Reading and writing large numbers
A bus-stand board shows that 63,497 passengers used the station this month. How do we read such a big number? We use the place value chart: ones, tens, hundreds, thousands and ten thousands.
Each place is 10 times the place on its right: 10 ones = 1 ten, 10 tens = 1 hundred, 10 hundreds = 1 thousand and 10 thousands = 1 ten thousand (10,000). A 5-digit number has the places TTh, Th, H, T, O (TTh = ten thousands).
Formula: Number name = (TTh Th part) thousand + (H digit) hundred + (last two digits).
In the figure: Each box is 10 times the box on its right. 63,497 has 6 in TTh, 3 in Th, 4 in H, 9 in T and 7 in O.
Rule 1: Read from the left — Read the part before the comma and say "thousand"; then read the last three digits.
Reading and writing large numbers
In the app you can move this figure with your finger.
Open in the app- Easy: 6,308 = six thousand three hundred eight.
- Medium: 37,040 = thirty-seven thousand forty.
- Exam-type: 80,106 = eighty thousand one hundred six.
Rule 2: Write from the name — Write the digits of each part you hear, and put 0 in every place that is not named.
- Easy: Fifteen thousand seven = 15,007 (hundreds, tens are 0).
- Medium: Thirty thousand six hundred = 30,600.
- Exam-type: Sixty thousand sixty = 60,060 (not 6,060 and not 60,600).
The comma only helps us read: put it after the thousands (63,497). It is not a digit.
Writing "ten thousand five" as 1,005 or 10,050. Count the places: 10,005 has five digits — 1, 0, 0, 0, 5.
Name the biggest part first (thousands), and never skip a place that holds 0. Memory clue: from the right say "O‑T‑H‑Th‑TTh" — every step to the left makes the value 10 times bigger.
Spot the difference
| Point | 10,005 | 10,050 |
|---|---|---|
| Number name | ten thousand five | ten thousand fifty |
| Tens digit | 0 | 5 |
| Ones digit | 5 | 0 |
Practice questions
- 1Write 5,914 in words.
- 2Which digit is in the ten thousands place of 63,497?
- 3Write "forty-one thousand five hundred twenty" in digits.
- 4What is the value of the digit 7 in 27,356?
- 5Write the number name of 70,008.
- 6Write "thirty-eight thousand two hundred sixty-four" in digits.
- 7A number has 8 in TTh, 0 in Th, 5 in H, 0 in T and 2 in O. Write it in digits and in words.
- 8Write the number that is 1 more than the greatest 4-digit number, and give its name.
Show answers (try them yourself first)
- 1five thousand nine hundred fourteen
- 26
- 341,520
- 47,000 (7 is in the thousands place)
- 5seventy thousand eight
- 638,264
- 780,502 = eighty thousand five hundred two
- 89,999 + 1 = 10,000 = ten thousand
Expanded form and regrouping
A cashier is handed ₹5,280. She thinks: 5 thousands, 2 hundreds and 8 tens. Breaking a number into place values like this makes counting notes easy.
Expanded form writes a number as the sum of the values of its digits. A number can also be regrouped: 1 hundred = 10 tens and 1 thousand = 10 hundreds = 100 tens.
Formula: n = (Th digit × 1,000) + (H digit × 100) + (T digit × 10) + (O digit).
In the figure: 63,497 = 60,000 + 3,000 + 400 + 90 + 7: each digit is written with its place value.
Rule 1: Write the expanded form — Write each non-zero digit with its place value and join the parts with +. Skip the zeros.
Expanded form and regrouping
In the app you can move this figure with your finger.
Open in the app- Easy: 3,092 = 3,000 + 90 + 2.
- Medium: 51,008 = 50,000 + 1,000 + 8.
- Exam-type: 20,603 = 20,000 + 600 + 3 (no thousands, no tens).
Rule 2: Regroup into tens and hundreds — To count in tens divide by 10; to count in hundreds divide by 100. What is left over stays as ones (or tens).
- Easy: ₹3,400 in ₹100 notes: 3,400 ÷ 100 = 34 notes.
- Medium: 783 = 70 tens + 83 ones (70 tens = 700).
- Exam-type: 5,926 = 3 thousands + 29 hundreds + 2 tens + 6 ones (3,000 + 2,900 + 20 + 6).
Zero digits are left out of the expanded form: 3,092 has no hundreds part.
Writing 9,053 as 9,000 + 500 + 3. The digit 5 is in the tens place, so it is 50.
Number of ₹10 notes: drop the last digit. Number of ₹100 notes: drop the last two digits. Memory clue: "one zero off for tens, two zeros off for hundreds" — chop digits from the right.
Spot the difference
| Point | Expanded form | Regrouping |
|---|---|---|
| Example | 9,053 = 9,000 + 50 + 3 | 9,053 = 905 tens + 3 ones |
| Uses | place value of each digit | a bigger unit (tens, hundreds) |
| Operation | add the parts | divide by 10 or 100 |
Practice questions
- 1Write 7,408 in expanded form.
- 2How many ₹10 notes make ₹830?
- 374 = ___ tens + 4 ones. Fill in the blank.
- 4Write 60,309 in expanded form.
- 5592 = 50 tens + ___ ones. Fill in the blank.
- 6How many ₹100 notes make ₹8,600?
- 77,342 = 5 thousands + ___ hundreds + 4 tens + 2 ones. Find the missing number.
- 8Which is more: 28 hundreds or 285 tens?
Show answers (try them yourself first)
- 17,000 + 400 + 8
- 283
- 37
- 460,000 + 300 + 9
- 592 (50 tens = 500, and 592 − 500 = 92)
- 686
- 723 (7,342 − 5,000 − 42 = 2,300 = 23 hundreds)
- 8285 tens = 2,850 is more than 28 hundreds = 2,800
Comparing and ordering numbers
Meena's school bus has run 48,350 km and Ravi's bus has run 9,980 km. Which bus has run more? We compare the two numbers digit by digit.
Step 1: count the digits — the number with more digits is bigger. Step 2: if the digits are equal, compare from the left; the first different digit decides. Ascending = small to big, descending = big to small. Signs: >, <, =.
In the figure: 48,350 has 5 digits and 9,980 has 4 digits (the TTh box is empty), so 48,350 is greater: 4 ten thousands against none.
Rule 1: Count digits, then compare from the left — Different number of digits: the longer number wins. Same number of digits: the left-most different digit decides.
Comparing and ordering numbers
In the app you can move this figure with your finger.
Open in the app- Easy: 9,980 < 48,350 because 4 digits are fewer than 5 digits.
- Medium: 31,246 < 31,264: the first difference is in the tens place (4 < 6).
- Exam-type: 64,712 < 65,102: ten thousands are equal (6 = 6); thousands decide (4 < 5).
Rule 2: Ascending, descending and digit swaps — Find the smallest (or greatest) each time and write it next. For a swap, bring the biggest digit to the highest place.
- Easy: Ascending: 570, 5,070, 50,700 (3, 4 and 5 digits).
- Medium: Descending: 62,150, 62,051, 26,510, 26,105.
- Exam-type: Biggest by swapping two digits of 37,926: swap 3 and 9 → 97,326.
A number with more digits is always greater (as long as there is no leading zero).
Saying 9,980 is greater than 48,350 because 9 is greater than 4. First count the digits!
Write both numbers in the place value chart; the first column (from the left) where the digits differ decides. Memory clue: "Longer wins, then Left-most difference wins" — L and L.
Spot the difference
| Point | Ascending | Descending |
|---|---|---|
| Direction | small → big | big → small |
| Starts with | smallest number | greatest number |
| Numbers 3, 5, 9 | 3, 5, 9 | 9, 5, 3 |
Practice questions
- 1Which is greater: 7,520 or 39,410?
- 2Put >, < or = : 70,926 ___ 70,962.
- 3Arrange in ascending order: 8,030; 830; 80,300.
- 4Which is the greatest: 51,832; 51,382; 58,132; 58,312?
- 5Arrange in descending order: 91,206; 19,602; 92,106; 29,610.
- 6Write any 3 numbers between 31,420 and 32,075.
- 7Interchange two digits of 25,874 to make the greatest possible number.
- 8How many numbers between 38,200 and 46,900 have 0 in the ones, tens and hundreds places?
Show answers (try them yourself first)
- 139,410 (it has more digits)
- 270,926 < 70,962 (2 tens < 6 tens)
- 3830, 8,030, 80,300
- 458,312
- 592,106, 91,206, 29,610, 19,602
- 6For example 31,500; 32,000; 32,050
- 785,274 (swap 2 and 8)
- 88 (39,000, 40,000, … , 46,000)
Nearest ten, hundred and thousand
A shopkeeper says there are "about 3,300 mangoes" in the godown when the exact count is 3,284. Using a handy nearby number like this is called rounding.
To round to the nearest ten, hundred or thousand, look at the digit just to the right of that place. 0 to 4: keep the place digit and make the rest 0 (round down). 5 to 9: add 1 to the place digit and make the rest 0 (round up).
Formula: Nearest ten: look at the ones digit. Nearest hundred: look at the tens digit. Nearest thousand: look at the hundreds digit.
In the figure: 4,628 lies between 4,620 and 4,630. It is nearer to 4,630, only 2 jumps away, so the nearest ten is 4,630.
Rule 1: Find the two neighbours — The number lies between two round numbers. The nearer one is the answer; a number exactly in the middle goes up.
Nearest ten, hundred and thousand
In the app you can move this figure with your finger.
Open in the app- Easy: 4,628: neighbours 4,620 and 4,630 → nearest ten 4,630 (2 jumps).
- Medium: 6,953: neighbours 6,950 and 6,960 → nearest ten 6,950 (3 jumps).
- Exam-type: 7,405 is exactly in the middle of 7,400 and 7,410, so it goes up to 7,410.
Rule 2: Round to hundreds and thousands — Decide with one digit: the tens digit for hundreds, the hundreds digit for thousands. Then make the digits to the right 0.
- Easy: 6,953 to the nearest hundred: the tens digit is 5, so round up → 7,000.
- Medium: 3,284: nearest hundred 3,300 (tens digit 8); nearest thousand 3,000 (hundreds digit 2).
- Exam-type: A number with nearest hundred 5,000 and nearest thousand 5,000: for example 4,980 or 5,030 (anything from 4,950 to 5,049).
Only one digit decides: the digit just to the right of the place you are rounding to.
Rounding 3,284 to the nearest hundred as 3,200 because the hundreds digit is 2. Look at the tens digit 8: 8 is 5 or more, so the answer is 3,300.
5 or more: go up. 4 or less: stay down. Memory clue: "Five or more, raise the score; four or less, let it rest."
Spot the difference
| Point | Nearest ten | Nearest hundred | Nearest thousand |
|---|---|---|---|
| Digit to look at | ones | tens | hundreds |
| 3,284 becomes | 3,280 | 3,300 | 3,000 |
| Digit checked in 3,284 | 4 | 8 | 2 |
Practice questions
- 1Round 5,172 to the nearest ten.
- 2Round 4,380 to the nearest hundred.
- 3Round 6,240 to the nearest thousand.
- 4Round 8,962 to the nearest hundred.
- 5A rabbit is at 7,396. How many jumps to the nearest ten?
- 6Round 2,508 to the nearest hundred and to the nearest thousand.
- 7Write two numbers whose nearest hundred and nearest thousand are both 8,000.
- 8Priya rounds 6,497 to the nearest hundred; Arun rounds it to the nearest thousand. Whose answer is bigger, and by how much?
Show answers (try them yourself first)
- 15,170
- 24,400
- 36,000
- 49,000 (tens digit 6 → up)
- 54 jumps (to 7,400)
- 62,500 and 3,000 (the hundreds digit is 5, so up)
- 7For example 7,980 and 8,020 (any number from 7,950 to 8,049)
- 8Priya: 6,500; Arun: 6,000. Priya's answer is bigger by 500.
Travel and large-number problems
A school trip needs vehicles for 600 children. A bus carries 50 children and travels 40 km every hour. How many buses are needed, and how far will they go in 3 hours? Large numbers appear in every journey.
Distance = speed × time. Time = distance ÷ speed. To find the number of vehicles divide the people by the seats; if some people are left over, one more vehicle is needed. 1 km = 1,000 m.
Formula: Distance = speed × time ; Vehicles = people ÷ seats (round up).
In the figure: A cyclist who rides 18 km each hour reaches 18, 36, 54 and 72 km: 4 hours × 18 km = 72 km.
Rule 1: Use distance = speed × time — Multiply the distance covered in one hour by the number of hours. To find the time, divide.
- Easy: A train goes 80 km in 1 hour: in 5 hours it goes 80 × 5 = 400 km.
- Medium: At 60 km an hour, 360 km takes 360 ÷ 60 = 6 hours.
- Exam-type: An aeroplane flies 800 km an hour: in 3 hours it covers 800 × 3 = 2,400 km.
Rule 2: Find the number of vehicles — Add up all the people, divide by the seats in one vehicle, and if some are left over take one more vehicle.
- Easy: 120 people, 4 in a car: 120 ÷ 4 = 30 cars.
- Medium: 145 people, 20 in a boat: 145 ÷ 20 = 7 remainder 5, so 8 boats.
- Exam-type: 260 girls and 190 boys, 25 in a minibus: 450 ÷ 25 = 18 minibuses exactly.
Always check the unit: speed in km per hour gives distance in km. 1 km = 1,000 m.
Ignoring the leftover people: 145 ÷ 20 = 7 remainder 5 does NOT mean 7 boats. The 5 people still need a boat, so 8.
Speed, time and distance: cover the one you want in the triangle D = S × T and read the other two. Memory clue: "Left over? One more!" — a remainder always means one extra vehicle.
Spot the difference
| Question | Distance | Number of vehicles |
|---|---|---|
| Operation | multiply speed × time | divide people ÷ seats |
| Remainder | not needed | add one more vehicle |
| Example | 40 × 3 = 120 km | 600 ÷ 50 = 12 buses |
Practice questions
- 1A bus goes 40 km in 1 hour. How far in 3 hours?
- 2How many metres are there in 7 km?
- 3A ship sails 36 km in 1 hour. How far in 5 hours?
- 4A motorbike rider goes 50 km every hour. How long to cover 300 km?
- 5180 people are going on a trip. A tempo traveller seats 10. How many are needed?
- 6A book has about 120 pages with about 80 words on each page. About how many words are there?
- 7A school has 238 girls and 192 boys going on a trip. A minibus seats 25. How many minibuses are needed?
- 8A spacecraft travels at least 28,000 km every hour. What is the least distance in 3 hours? Is it a 5-digit number?
Show answers (try them yourself first)
- 1120 km
- 27,000 m
- 3180 km
- 4300 ÷ 50 = 6 hours
- 5180 ÷ 10 = 18
- 6120 × 80 = 9,600 words
- 7238 + 192 = 430; 430 ÷ 25 = 17 remainder 5, so 18 minibuses
- 828,000 × 3 = 84,000 km; yes, it has 5 digits