Class 9 Maths Ganita Manjari Chapter 11 Exercise 11.1 Solutions The World of Algorithms
The Class 9 Maths Ganita Manjari Chapter 11 Exercise 11.1 Solutions – The World of Algorithms help students understand how step-by-step instructions are used to solve mathematical problems accurately. This exercise mainly focuses on algorithms through familiar concepts such as addition, place value and logical sequencing of steps.
The solutions are explained in a simple and systematic manner so that students can understand the reasoning behind each step, identify missing instructions and learn how an algorithm can be improved for different situations. Students can also access the Class 9 Maths Ganita Manjari Chapter 11 Exercise 11.1 Solutions PDF for quick revision and convenient offline study.
NCERT Class 9 Maths Ganita Manjari Chapter 11 Exercise Set 11.1
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:
The printed steps work for some pairs of numbers, but they do not cover every possible column total.
For example:
2468 + 1357
Units: 8 + 7 = 15. Write 5 and carry 1.
Tens: 6 + 5 + 1 = 12. Write 2 and carry 1.
Hundreds: 4 + 3 + 1 = 8. Write 8 and set the carry to 0.
Thousands: 2 + 1 = 3. Write 3.
Therefore,
2468 + 1357 = 3825.
However, consider:
4586 + 3414
In the units column,
6 + 4 = 10.
The printed algorithm gives one instruction when the total is less than 10 and another when it is more than 10. It does not tell us what to do when the total is exactly 10.
Therefore, if we follow only the written instructions, we cannot complete the addition.
Correction: Replace “more than 10” with “10 or more.” When the total is 10, write 0 in that column and carry 1 to the next column.
With this correction,
4586 + 3414 = 8000.
2. What happens if you add a 5-digit number to a 3-digit number? Do our steps handle this situation correctly?
Answer:
If we add a 5-digit number to a 3-digit number, the numbers must be aligned according to place value.
For example:
45,672 + 389
Write them as:
45,672
389
The units, tens, and hundreds digits line up. Since 389 has no thousands or ten-thousands digits, those missing places can be treated as 0.
Units: 2 + 9 = 11. Write 1 and carry 1.
Tens: 7 + 8 + 1 = 16. Write 6 and carry 1.
Hundreds: 6 + 3 + 1 = 10. Write 0 and carry 1.
Thousands: 5 + 0 + 1 = 6.
Ten-thousands: 4 + 0 = 4.
Therefore,
45,672 + 389 = 46,061.
If our written steps only explain how to add numbers with the same number of digits, then they do not completely handle this situation.
Correction: The algorithm should say to align the numbers by their units place and treat any missing digits on the left as 0. Then continue adding until all place-value columns have been processed.
3. Why is it important to align the columns from right to left?
Answer:
It is important to align the columns from right to left because each digit must be placed under the digit with the same place value.
The rightmost digit is always the units digit. So, when numbers are aligned from the right, units come under units, tens under tens, hundreds under hundreds, and so on.
For example, when adding 45,672 and 389, we write:
45,672
389
This ensures that 2 and 9 are both in the units column, 7 and 8 are in the tens column, and 6 and 3 are in the hundreds column.
If we align the numbers from the left instead, digits with different place values may be added together, which gives an incorrect result.
Therefore, aligning from right to left preserves place value and makes the addition algorithm work correctly.
4. In Step 3, why cannot the value of carry be more than 1?
Answer:
In Step 3, we are adding two digits and possibly a carry from the previous column.
The largest possible digit is 9, and the carry can be at most 1. So the greatest possible total in any column is:
9 + 9 + 1 = 19
Since 19 contains only 1 group of ten, the carry to the next column can only be 1.
Therefore, when adding two numbers, the value of the carry can never be more than 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.
Answer:
If we do not include the fifth step, the algorithm will fail whenever there is a carry left after adding the leftmost column.
For example, consider:
2468 + 1357
The addition gives:
2468
+1357
─────
3825
There is no carry left after adding the thousands column, so the algorithm still gives the correct answer even without the fifth step.
Now consider:
6789 + 5678
Units: 9 + 8 = 17. Write 7 and carry 1.
Tens: 8 + 7 + 1 = 16. Write 6 and carry 1.
Hundreds: 7 + 6 + 1 = 14. Write 4 and carry 1.
Thousands: 6 + 5 + 1 = 12. Write 2 and carry 1.
If the fifth step is missing, there is no instruction to write the final carry 1. The algorithm would give 2467 instead of the correct answer, 12,467.
Therefore, the fifth step is necessary because it tells us to write any carry that remains after the leftmost column has been added.
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