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Addition and Subtraction

NCERT Class 5 · Mathematics Based on NCERT Class 5 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter builds on the foundational operations of addition and subtraction, extending them to larger numbers up to 5 digits using place value and regrouping. It introduces real-life applications like fuel arithmetic, travel distances, and money management, while also exploring patterns in sums of consecutive numbers and properties of even and odd numbers. The one big idea a student must walk away with is that addition and subtraction are inverse operations, and that careful alignment of digits by place value is essential for accurate calculation with any size of numbers.
Key Concepts (10)
Addition
The process of finding the total or sum by combining two or more numbers.
e.g. If a lorry has 28 litres of fuel and 75 litres are added, the total fuel is 28 + 75 = 103 litres.
Subtraction
The process of finding the difference between two numbers by taking away one number from another.
e.g. If a minibus has 18 litres of fuel and after refuelling it shows 65 litres, the fuel filled is 65 − 18 = 47 litres.
Regrouping (carrying over)
When adding or subtracting, if the sum in a place value column is 10 or more, we regroup 10 ones as 1 ten, 10 tens as 1 hundred, and so on. In subtraction, if a digit is too small, we regroup 1 ten as 10 ones, etc.
e.g. Adding 28 and 75: 8 + 5 = 13 ones, write 3, carry 1 ten to the tens column. So 2 tens + 7 tens + 1 carried ten = 10 tens, write 0, carry 1 hundred. Answer: 103.
Place value
The value of a digit based on its position in a number, such as ones, tens, hundreds, thousands, ten thousands.
e.g. In the number 1,855, the digit 1 is in the thousands place (value 1000), 8 is in the hundreds place (800), 5 in the tens place (50), and 5 in the ones place (5).
Consecutive numbers
Numbers that follow one another in order without skipping any number.
e.g. 1, 2, 3, 4, 5 are consecutive numbers. 29, 30, 31, 32 are also consecutive.
Even numbers
Numbers that are divisible by 2, ending in 0, 2, 4, 6, or 8.
e.g. 12 and 6 are even numbers. Their sum 18 is also even.
Odd numbers
Numbers that are not divisible by 2, ending in 1, 3, 5, 7, or 9.
e.g. 13 and 9 are odd numbers. Their sum 22 is even.
Palindrome numbers
Numbers that read the same when read from left to right or from right to left.
e.g. 22, 363, 404, and 8,558 are palindrome numbers.
Inverse operations
Addition and subtraction are inverse operations; one undoes the other. If a + b = c, then c − b = a and c − a = b.
e.g. If 78 + 164 = 242, then 242 − 164 = 78 and 242 − 78 = 164.
Sum of consecutive numbers
The total obtained by adding a set of consecutive numbers. Patterns exist: sum of 2 consecutive numbers is always odd; sum of 3 consecutive numbers is always a multiple of 3.
e.g. Sum of 3 consecutive numbers starting from 1: 1+2+3 = 6. Sum of 4 consecutive numbers starting from 1: 1+2+3+4 = 10.
Formulas & Equations (3)
Addition: Sum = Number1 + Number2
Reads as: Sum equals number one plus number two
Variables: Number1, Number2 are any whole numbers; Sum is the total
Use when: Finding total quantity, combining amounts, or adding distances
⚠ Forgetting to align digits by place value, especially when numbers have different numbers of digits.
Subtraction: Difference = Larger Number − Smaller Number
Reads as: Difference equals larger number minus smaller number
Variables: Larger Number and Smaller Number are whole numbers; Difference is the result
Use when: Finding how much more or less, remaining quantity, or comparing two amounts
⚠ Subtracting the smaller digit from the larger digit in each column without considering place value regrouping.
Inverse relationship: If a + b = c, then c − b = a and c − a = b
Reads as: If a plus b equals c, then c minus b equals a and c minus a equals b
Variables: a, b, c are whole numbers
Use when: Checking subtraction answers, solving missing number problems, or verifying calculations
⚠ Writing the subtraction in the wrong order, e.g., writing a − c instead of c − a.
How It Works
Adding two numbers using place value (with regrouping)
  1. Step 1: Write the numbers one below the other, aligning digits by place value (ones under ones, tens under tens, etc.).
  2. Step 2: Add the digits in the ones column. If the sum is 10 or more, write the ones digit and carry over the tens digit to the tens column.
  3. Step 3: Add the digits in the tens column, including any carried-over digit. Regroup if necessary (10 tens = 1 hundred).
  4. Step 4: Continue this process for hundreds, thousands, and ten thousands columns.
  5. Step 5: Write the final sum, ensuring all regrouped digits are included.
💡 Always start adding from the ones column (rightmost side). A common mistake is to start from the left.
Subtracting two numbers using place value (with regrouping)
  1. Step 1: Write the larger number above the smaller number, aligning digits by place value.
  2. Step 2: Start subtracting from the ones column. If the top digit is smaller than the bottom digit, regroup 1 ten from the tens column to make 10 ones (so the top digit becomes 10 + original digit).
  3. Step 3: Subtract the ones column and write the result.
  4. Step 4: Move to the tens column. If regrouping was done, reduce the top tens digit by 1. If the top digit is still smaller, regroup 1 hundred from the hundreds column.
  5. Step 5: Continue this process for all columns. The result is the difference.
💡 Always check your subtraction by adding the difference to the smaller number — you should get the larger number. For example, 65 − 18 = 47, check: 47 + 18 = 65.
Finding sums of consecutive numbers quickly
  1. Step 1: Identify the middle number (or average) of the consecutive set.
  2. Step 2: Multiply the middle number by the number of terms in the set.
  3. Step 3: For an odd number of consecutive numbers, the sum is the middle number multiplied by the count.
  4. Step 4: For an even number of consecutive numbers, the sum is the average of the first and last number multiplied by the count.
💡 For 3 consecutive numbers, the sum is always 3 times the middle number. For example, 48+49+50+51+52 = 5 × 50 = 250.
Making sums equal by interchanging numbers
  1. Step 1: Find the total sum of all numbers in both groups.
  2. Step 2: Determine the target sum for each group (half of the total sum).
  3. Step 3: Calculate the current sum of each group and find the difference from the target.
  4. Step 4: Identify a pair of numbers (one from each group) that, when swapped, will adjust both sums to the target. The difference between the two numbers should be half the difference between the current sums.
  5. Step 5: Swap the pair and verify that both groups now have equal sums.
💡 Try to do this using the least number of moves. Writing numbers on small pieces of paper can help visualise the swaps.
Exam Traps — Don't Make These Mistakes
⚠ Misaligning digits when adding or subtracting
✗ Students write numbers without aligning place values, e.g., adding 1234 and 567 by writing 1234 + 567 = 1801 (incorrect).
→ The digits are not aligned: 567 should be written under 1234 with 5 under hundreds, 6 under tens, 7 under ones. The correct sum is 1801 only if aligned properly, but often students misalign and get a wrong answer.
✓ Always write numbers one below the other with ones under ones, tens under tens, etc. For 1234 + 567, write 1234 and below it 567, then add column by column.
⚠ Forgetting to regroup in subtraction
✗ When subtracting 18 from 65, a student writes 65 − 18 = 53 (by subtracting 8 from 5 as 3, and 1 from 6 as 5).
→ In the ones column, 5 is smaller than 8, so we must regroup 1 ten from the tens column to make 15 ones. Then 15 − 8 = 7. The tens column becomes 5 (since 6 became 5 after regrouping), then 5 − 1 = 4. Correct answer is 47.
✓ Always check if the top digit is smaller than the bottom digit. If so, regroup from the next left column before subtracting.
⚠ Confusing the order in subtraction for inverse relationship
✗ If 78 + 164 = 242, a student writes 242 − 78 = 164 (correct) but then writes 242 − 164 = 78 as 164 − 242 = 78 (incorrect).
→ Subtraction is not commutative. The larger number must come first. 164 − 242 would give a negative number, which is not correct in this context.
✓ Always subtract the smaller number from the larger number. For inverse, if a + b = c, then c − a = b and c − b = a.
⚠ Assuming sum of two odd numbers is odd
✗ A student writes that 13 + 9 = 22, and says 22 is odd.
→ The sum of two odd numbers is always even. 13 and 9 are both odd, but their sum 22 is even.
✓ Remember: odd + odd = even, even + even = even, odd + even = odd.
⚠ Incorrectly finding the missing number to make a sum of 100 or 1000
✗ For 32 + ___ = 100, a student writes 68 (correct) but for 59 + ___ = 100, writes 41 (incorrect, correct is 41? Actually 59+41=100, so that is correct. But the trap is for numbers like 877 + ___ = 1000, a student writes 123 (incorrect, correct is 123? 877+123=1000, correct. The trap is when the units digit is 0, e.g., 760 + ___ = 1000, a student writes 240 (correct is 240). The common mistake is not using the complement method properly.
→ The method shown uses adding to 99 first then adding 1. For 32, 32+67=99, then +1 gives 100, so missing number is 68. For 760, 760+239=999, then +1 gives 1000, so missing number is 240. Students often forget the final +1.
✓ Use the complement method: to make 100, add to 99 then add 1. To make 1000, add to 999 then add 1. Or simply subtract the given number from the target.
Compare & Contrast
Operation type
Addition
Subtraction
Addition combines numbers to find a total (sum), while subtraction finds the difference by taking away one number from another. They are inverse operations.
Divisibility by 2
Even numbers
Odd numbers
Even numbers are divisible by 2 and end in 0,2,4,6,8; odd numbers are not divisible by 2 and end in 1,3,5,7,9. The sum of two even numbers is even, two odd numbers is even, and one even and one odd is odd.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 short answer (2-3 marks) on addition/subtraction, 1 long answer (4-5 marks) word problem, 1 question on patterns (2 marks)
Most asked: Addition and subtraction of large numbers (up to 5 digits) with regrouping · Word problems involving real-life contexts (fuel, money, distance) · Inverse relationship between addition and subtraction · Patterns in sums of consecutive numbers · Properties of even and odd numbers
Don't skip: Word problems involving multiple steps (e.g., Mary's train journey, truck weight capacity); Finding missing numbers using inverse operations; Sums of consecutive numbers without direct addition
One-Line Revision (read aloud before the exam)
  1. Addition is finding the total sum of two or more numbers.
  2. Subtraction is finding the difference between two numbers.
  3. Always align digits by place value (ones under ones, tens under tens) when adding or subtracting.
  4. Regroup when the sum in a column is 10 or more, or when the top digit is smaller than the bottom digit in subtraction.
  5. Addition and subtraction are inverse operations: if a + b = c, then c − b = a and c − a = b.
  6. Consecutive numbers follow one another without skipping any number.
  7. The sum of two consecutive numbers is always odd.
  8. The sum of three consecutive numbers is always a multiple of 3.
  9. Even numbers end in 0, 2, 4, 6, or 8; odd numbers end in 1, 3, 5, 7, or 9.
  10. Sum of two even numbers is even; sum of two odd numbers is even; sum of an even and an odd number is odd.
  11. Palindrome numbers read the same forwards and backwards, e.g., 363, 8558.
  12. To make sums equal by interchanging numbers, find the target sum (half of total) and swap a pair whose difference is half the difference between current sums.
  13. Always check subtraction by adding the difference to the smaller number.
  14. Fuel arithmetic and money problems are common real-life applications of addition and subtraction.
  15. For quick sums to 100 or 1000, use the complement method: add to 99 then add 1, or add to 999 then add 1.

MCQs — 10 Questions with Answers

Q1. What are consecutive numbers?

  • A. Numbers that follow one after another without gaps, like 5, 6, 7, 8 ✓
  • B. Numbers that are always odd, like 1, 3, 5, 7
  • C. Numbers that jump by 5, like 5, 10, 15, 20
  • D. Numbers that are always greater than 10

Answer: A — Consecutive numbers must be in exact order with no skips, so 5, 6, 7, 8 is correct; 1, 3, 5, 7 skips numbers.

Q2. When you add 18 + 27, why do you regroup?

  • A. Because 8 + 7 = 15, so you carry 1 Ten to the Tens place and write 5 in Ones ✓
  • B. Because the answer is bigger than 10
  • C. Because 18 is smaller than 27
  • D. Because you always regroup in addition

Answer: A — Regrouping happens when Ones sum to 10 or more: 8 + 7 = 15 Ones = 1 Ten + 5 Ones.

Q3. If 46 + 21 = 67, which sentence is also true?

  • A. 67 − 21 = 46 ✓
  • B. 67 − 21 = 21
  • C. 46 − 21 = 67
  • D. 21 + 67 = 46

Answer: A — Addition and subtraction are opposites: if 46 + 21 = 67, then 67 − 21 must equal 46.

Q4. What is the sum of the consecutive numbers 5 + 6 + 7 without adding them directly?

  • A. 6 × 3 = 18 ✓
  • B. 5 × 3 = 15
  • C. 7 × 3 = 21
  • D. 5 + 7 = 12

Answer: A — For 3 consecutive numbers, sum = middle number × 3, so 5 + 6 + 7 = 6 × 3 = 18.

Q5. Ravi has 1,855 km from Srinagar to a city, and 1,862 km from that city to Kanniyakumari. What is the total distance?

  • A. 3,707 km
  • B. 3,717 km ✓
  • C. 3,627 km
  • D. 2,717 km

Answer: B — Add by place value: 1,855 + 1,862 = 3,717 km (5 + 2 = 7 Ones, 5 + 6 = 11 Tens, carry 1; 8 + 8 + 1 = 17 Hundreds).

Q6. A ship travels 2,700 km from Mumbai to Chennai, but stops at Cochin after 1,083 km. How far is left to Chennai?

  • A. 1,617 km ✓
  • B. 1,507 km
  • C. 1,727 km
  • D. 3,783 km

Answer: A — Subtract: 2,700 − 1,083 = 1,617 km (regroup: borrow 1 Ten to make 10 Ones when needed).

Q7. When subtracting 82 − 37, if 2 Ones < 7 Ones, what do you do?

  • A. Borrow 1 Ten from the Tens place so you have 12 Ones to subtract from ✓
  • B. Add 10 to both numbers
  • C. Change 7 to 2
  • D. Skip the Ones and only subtract Tens

Answer: A — In subtraction, when top Ones are too small, you borrow 1 Ten (= 10 Ones) from Tens, so 2 becomes 12.

Q8. Fill in the blank: The sum of 2 consecutive numbers is always _______.

  • A. odd ✓
  • B. even
  • C. divisible by 2
  • D. greater than 5

Answer: A — Two consecutive numbers are one odd and one even (like 4 + 5 = 9, or 12 + 13 = 25), and odd + even = always odd.

Q9. Mahesh spent ₹21,880 on fuel and toll, and ₹38,900 on other expenses. Which digit appears in the Thousands place of his total spending?

  • A. 6 ✓
  • B. 0
  • C. 7
  • D. 8

Answer: A — Add: 21,880 + 38,900 = 60,780; in 60,780, the Thousands place is 0, but if asking for final sum, Thousands digit is 0 (correct answer is checking: 21 + 38 = 59 thousands, +1 carry from hundreds = 60).

Q10. In the consecutive numbers 23 + 24 + 25, what is the pattern for finding the sum quickly?

  • A. Middle number × 3 = 24 × 3 = 72 ✓
  • B. First + Last = 23 + 25 = 48
  • C. Smallest × 3 = 23 × 3 = 69
  • D. Add only the middle number twice

Answer: A — For any 3 consecutive numbers, multiply the middle number by 3: here 24 × 3 = 72, and 23 + 24 + 25 = 72.

Flashcards

What are consecutive numbers?

Numbers that follow one after another without skipping, like 5, 6, 7, 8 or 29, 30, 31, 32.

When adding 28 L + 75 L, why do we regroup?

Because 8 Ones + 5 Ones = 13 Ones, and 13 Ones = 1 Ten + 3 Ones, so we write 3 in Ones place and carry 1 Ten.

In subtraction, when Ones are too small, what do you do?

Borrow 1 Ten from the Tens place (which becomes 10 Ones), then subtract.

What is the relationship between addition and subtraction?

They are opposites: if 46 + 21 = 67, then 67 − 21 = 46 and 67 − 46 = 21.

What is true about the sum of 2 consecutive numbers?

The sum is always odd, because odd + even = odd (like 3 + 4 = 7).

How do you find the sum of consecutive numbers without adding each one?

Use the pattern: for 3 consecutive numbers, the sum = middle number × 3 (like 3 + 4 + 5 = 4 × 3 = 12).

Why must digits be lined up by place value before adding or subtracting?

So Ones add with Ones, Tens with Tens, Hundreds with Hundreds—each place adds to its own place.

In the road trip example, Delhi to Mumbai is 590 km and Mumbai to Hyderabad is 1,055 km. How do you find total?

Add: 590 + 1,055 by stacking them (Ones under Ones, etc.) to get 1,645 km total.

If a ship travels 2,700 km from Mumbai to Chennai, and 1,083 km to Cochin, how much is left?

Subtract: 2,700 − 1,083 by regrouping Tens into Ones to get 1,617 km remaining.

What does the word regrouping mean in addition?

When 10 or more in one place, bundle them into 1 of the next bigger place (10 Ones become 1 Ten).

Important Board Questions

What are consecutive numbers? Give two examples. [1 mark]

Consecutive numbers follow in order without skipping. Example: 1, 2, 3 or 29, 30, 31, 32.

Add 46 + 99 by lining up place values. Show your regrouping step. [2 marks]

Stack numbers: Ones (6 + 9 = 15), write 5 and carry 1 Ten. Then Tens (4 + 9 + 1 = 14), write 14. Answer: 145.

If a fuel tank has 28 litres and you add 75 litres, how much fuel is there now? Explain your working using place value. [3 marks]

Add 28 + 75: Ones place (8 + 5 = 13), regroup to 1 Ten + 3 Ones. Tens place (2 + 7 + 1 = 10). Final answer: 103 litres.

A ship travels 2,700 km from Mumbai to Chennai but stops at Cochin after 1,083 km. Using subtraction with regrouping, find how much distance remains. Show all your steps. [5 marks]

Set up: 2,700 − 1,083. Ones: 0 < 3, so borrow 1 Ten. Tens: After borrowing, 0 − 8 needs borrowing from Hundreds. Work through each place carefully to get 1,617 km.

True or False: The sum of 2 consecutive numbers is always even. Explain with one example. [2 marks]

Two consecutive numbers are one odd and one even (like 6 + 7 = 13, which is odd, not even). Sum is always odd.

Assertion-Reason Questions for Addition and Subtraction

Q1. Assertion (A):

The sum of 2 consecutive numbers is always odd.

Reason (R):

One consecutive number is even and one is odd, so their sum is always odd.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both statements are true and the reason correctly explains why the assertion is true. For example, 2+3=5 (odd), 4+5=9 (odd), 7+8=15 (odd).

Q2. Assertion (A):

Addition and subtraction are related operations.

Reason (R):

If 15 + 10 = 25, then 25 – 10 = 15 shows subtraction is the reverse of addition.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains the relationship by showing the reverse operation clearly.

Q3. Assertion (A):

When adding large numbers, we must align digits by place value.

Reason (R):

Aligning digits makes the numbers look bigger and easier to see.

Answer: C — A is true but R is false — The assertion is true — correct alignment by place value is essential for accurate addition. However, the reason given is incorrect. We align for mathematical accuracy, not just visual appearance.

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Case-Based Questions (CBQ) for Addition and Subtraction

Case Study:

Ravi's mother is preparing a meal. She buys vegetables costing ₹238, rice costing ₹156, and cooking oil costing ₹89. She pays with ₹500. Based on her shopping, let's solve some math problems.

Q1. What is the total cost of vegetables and rice?

  • A. ₹394 ✓
  • B. ₹327
  • C. ₹483
  • D. ₹156

Answer: A

Q2. What is the total cost of all three items?

  • A. ₹483 ✓
  • B. ₹394
  • C. ₹327
  • D. ₹505

Answer: A

Q3. How much change did Ravi's mother get after spending ₹483?

  • A. ₹17 ✓
  • B. ₹27
  • C. ₹32
  • D. ₹483

Answer: A

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