IGKO is TODAY!
Mock Test ₹150 View Schedule

Ganita Manjari · Class 9 · Part II · Chapter 11

Ganita Manjari Class 9 Maths Chapter 11 The World of Algorithms Exercise 11.1 Solutions

All 5 questions on the digit-by-digit addition algorithm. Each answer follows the steps exactly, tests them on real numbers, and points out where the steps need a fix.

Last updated 6 October 2026

  • 5Questions
  • 1Diagrams
  • Ch 11The World of Algorithms

How to answer every question

  1. Read the steps exactly Do only what is written. Do not fill in steps from memory.
  2. Trace an example Carry out every step on real numbers and write what happens in each column.
  3. Spot a missing case Look for a situation the steps do not cover, such as a sum of exactly 10.
  4. Fix it and state why Say what to add, and check the repaired steps on the same example.

Algorithm: a step-by-step procedure to arrive at an answer.

Carry: the full ten passed to the column on the left when a column sum is 10 or more.

Place value: what a digit counts: units, tens, hundreds and so on.

Following the steps

Chapter 11 Exercise 11.1 Question 1 Solution Add two 4-digit numbers using the steps of the addition algorithm. Follow the steps precisely and do nothing that is not written. Do you get the correct result?

Answer: 8452, with a gap
Step 1 · Write the numbers one below the other

Take 4758 + 3694. Align the digits from the right: units under units, tens under tens, and so on.

Steps 2 to 4 · Add column by column from the right
ColumnDigitsCarry inSumWriteCarry out
Units8 + 401221
Tens5 + 911551
Hundreds7 + 611441
Thousands4 + 31880
Column addition 4758 + 3694 with carriesCarries of 1 from the units, tens and hundreds columns; the answer is 8452.Q1 · 4758 + 3694, adding from the rightThousandsHundredsTensUnits111carry47583694+8452
Q1 · 4758 + 3694 = 8452, with a carry of 1 from each of the first three columns
Step 5 · Check the final carryCorrect

The last carry is 0, so Step 5 adds nothing. The answer is 8452. Check: 4758 + 3694 = 8452.

A gap in the stepsGap at exactly 10

The steps say: if the sum is less than 10 write it, and if it is more than 10 write the units digit and carry 1. Nothing is said about a sum of exactly 10.

Try 4755 + 3695. The units column gives 5 + 5 = 10, which is neither less than 10 nor more than 10, so a strict reader is stuck. The introduction of the chapter says to carry when the total is *equal to or more than* 10, so the steps should say 10 or more. With that fix: units write 0 and carry 1; tens 5 + 9 + 1 = 15, write 5, carry 1; hundreds 7 + 6 + 1 = 14, write 4, carry 1; thousands 4 + 3 + 1 = 8. The answer is 8450.

Following the steps gives the correct result (8452 for 4758 + 3694) whenever no column adds up to exactly 10. When a column sum is exactly 10 the steps are silent, so "10 or more" is needed.

Chapter 11 Exercise 11.1 Question 2 Solution What happens if you add a 5-digit number to a 3-digit number? Do the steps handle this correctly?

Answer: not quite; needs a fix
Step 1 · Try 52379 + 786

Align from the right. The first three columns have two digits each.

ColumnDigitsCarry inSumWriteCarry out
Units9 + 601551
Tens7 + 811661
Hundreds3 + 711111
Thousands2 and nothing1???
Ten-thousands5 and nothing????
Step 2 · Where the steps failSteps are not enough

In the thousands column the 3-digit number has no digit. Step 3 says "add the two digits", but there is only one digit, and the steps do not say what to do. Step 4 says to repeat "until there are no more digits on the left", which could be read as stopping as soon as one number runs out. Then the carry and the digits 2 and 5 would be lost, giving a wrong answer.

Step 3 · The fixWorks with zeros

Write the shorter number with leading zeros: 52379 + 00786. Now every column has two digits.

ColumnDigitsCarry inSumWriteCarry out
Thousands2 + 01330
Ten-thousands5 + 00550

The answer is 53165. Check: 52379 + 786 = 53165.

As written, the steps do not handle numbers of different lengths. Treat the missing digits as 0 (write leading zeros) and the algorithm works.

Why the steps work

Chapter 11 Exercise 11.1 Question 3 Solution Why is it important to align the columns from right to left?

Answer: place values
Reason 1 · Digits of the same place value must be added together

In a number, the rightmost digit counts units, the next counts tens, then hundreds, and so on. We may add only things of the same kind: units with units, tens with tens. Aligning from the right does exactly that, whatever the length of the numbers.

Example · What goes wrong if we align from the left
AlignmentWritten asSumCorrect?
From the right52379 + 786 (786 under 379)53165Yes
From the left52379 + 78600 (786 under 523)130979No, 786 became 78600

Aligned from the left, the 7 of 786 is added to the 5 of ten-thousands, so the hundreds are treated as ten-thousands.

Reason 2 · Carries move to the left

A carry from one column goes to the column on its left. If we start at the left we do not yet know the carry that comes from the right. Starting at the right, every column already has its carry ready.

Aligning from the right matches equal place values, and it lets each carry travel leftwards to a column that has not been added yet.

Chapter 11 Exercise 11.1 Question 4 Solution In Step 3, why can the value of the carry not be more than 1?

Answer: the largest sum is 19
Step 1 · Find the largest possible column sum

Each digit is at most 9, and the carry coming in is at most 1 (we will show this works by going column by column).

largest sum = 9 + 9 + 1 = 19
Step 2 · What the carry means

The carry is the number of full tens in the column sum. The sum is written as tens and units. Any sum from 10 to 19 is 1 ten and some units, and 20 would be needed for 2 tens.

Column sumTens (carry)Units (written)
1212
1818
1919
Step 3 · Why this keeps going

The first column has no carry in, so its sum is at most 9 + 9 = 18, and the carry is at most 1. Then the next column has a sum of at most 9 + 9 + 1 = 19, so its carry is again at most 1. The same argument repeats for each column.

The biggest possible sum in a column is 9 + 9 + 1 = 19, which is less than 20, so the carry is at most 1.

Testing the last step

Chapter 11 Exercise 11.1 Question 5 Solution What happens if the fifth step ("if the carry is 1, write 1 to the left of the bottom row") is left out? Give examples where the algorithm works and where it fails.

Answer: leading digit gets lost
Step 1 · When the last carry is 0Works

If the left-most column does not produce a carry, nothing is left over. Step 5 does nothing, so leaving it out does no harm.

Example: 321 + 456. Units 1 + 6 = 7, tens 2 + 5 = 7, hundreds 3 + 4 = 7. The answer is 777, which is correct.

Step 2 · When the last carry is 1Fails

If the left-most column gives a carry of 1, that 1 has nowhere to go. The book's own working of 473 + 695 shows it: the carries are the small 1s, and the last one becomes the leading 1 of 1168.

Working of 473 plus 695 in three stages: units give 8, tens give 68 with a carry of 1, and hundreds give 1168 with another carry. Above it are 5 dots and 7 dots, which make 12.
Q5 · the book's working of 473 + 695 (from the book)
SumCorrectWithout Step 5Missing
473 + 6951168168the leading 1
99 + 9919898the leading 1
87 + 6515252the leading 1

Check 473 + 695: units 3 + 5 = 8, tens 7 + 9 = 16 (write 6, carry 1), hundreds 4 + 6 + 1 = 11 (write 1, carry 1). Without Step 5 the answer stops at 168, but the true sum is 1168.

Without Step 5, the algorithm works when the last carry is 0 (321 + 456 = 777) and fails when it is 1 (473 + 695 gives 168 instead of 1168).

Answers at a glance

Every answer comes from tracing the steps on a real example.

QuestionWhat is askedKey ideaAnswer
Q1 Add two 4-digit numbersFollow the steps exactly4758 + 3694 = 8452; gap at a sum of 10
Q2 5-digit plus 3-digitMissing digitsNeeds leading zeros
Q3 Why align from the rightPlace values and carriesSame place value; carries go left
Q4 Why carry is at most 19 + 9 + 1 = 19Less than 20
Q5 Skip Step 5Final carryFails when the last carry is 1

Other chapters of Ganita Manjari (Class 9)

Source of the questions: NCERT, Ganita Manjari, Grade 9, Part II, Chapter 11, Exercise Set 11.1. The solutions, explanations and diagrams on this page are our own working.

Call WhatsApp Book Demo