Next column (tens): 6 + 2 + 1 (carry) = 9. Less than 10. Write 9. Carry = 0.
4. Repeat (hundreds): 5 + 9 + 0 = 14. More than 10. Write 4. Carry = 1.
5. Repeat (thousands): 4 + 8 + 1 (carry) = 13. More than 10. Write 3. Carry = 1.
6. No more digits on left. Step 5: Carry is 1, so write 1 to the left.
Result: 13491. Yes, the correct result is obtained.
What happens if you add a 5-digit number to a 3-digit number. Do our steps handle this situation correctly?
Answer: Our steps, as strictly written, do not explicitly mention what to do when one number runs out of digits while the other still has digits left. To handle this correctly, we must assume that the missing columns of the 3-digit number are filled with leading zeros so that the "add the two digits" instruction in Step 3 still applies.
Why is it important to align the columns from right to left?
Answer: Aligning from right to left ensures that digits of the same place value (units with units, tens with tens, hundreds with hundreds) are added together. If aligned left to right, we might mistakenly add thousands to hundreds or tens, resulting in an incorrect sum.
In Step 3, why cannot the value of carry be more than 1?
Answer: The maximum single digit is 9. If we add two 9s and a previous carry of 1, the maximum possible sum in any column is $9 + 9 + 1 = 19$. Since 19 is less than 20, the tens digit (which becomes the new carry) can never exceed 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.
Answer: Step 5 adds the final leftover carry to the front of the answer.
Where it works correctly: It works for $32 + 45 = 77$. The final column is $3 + 4 = 7$ (carry 0). Since carry is not 1, Step 5 wouldn't trigger anyway.
Where it fails: It fails for $85 + 42$.
- Units: $5 + 2 = 7$, carry = 0.
- Tens: $8 + 4 + 0 = 12$. Write 2, carry = 1.
Without Step 5, the algorithm ends and gives the answer as 27 instead of 127.
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