Numbers — Class 5 Mathematics (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 5 Mathematics, Unit 2. Numbers beyond 10,000, place value, comparison, ordering and operations. Taught in June and July.
1. Beyond ten thousand
Last year the largest number you handled was 10,000. Real life needs bigger ones.
A television costs ₹18,500. A mobile phone ₹15,250. An LPG cylinder ₹975. A wooden cot ₹30,000. A car ₹4,50,000.
To go further we need two new words.
One lakh = 1,00,000 and one crore = 1,00,00,000
A lakh can be seen several ways: it is 10 ten-thousands, or 100 thousands, or 1000 hundreds, or 100000 ones. A crore is 100 lakhs.
2. Commas, the Indian way
Big numbers are hard to read in a solid block of digits, so we group them with commas. The Indian system groups them into periods.
Working from the right, the first comma comes after three digits, and every comma after that comes after two.
- 341964 becomes 3,41,964 — three lakh forty one thousand nine hundred and sixty four
- 347810 becomes 3,47,810
- 15731997 becomes 1,57,31,997
3. The place value chart
| C | T.L | L | T.Th | Th | H | T | O |
|---|---|---|---|---|---|---|---|
| Crore | Ten lakhs | Lakhs | Ten thousands | Thousands | Hundreds | Tens | Ones |
Reading 34,284. Number name: thirty four thousand two hundred and eighty four. Expanded form: 3 ten thousands + 4 thousands + 2 hundreds + 8 tens + 4 ones = 30,000 + 4,000 + 200 + 80 + 4
Reading 1,21,35,211. Number name: one crore twenty one lakhs thirty five thousand two hundred and eleven. Expanded form: 1,00,00,000 + 20,00,000 + 1,00,000 + 30,000 + 5,000 + 200 + 10 + 1
4. Place value of each digit
The place value of a digit is the digit multiplied by the value of its column.
Take 34,56,789:
| Digit | Column | Place value |
|---|---|---|
| 9 | Ones | 9 × 1 = 9 |
| 8 | Tens | 8 × 10 = 80 |
| 7 | Hundreds | 7 × 100 = 700 |
| 6 | Thousands | 6 × 1000 = 6,000 |
| 5 | Ten thousands | 5 × 10000 = 50,000 |
| 4 | Lakhs | 4 × 100000 = 4,00,000 |
| 3 | Ten lakhs | 3 × 1000000 = 30,00,000 |
5. Comparing numbers
Rule 1 — count the digits. A number with more digits is larger.
3241 has 4 digits and 20344 has 5, so 3241 < 20344. You do not need to look at a single digit beyond that.
Rule 2 — same number of digits? Then compare from the highest place and move right, stopping at the first column where they differ.
Compare 73652 and 56372: both have 5 digits, and 7 is greater than 5 in the ten-thousands column, so 73652 is greater.
- 54,689 < 54,869 (the hundreds decide: 6 against 8)
- 75,432 > 75,412 (the tens decide: 3 against 1)
6. Ascending and descending order
Ascending order arranges numbers from smallest to greatest. Descending order goes from greatest to smallest.
Arrange 413, 43, 986, 38490, 8490 in ascending order. Sort by number of digits first, then within each group:
Ascending: 43, 413, 986, 8490, 38490
7. Addition
Line the numbers up by place value and start from the ones column on the right. Working right to left is what makes carrying possible.
Ananthan's father buys wedding clothes: gents ₹25,050, ladies ₹47,025, kids ₹7,125, and bride and groom ₹17,500.
25050 + 47025 + 7125 + 17500 = ₹96,700
8. Subtraction
Subtraction finds the difference between two numbers, and the answer is called the difference.
78,347 − 59,475 = 18,872
Ravi has 4021 stamps and Rahul has 3289. Ravi has 4021 − 3289 = 732 more.
9. Multiplication
Multiply by the ones digit, then by the tens digit, and add the two results. Remember the placeholder zero on the second line.
350 × 35:
- 350 × 5 = 1750
- 350 × 30 = 10500
- 1750 + 10500 = 12,250
A three-digit number by a two-digit number. Raveena plants 15 rows of coconut trees with 112 trees in each row.
Total trees = 112 × 15 = 1,680
10. Division
Division is the odd one out. For addition, subtraction and multiplication we start at the ones place; for division we start from the left, with the highest place.
53675 ÷ 8:
Quotient = 6709, remainder = 3.
Check it with the rule that ties division together:
Dividend = Divisor × Quotient + Remainder
Here 8 × 6709 + 3 = 53672 + 3 = 53675. Correct.
A car factory makes 3750 cars in a month of 30 days. Cars per day = 3750 ÷ 30 = 125, with no remainder.
4327 ÷ 18 gives quotient 240 and remainder 7. Check: 18 × 240 + 7 = 4320 + 7 = 4327.
11. Factors
Kavitha brings home 12 laddus and wants to share them equally with none left over. She could give 1 each to 12 friends, 2 each to 6 friends, 3 each to 4 friends, 4 each to 3 friends, or 6 each to 2 friends. The numbers that divide 12 exactly are 1, 2, 3, 4, 6 and 12.
A factor of a number is a number that divides it with no remainder.
The reliable way to list factors is to hunt in pairs, starting at 1 and working inwards:
| Number | Pairs that multiply to it | Factors |
|---|---|---|
| 18 | 1 × 18, 2 × 9, 3 × 6 | 1, 2, 3, 6, 9, 18 |
| 20 | 1 × 20, 2 × 10, 4 × 5 | 1, 2, 4, 5, 10, 20 |
| 24 | 1 × 24, 2 × 12, 3 × 8, 4 × 6 | 1, 2, 3, 4, 6, 8, 12, 24 |
Stop as soon as the pairs begin to repeat themselves. Every number has 1 and itself as factors, so those two come free, and no factor is ever larger than the number.
12. Multiples
A rabbit jumps 4 steps at a time. After ten jumps it has reached 4, 8, 12, 16, 20, 24, 28, 32, 36, 40. These are the multiples of 4, and they are exactly the answers in the 4 times table.
Notice the difference. A number has only a fixed handful of factors, but its multiples go on for ever.
The multiples two numbers share are their common multiples.
- Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
- Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108
Both lists contain 36 and 72, so those are common multiples of 9 and 12.
13. Prime and composite numbers
Count how many factors a number has and it tells you which family it belongs to.
- A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19.
- A composite number has more than two factors: 4, 6, 8, 9, 12, 60.
- 1 is neither prime nor composite. It has only one factor, so it fails both tests.
2 is the only even prime number. Every other even number has 2 as a factor on top of 1 and itself, which already makes three factors.
14. Lowest common multiple
The L.C.M. of two or more numbers is the smallest number that all of them divide into exactly.
To find the L.C.M. of 4 and 6, list the multiples of each until a number appears in both lists.
- Multiples of 4: 4, 8, 12, 16, 20, 24
- Multiples of 6: 6, 12, 18, 24, 30
12 is the first number common to both, so the L.C.M. of 4 and 6 is 12.
For larger numbers, listing takes too long and splitting into prime factors is faster. Since 8 = 2 × 2 × 2 and 12 = 2 × 2 × 3, take each factor the greatest number of times it appears in either number: 2 × 2 × 2 × 3 = 24.
L.C.M. answers real questions. If one bus leaves a stand every 4 minutes and another every 6 minutes, and they leave together now, they will next leave together 12 minutes from now.
15. Estimation
Look at a bunch of grapes and say roughly how many there are before counting them. That is an estimate, an answer found quickly and accepted as close rather than exact.
Estimating is the right tool when the precise figure does not matter, and it is also a check on your own arithmetic. To estimate 297 + 412, round each to the nearest hundred: 300 + 400 = 700. The true answer is 709.
Work the estimate out first, then calculate properly. If the two are far apart, one of them is wrong and it is worth finding out which.
16. Systematic ordering
Some questions are answered not by calculating but by running through the possibilities in a sensible order. Having the standard sequences by heart makes this quick:
| Sequence | First terms |
|---|---|
| Natural numbers | 1, 2, 3, 4, 5 |
| Odd numbers | 1, 3, 5, 7, 9 |
| Even numbers | 2, 4, 6, 8, 10 |
| Square numbers | 1, 4, 9, 16, 25 |
| Prime numbers | 2, 3, 5, 7, 11 |
Suppose a number is wanted that is less than 20, even, and a multiple of 6. Go along the multiples of 6 in order, 6, 12, 18, 24, and stop at 18. Searching in order is what stops you missing a case or testing the same one twice.
17. Worked examples
Example 1. Place commas in 341964 and write its number name.
Solution: 3,41,964 — three lakh forty one thousand nine hundred and sixty four.
Example 2. Write the place value of 5 in 5,86,012.
Solution: 5 is in the lakhs column, so its place value is 5,00,000.
Example 3. Write 63,570 in expanded form.
Solution: 60,000 + 3,000 + 500 + 70 + 0 = 60,000 + 3,000 + 500 + 70.
Example 4. Which is greater, 43,731 or 44,371?
Solution: Both have 5 digits and the same ten-thousands digit. The thousands decide: 4 against 3, so 44,371 is greater.
Example 5. Find the sum of the greatest 4-digit number and the smallest 5-digit number.
Solution: 9999 + 10000 = 19,999.
Example 6. Multiply 473 × 48.
Solution: 473 × 8 = 3784, and 473 × 40 = 18920. Total = 22,704.
Example 7. There are 55 mangoes in a basket and one mango costs ₹15. Find the total cost.
Solution: 55 × 15 = ₹825.
Example 8. Find the quotient and remainder for 5732 ÷ 9.
Solution: Quotient 636, remainder 8. Check: 9 × 636 + 8 = 5724 + 8 = 5732.
18. Practice
- Write 1,00,000 as a number of ten thousands.
- Place commas in 29121972 and write its number name.
- Write the place value of 5 in 2,87,500.
- Write 30000 + 3000 + 300 + 30 + 3 in standard notation.
- Which is smaller, 20344 or 3241?
- Arrange in ascending order: 33,270; 1,078; 137; 27,935.
- Add 19732 + 24105 + 525 + 48.
- Subtract 3,748 from 67,056.
- Multiply 854 × 21.
- Find the quotient and remainder for 2532 ÷ 4.
- List all the factors of 36.
- Find the L.C.M. of 8 and 6.
- Which of these are prime numbers: 15, 17, 21, 23?
- Estimate 497 + 312 by rounding to the nearest hundred, then find the exact answer.
19. Answers
- 10 ten thousands.
- 2,91,21,972 — two crore ninety one lakh twenty one thousand nine hundred and seventy two.
- 5 is in the hundreds column, so its place value is 500.
- 33,333.
- 3241, because it has 4 digits while 20344 has 5.
- 137; 1,078; 27,935; 33,270.
- 44,410.
- 63,308.
- 17,934.
- Quotient 633, remainder 0. Check: 4 × 633 = 2532.
- 1, 2, 3, 4, 6, 9, 12, 18 and 36.
- Multiples of 8 are 8, 16, 24; multiples of 6 are 6, 12, 18, 24. The L.C.M. is 24.
- 17 and 23 are prime. 15 = 3 × 5 and 21 = 3 × 7, so both are composite.
- Estimate 500 + 300 = 800. The exact answer is 809.
20. Summary
- One lakh is 1,00,000 and one crore is 1,00,00,000; a lakh is 10 ten-thousands.
- Indian commas fall after the first three digits from the right, then every two digits.
- The place value chart runs C, T.L, L, T.Th, Th, H, T, O.
- Place value is the digit multiplied by its column value.
- More digits means a larger number; with equal digits, compare from the highest place.
- Ascending goes smallest to greatest, descending the other way.
- Add, subtract and multiply starting from the ones place; divide starting from the left.
- In multiplication, write the placeholder zero before multiplying by the tens digit.
- Dividend = Divisor × Quotient + Remainder is the check for every division.
