Chapter 3 exercise answers: Number Play

Open NCERT p. 57Matches NCERT’s answer

1. A cell is a supercell only if it beats every number sitting right next to it.
2. An end cell (first or last) has only one neighbour, so it only has to beat that one number.
3. 6828 is the first cell. Its only neighbour is 670. 6828 > 670, so 6828 is a supercell.
4. 670 has neighbours 6828 and 9435. It is smaller than both, so it is not a supercell.
5. 9435 has neighbours 670 and 3780. It beats both, so 9435 is a supercell.
6. 3780 has neighbours 9435 and 3708. 3780 < 9435, so it is not a supercell.
7. 3708 has neighbours 3780 and 7308. 3708 < 7308, so it is not a supercell.
8. 7308 has neighbours 3708 and 8000. 7308 < 8000, so it is not a supercell.
9. 8000 has neighbours 7308 and 5583. It beats both, so 8000 is a supercell.
10. 5583 has neighbours 8000 and 52. 5583 < 8000, so it is not a supercell.
11. 52 is the last cell. Its only neighbour is 5583. 52 < 5583, so it is not a supercell.

Answer6828, 9435 and 8000 are the supercells.

Watch this explained “Walking the printed row, cell by cell”, 2:17 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 57Checked by computerAnswers can differ: one example

1. On the page three cells are coloured: the 2nd cell, the 4th cell (1258) and the last cell. These, and only these, must be supercells.
2. 1258 is coloured, so it must beat both its neighbours. The 3rd and 5th cells must therefore be 4-digit numbers smaller than 1258: use 1000 and 1100.
3. The 2nd cell is coloured, so it must beat 5346 and 1000. Use 6000. Then 5346 < 6000, so 5346 (white) is not a supercell, as required.
4. The 3rd cell, 1000, is smaller than both 6000 and 1258, so it is not a supercell.
5. The 5th, 6th and 7th cells are white. Fill them with 1100, 2000, 3000: each is smaller than the cell on its right (1100 < 2000 < 3000 < 9635), so none of them is a supercell.
6. 9635 is white, and its left neighbour 3000 is smaller, so its right neighbour must be bigger than 9635. The last cell is coloured and has only one neighbour, so use 9999: 9999 > 9635.
7. Check the whole row 5346, 6000, 1000, 1258, 1100, 2000, 3000, 9635, 9999: the supercells are 6000, 1258 and 9999 — exactly the coloured cells.

AnswerFill the blanks with 6000, 1000, 1100, 2000, 3000, 9999, giving the row 5346, 6000, 1000, 1258, 1100, 2000, 3000, 9635, 9999. Many other fillings also work.

Watch this explained “Table 2 backwards — the shading forces the corner”, 8:13 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 57Checked by computerAnswers can differ: one example

1. Two supercells can never sit side by side: if two neighbouring cells were both supercells, each would have to be bigger than the other, which is impossible.
2. So in a row of 9 cells, at most every other cell can be a supercell: cells 1, 3, 5, 7 and 9. That is 5 cells.
3. To make all five of them win, put a big number in each of them and a small number in each cell between them.
4. Big numbers 990, 980, 970, 960, 950 go in cells 1, 3, 5, 7, 9. Small numbers 110, 120, 130, 140 go in cells 2, 4, 6, 8.
5. Every big number beats the small numbers beside it, and every small number loses to the big numbers beside it. All nine numbers are between 100 and 1000, and none is repeated.

Answer990, 110, 980, 120, 970, 130, 960, 140, 950. This gives 5 supercells (990, 980, 970, 960, 950), the most possible in 9 cells. Many other fillings also work.

Watch this explained “And five is reachable: high, low, high, low”, 5:40 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 57Matches NCERT’s answer

1. “The table above” is the table of question 3, printed just above this question. It was filled to get as many supercells as possible.
2. Two supercells can never be next to each other: if two neighbouring cells were both supercells, each would have to be bigger than the other, which is impossible.
3. So there is at least one ordinary cell between any two supercells. In 9 cells the most we can choose this way is cells 1, 3, 5, 7 and 9 — that is 5 cells.
4. 5 is really reached: in the filling 990, 110, 980, 120, 970, 130, 960, 140, 950, the supercells are 990, 980, 970, 960 and 950.

Answer5 supercells.

Watch this explained “So nine cells can never hold more than five”, 4:56 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 57One way to think about it

1. Two supercells can never be next-door neighbours, because whichever of the two is smaller cannot beat the other.
2. So supercells must always have at least one non-supercell between them.
3. For 1 cell, the greatest number of supercells is 1. For 2 cells, it is 1. For 3 cells, it is 2. For 4 cells, it is 2. For 5 cells, it is 3.
4. The pattern: for n cells, the greatest number of supercells is n divided by 2, rounded up.
5. Method to reach that many: put a big number in cells 1, 3, 5, 7, … and a small number in cells 2, 4, 6, …, so every big cell beats the small cells on either side of it.

In shortThe greatest number of supercells in n cells is n ÷ 2 rounded up. Get it by alternating high, low, high, low, … along the row.

Watch this explained “So nine cells can never hold more than five”, 4:56 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 58One way to think about it

1. Look at the cell holding the single biggest number in the whole table.
2. Since no number repeats, every neighbour of that cell holds a smaller number.
3. So that cell always beats every neighbour it has — it is always a supercell.
4. This is true no matter how the numbers are arranged, so a table with no repeated numbers can never have zero supercells.

In shortNo, it is not possible. The cell with the overall biggest number is always bigger than its neighbours, so it is always a supercell.

Watch this explained “The largest always wins its cell; the smallest never does”, 6:34 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 58One way to think about it

1. The largest number in the table is bigger than every other number, including whichever numbers sit next to it.
2. So the cell with the largest number always beats its neighbour or neighbours — it is always a supercell.
3. The smallest number in the table is smaller than every other number, including its neighbours.
4. A supercell must beat every neighbour it has, but the smallest number cannot beat any neighbour — so it can never be a supercell.

In shortYes, the largest number is always a supercell. No, the smallest number can never be a supercell.

Watch this explained “The largest always wins its cell; the smallest never does”, 6:34 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 58Checked by computerAnswers can differ: one example

1. Put the second largest number right next to the largest number.
2. Use 900 as the largest and 800 as the second largest, with 800 sitting beside 900.
3. 800 cannot beat 900, so 800 is not a supercell, however the rest of the table is filled.
4. Table: 500, 900, 800, 200, 100. Only 900 is a supercell; 800 loses to its neighbour 900.

Answer500, 900, 800, 200, 100 — the second largest number, 800, sits next to the largest, 900, so it loses and is not a supercell.

Watch this explained “The rule: a cell wins if it beats everyone beside it”, 0:49 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 58Checked by computerAnswers can differ: one example

1. For the second smallest number to be a supercell, it must beat every neighbour it has — easiest if it sits at an end, with only one neighbour, and that neighbour is the smallest number.
2. Put 20 (second smallest) at one end and 10 (the smallest) right next to it: 20 beats 10, so 20 is a supercell.
3. For the second largest number to not be a supercell, sit it next to the largest number so it loses.
4. Put 100 (largest) and 90 (second largest) at the other end, with 90 next to 100: 90 loses to 100.
5. Table: 20, 10, 30, 100, 90. Supercells: 20 (beats 10) and 100 (beats 30 and 90). 90 is not a supercell, and 20 is — exactly what was asked.

AnswerYes, it is possible. Example: 20, 10, 30, 100, 90.

Watch this explained “The second row, and the two cases that catch people out”, 3:06 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 58One way to think about it

1. Try a table with two rows and two columns, where each cell compares with the cell above, below, left and right of it, not just left and right.
2. Try asking for the fewest possible supercells instead of the most.
3. Try a circular row, where the first and last cells are also neighbours of each other.

In shortFor example: make a grid version (comparing up/down as well as left/right), or ask for the smallest possible number of supercells instead of the largest.

Watch the lesson Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 59Matches NCERT’s answer

(a) 2010 2020

1. There are 10 equally spaced marks. 2010 is on the 5th mark and 2020 is on the 7th mark, 2 marks apart.
2. 2020 − 2010 = 10, spread over 2 marks, so each mark is 10 ÷ 2 = 5 apart.
3. Counting by 5s from the 1st mark: 1990, 1995, 2000, 2005, 2010, 2015, 2020, 2025, 2030, 2035.

AnswerRemaining marks: 1990, 1995, 2000, 2005, 2015, 2025, 2030, 2035. Smallest: 1990. Largest: 2035.

(b) 9996 9997

1. 9996 and 9997 are on two marks right next to each other, so each mark is 9997 − 9996 = 1 apart.
2. Counting by 1s from the 1st mark: 9993, 9994, 9995, 9996, 9997, 9998, 9999, 10000, 10001, 10002.
3. The line crosses from a 4-digit number into a 5-digit number between the 7th and 8th marks.

AnswerRemaining marks: 9993, 9994, 9995, 9998, 9999, 10000, 10001, 10002. Smallest: 9993. Largest: 10002.

(c) 15,077 15,078 15,083

1. 15,077 and 15,078 are on two marks next to each other, so each mark is 1 apart.
2. 15,083 is 5 marks after 15,078, and 15,083 − 15,078 = 5, which matches — a good check that the step really is 1.
3. Counting by 1s from the 1st mark: 15,077 up to 15,086.

AnswerRemaining marks: 15,079, 15,080, 15,081, 15,082, 15,084, 15,085, 15,086. Smallest: 15,077. Largest: 15,086.

(d) 86,705 87,705

1. 86,705 and 87,705 are on two marks next to each other, so each mark is 87,705 − 86,705 = 1,000 apart.
2. Counting by 1,000s from the 1st mark: 83,705 up to 92,705.

AnswerRemaining marks: 83,705, 84,705, 85,705, 88,705, 89,705, 90,705, 91,705, 92,705. Smallest: 83,705. Largest: 92,705.

Watch this explained “Turning it round: two labels, and the step falls out”, 6:03 into Placing a large number on the number line by feel · हिंदी में देखें

Open NCERT p. 60Checked by computerAnswers can differ: one example

(a) Write other numbers whose digits add up to 14.

1. The book already shows 68, 176 and 545, so choose different numbers whose digits add up to 14.
2. 7 + 7 = 14, 8 + 6 = 14, 9 + 5 = 14, 1 + 4 + 9 = 14, 2 + 3 + 9 = 14.

Answer77, 86, 95, 149 and 239 (there are many more).

(b) What is the smallest number whose digit sum is 14?

1. A number with fewer digits is smaller, so use as few digits as possible.
2. The biggest single digit is 9, which is less than 14, so we need at least 2 digits.
3. With 2 digits adding to 14, the tens digit should be as small as possible. The units digit can be at most 9, so the tens digit is at least 14 − 9 = 5. That gives 59.

Answer59

(c) What is the largest 5-digit whose digit sum is 14?

1. To make a 5-digit number as large as possible, make the leftmost digit as big as possible first.
2. The first digit can be 9, leaving 14 − 9 = 5 for the other 4 digits.
3. The second digit takes all of that 5, so the last three digits are 0.
4. 9, 5, 0, 0, 0 gives 95,000, and 9 + 5 + 0 + 0 + 0 = 14.

Answer95,000

(d) How big a number can you form … digit sum of 14? …

1. 95,000 has digit sum 9 + 5 = 14.
2. Put one more 0 at the end: 9,50,000. A zero adds nothing to the digit sum, so it is still 14, and the number is ten times bigger.
3. We can keep adding zeros (95,00,000 and so on), so there is no biggest number with digit sum 14 — we can always make an even bigger one.

Answer95,000 has digit sum 14, and 9,50,000 is bigger with the same digit sum. Adding zeros never stops, so there is no largest such number.

Watch this explained “The smallest number adding to 14, and why it is 59”, 5:24 into How many numbers have a given number of digits, and why digit sums behave · हिंदी में देखें

Open NCERT p. 60Checked by computer

1. From 40 to 49, the tens digit stays 4 and the units digit goes 0 to 9, so the digit sum goes up by 1 each time: 4, 5, 6, …, 13.
2. At 50, the tens digit becomes 5 and the units digit resets to 0, so the digit sum drops back down to 5, then climbs again: 5, 6, …, 14.
3. At 60, the same thing happens again: the digit sum drops to 6 and climbs to 15.
4. 70 is the last number, with digit sum 7.

Answer4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7.

Watch this explained “Counting 40 to 70: up one, up one, then down eight”, 7:19 into How many numbers have a given number of digits, and why digit sums behave · हिंदी में देखें

Open NCERT p. 60Checked by computerReads two ways: both answers shown

1. 3-digit numbers whose digits go up by 1: 123, 234, 345, 456, 567, 678, 789. There are no more, because the next one would need a digit 10.
2. Digit sums: 1 + 2 + 3 = 6, 2 + 3 + 4 = 9, 3 + 4 + 5 = 12, then 15, 18, 21, 24.
3. Pattern: each sum is 3 more than the one before. Also, each sum is 3 times the middle digit, because the first digit is 1 less and the last digit 1 more than the middle: 3 + 4 + 5 = 4 + 4 + 4 = 3 × 4 = 12. So every sum is a multiple of 3.
4. If you also count digits going down by 1 (210, 321, …, 987), you get the same sums and one more: 2 + 1 + 0 = 3.
5. The book does not say which pattern 'this pattern' means, so here are both. Read as 'will the list of sums keep going up by 3?': no. It stops at 24 (for 789), because digits stop at 9.
6. Read as 'will the rule keep working?': yes. For every 3-digit number with consecutive digits, the digit sum is 3 × the middle digit, always a multiple of 3.

AnswerDigit sums: 6, 9, 12, 15, 18, 21, 24 (counting falling digits too, such as 210, also 3). Read as 'does the list keep going up by 3?': no, it stops at 24 (789). Read as 'does the rule keep working?': yes, every sum is 3 × the middle digit, a multiple of 3.

Watch this explained “Consecutive digits: every sum a three, and only seven exist”, 8:13 into How many numbers have a given number of digits, and why digit sums behave · हिंदी में देखें

Open NCERT p. 64Checked by computerAnswers can differ: one example

(a) the difference between the largest and smallest numbers greater than 5085

1. A bigger spread between the digits gives a bigger difference.
2. Digits 9, 8, 2, 1: largest = 9821, smallest = 1289, difference = 9821 − 1289 = 8532, which is more than 5085.

AnswerDigits 9, 8, 2, 1 (difference 8,532).

(b) the difference between the largest and smallest numbers less than 5085

1. Digits that are close together give a smaller difference.
2. Digits 5, 6, 7, 8: largest = 8765, smallest = 5678, difference = 8765 − 5678 = 3087, which is less than 5085.

AnswerDigits 5, 6, 7, 8 (difference 3,087).

(c) the sum of the largest and smallest numbers greater than 9779

1. Bigger digits overall give a bigger sum.
2. Digits 9, 8, 7, 6: largest = 9876, smallest = 6789, sum = 9876 + 6789 = 16,665, which is more than 9779.

AnswerDigits 9, 8, 7, 6 (sum 16,665).

(d) the sum of the largest and smallest numbers less than 9779

1. Smaller digits overall give a smaller sum.
2. Digits 1, 2, 3, 4: largest = 4321, smallest = 1234, sum = 4321 + 1234 = 5555, which is less than 9779.

AnswerDigits 1, 2, 3, 4 (sum 5,555).

Watch this explained “Pratibha's four digits: 999 and 90, the lever in the puzzle”, 9:06 into Clock times and calendar dates: patterns the format lets you have · हिंदी में देखें

Open NCERT p. 65Matches NCERT’s answer

1. A 5-digit palindrome reads the same forwards and backwards, so it looks like AB C BA, where A cannot be 0.
2. To make it as small as possible, make A as small as it can be (1, since it can't be 0) and B, C as small as possible (0): 1 0 0 0 1 = 10,001.
3. To make it as large as possible, make every digit as big as possible: 9 9 9 9 9 = 99,999.
4. Sum: 10,001 + 99,999 = 1,10,000.
5. Difference: 99,999 − 10,001 = 89,998.

AnswerSum = 1,10,000. Difference = 89,998.

Watch this explained “The property is about the writing, not the size”, 0:46 into Palindromes, and why reverse-and-add usually lands on one · हिंदी में देखें

Open NCERT p. 65Matches NCERT’s answer

1. 10:01 itself reads the same forwards and backwards (1, 0, 0, 1), but the question wants the next one after now.
2. Checking each later time minute by minute, the next palindromic time is 11:11 (1, 1, 1, 1) — 70 minutes after 10:01.
3. Continuing on, the one after that is 12:21 (1, 2, 2, 1) — another 70 minutes later, so 140 minutes after 10:01 in total.

Answer70 minutes to 11:11, and 140 minutes in total to 12:21.

Watch this explained “From 10:01, how long until the next? Twice seventy minutes”, 4:30 into Clock times and calendar dates: patterns the format lets you have · हिंदी में देखें

Open NCERT p. 65Checked by computer

1. Round 1: 5683 → largest 8653, smallest 3568, 8653 − 3568 = 5085.
2. Round 2: 5085 → largest 8550, smallest 0558, 8550 − 558 = 7992.
3. Round 3: 7992 → largest 9972, smallest 2799, 9972 − 2799 = 7173.
4. Round 4: 7173 → largest 7731, smallest 1377, 7731 − 1377 = 6354.
5. Round 5: 6354 → largest 6543, smallest 3456, 6543 − 3456 = 3087.
6. Round 6: 3087 → largest 8730, smallest 0378, 8730 − 378 = 8352.
7. Round 7: 8352 → largest 8532, smallest 2358, 8532 − 2358 = 6174 — reached the Kaprekar constant.
8. The answer key at the back of the book prints 8 rounds; it makes the smallest number from 5, 0, 8, 5 as 5058, but the smallest is 0558 (= 558), so 8550 − 558 = 7992 and the answer is 7 rounds.

Answer7 rounds.

Watch this explained “The zero the chapter never mentions”, 6:43 into Kaprekar's 6174: a process that always ends in the same place · हिंदी में देखें

Open NCERT p. 66Checked by computerAnswers can differ: one example

1. The smallest 5-digit number is 10,000 and the largest is 99,999 (and the smallest/largest 3- and 4-digit numbers are 100/999 and 1,000/9,999).
2. For each box, work out the smallest and largest value the sum or difference could ever be, using those extreme numbers.
3. If the number asked for lies inside that smallest-to-largest range, pick two numbers that actually land on it. If it lies outside the range, no example can exist.

4. Scenario	Example, or why it is not possible
   5-digit + 5-digit → 5-digit sum more than 90,250	45,200 + 45,100 = 90,300
   5-digit + 3-digit → 6-digit sum	99,999 + 100 = 100,099
   4-digit + 4-digit → 6-digit sum	Not possible: the biggest such sum is 9,999 + 9,999 = 19,998, which has only 5 digits.
   5-digit + 5-digit → 6-digit sum	60,000 + 50,000 = 110,000
   5-digit + 5-digit → 18,500	Not possible: the smallest such sum is 10,000 + 10,000 = 20,000, which is already more than 18,500.
   5-digit − 5-digit → difference less than 56,503	30,000 − 25,000 = 5,000
   5-digit − 3-digit → 4-digit difference	10,500 − 600 = 9,900
   5-digit − 4-digit → 4-digit difference	10,500 − 2,000 = 8,500
   5-digit − 5-digit → 3-digit difference	10,500 − 10,100 = 400
   5-digit − 5-digit → 91,500	Not possible: the biggest such difference is 99,999 − 10,000 = 89,999, which is less than 91,500.

Answer7 of the 10 boxes have a working example (shown in the table); the other 3 are impossible because the target number falls outside the smallest-to-largest range the sum or difference can ever reach.

Watch this explained “An example proves possible; only a bound proves impossible”, 5:58 into Rearranging a sum so it can be done in your head · हिंदी में देखें

Open NCERT p. 67Matches NCERT’s answer

(a) 5-digit number + 5-digit number gives a 5-digit number

1. Smallest possible sum: 10,000 + 10,000 = 20,000 — that is a 5-digit number.
2. Largest possible sum: 99,999 + 99,999 = 199,998 — that is a 6-digit number.
3. Since the sum can be either 5-digit or 6-digit, it is not always a 5-digit number.

AnswerOnly sometimes true.

(b) 4-digit number + 2-digit number gives a 4-digit number

1. Smallest possible sum: 1,000 + 10 = 1,010 — a 4-digit number.
2. Largest possible sum: 9,999 + 99 = 10,098 — a 5-digit number.
3. So the sum is sometimes 4-digit and sometimes 5-digit.

AnswerOnly sometimes true.

(c) 4-digit number + 2-digit number gives a 6-digit number

1. The biggest a 4-digit number can be is 9,999, and the biggest 2-digit number is 99.
2. So the biggest possible sum is 9,999 + 99 = 10,098, which has only 5 digits.
3. A 6-digit number must be at least 100,000, and this sum never gets close to that.

AnswerNever true.

(d) 5-digit number – 5-digit number gives a 5-digit number

1. 10,000 − 10,000 = 0, which is not a 5-digit number.
2. 99,999 − 10,000 = 89,999, which is a 5-digit number.
3. So the difference is sometimes 5-digit and sometimes not.

AnswerOnly sometimes true.

(e) 5-digit number – 2-digit number gives a 3-digit number

1. To make the difference as small as possible, subtract the biggest 2-digit number from the smallest 5-digit number: 10,000 − 99 = 9,901.
2. 9,901 already has 4 digits, and every other 5-digit-minus-2-digit difference is bigger still.
3. So the difference can never shrink down to only 3 digits.

AnswerNever true.

Watch this explained “An example proves possible; only a bound proves impossible”, 5:58 into Rearranging a sum so it can be done in your head · हिंदी में देखें

Open NCERT p. 70One way to think about it

a From the place you are sitting to the classroom door

1. This distance is short, so you can simply count while you walk.
2. Walk from your seat to the classroom door at your normal pace and count every step.

In shortCount your steps directly. The number is different for every student and classroom; for example, about 12 steps.

b Across the school ground from start to end

1. This distance is too long to count every step accurately, so first measure your own step length.
2. Mark out 10 m with a measuring tape and count how many normal walking steps it takes. Say it takes 14 steps, so one step is about 71 cm.
3. Estimate how many 10 m lengths fit across the ground, then multiply that number by 14. For example, a 60 m ground is 6 lengths of 10 m, so 6 × 14 = 84 steps.

In shortPace out 10 m to find your step length, then scale up. The answer differs from school to school; for a 60 m ground at 14 steps per 10 m, it is about 84 steps.

c From your classroom door to the school gate

1. Use the same measurement: 14 steps for every 10 m.
2. Estimate the distance from the classroom door to the school gate in lengths of 10 m.
3. Multiply. For example, 150 m is 15 lengths of 10 m, so 15 × 14 = 210 steps.

In shortIt depends on the school's layout; for a gate 150 m from the classroom door, about 210 steps.

d From your school to your home

1. This is usually the longest of the four distances, so counting every step is impractical.
2. Find the distance from school to home in metres or kilometres (ask an adult or use a map), or compare it with a distance you have already estimated, such as the length of the school ground.
3. Convert it to steps. For example, 1.5 km = 1,500 m = 150 lengths of 10 m, so 150 × 14 = 2,100 steps.

In shortIt is different for every student; for a home 1.5 km from school, about 2,100 steps.

Watch this explained “Pacing: measure ten metres, then multiply”, 3:54 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 70One way to think about it

a In a minute

1. Pick either blinking or breathing to count — it is too fast to count for a whole day, so count for a short time instead.
2. Watch a clock and count how many times it happens in exactly one minute.

In shortCount directly for one minute; for breathing this is commonly around 15 breaths a minute (it varies from person to person).

b In an hour

1. There are 60 minutes in an hour.
2. Multiply the one-minute count by 60.
3. 15 breaths a minute × 60 = 900 breaths an hour.

In shortMultiply the one-minute count by 60; 15 breaths a minute gives about 900 breaths an hour.

c In a day

1. There are 24 hours in a day, so multiply the one-hour estimate by 24.
2. 900 breaths an hour × 24 = 21,600 breaths a day.
3. This assumes the rate stays the same all day, which is not quite true — breathing slows and blinking almost stops during sleep — so the real count is a little lower. Say this assumption out loud with the answer.

In shortMultiply the one-hour estimate by 24; 900 breaths an hour gives about 21,600 breaths a day, though the true count is a little lower because the rate drops during sleep.

Watch this explained “A rate stretched twice: a minute, an hour, a day”, 4:44 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 70One way to think about it

a a few thousand in number

1. Many answers are correct: any objects whose number is in the thousands will do.
2. Think of an everyday object made up of many small identical pieces, and judge whether the count is in the thousands.
3. A bowl (katori) of cooked rice holds roughly two thousand grains.
4. The bricks in the walls of a small one-room house also number a few thousand.

In shortMany answers are possible. For example: the grains of rice in one bowl (about 2,000), or the bricks in the walls of a small room (a few thousand).

b more than ten thousand in number

1. Again many answers are correct; check that the count you are thinking of crosses ten thousand.
2. The hairs on an average person's head number about a lakh, well past ten thousand.
3. A large stadium filled with spectators can hold tens of thousands of people.

In shortMany answers are possible. For example: the hairs on a person's head (about 1,00,000), or the spectators filling a large stadium (tens of thousands).

Watch the lesson Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 70One way to think about it

1. Count the words on one page of your maths book that is mostly writing. A full page of this book has about 200 words.
2. This chapter, Number Play, alone runs from page 55 to page 73. That is 19 pages.
3. So this one chapter has about 19 × 200 = 3,800 words.
4. The book has many more chapters than this one, so the whole book has many times 3,800 words.

In shortMore than 5000. One chapter alone has about 19 × 200 = 3,800 words, and the book has many chapters, so the total is far more than 5,000. To check, count the words on one page of your own book and multiply by the number of pages.

Watch the lesson Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 70One way to think about it

1. Count how many children in your own class come to school by bus. Suppose it is 8.
2. Assume every section of every class is about the same. In Paromita's school there are Classes 6 to 10 with 3 sections each: 5 × 3 = 15 sections.
3. Estimate for the school: 8 × 15 = 120 children come by bus.
4. 120 is less than 200, so for this school the answer is 'less than 200'.

In shortIt depends on your school, so different answers can be right. For example, 8 bus riders in each of 15 sections gives 8 × 15 = 120, which is less than 200. A school where most children come by bus could come out at more than 200 by the same method.

Watch this explained “Five year groups, a hundred each: about 500”, 2:13 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 70One way to think about it

1. List what the ₹100 must pay for: milk, custard powder and sugar, and three kinds of fruit — enough for 5 people.
2. Use example prices (they change from place to place and season to season): about 1 litre of milk ≈ ₹60, custard powder and sugar ≈ ₹20.
3. That is already ₹60 + ₹20 = ₹80, which leaves only ₹20 for three kinds of fruit for 5 people.
4. Fruit for 5 people, for example bananas ≈ ₹30, apples ≈ ₹40 and grapes ≈ ₹40: ₹30 + ₹40 + ₹40 = ₹110.
5. Total ≈ ₹80 + ₹110 = ₹190.

In shortIt depends on local prices and on which fruits he buys, so give your reason, not just yes or no. With the example prices here the custard costs about ₹190, so ₹100 is too low and we do not agree with Roshan. It could come close to ₹100 only with less milk and very cheap fruit in season.

Watch this explained “Roshan's ₹100: “it depends” — but finish the sentence”, 7:15 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 70One way to think about it

1. Find both cities on a map of India: Gandhinagar lies in the west, in Gujarat, and Kohima lies in the far northeast, in Nagaland.
2. Use the map's scale bar with a ruler (or a piece of string laid between the two cities and then measured) to find the straight-line gap, and convert that length using the scale.
3. This straight-line ('as the crow flies') distance works out to roughly 2,200 km.
4. Roads are never perfectly straight: they bend around hills, rivers and towns, typically adding about a quarter to a half more distance, so the road distance is closer to about 3,000 km.
5. Different maps and routes give slightly different figures, so any estimate close to these is reasonable.

In shortAbout 2,200 km in a straight line on the map, and about 3,000 km by road; estimates close to these are also correct.

Watch this explained “Distance from a map, and testing the method on India's length”, 8:05 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 71Matches NCERT’s answer

1. Build the estimate the same way she must have: hours in a school day, times school days in a year, times years spent in school.
2. Take a school day as about 6 hours and a school year as about 220 school days: 6 × 220 = 1,320 hours a year.
3. A Grade 6 student has usually been in school for around 6–7 years: 1,320 × 7 = 9,240 hours.
4. 9,240 hours is well short of 13,000 hours, so her estimate looks too high under these ordinary assumptions.

AnswerNo — with a typical 6-hour school day and about 220 school days a year, a Grade 6 student has usually spent around 8,000–9,500 hours in school, well under Sheetal's 13,000. Her number would need almost 10 years of schooling to be right.

Watch this explained “Sheetal's 13,000 hours, and what would have to be true”, 6:05 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 71One way to think about it

(a) Your current location to one of your favourite places nearby.

1. A normal walking pace is about 5 km in one hour, so 1 km takes about 60 ÷ 5 = 12 minutes.
2. Suppose your favourite nearby place is 2 km away.
3. Time ≈ 2 × 12 = 24 minutes.

In shortAbout 24 minutes for a place 2 km away. Your answer depends on how far your place is: allow about 12 minutes for every km.

(b) Your current location to any neighbouring state’s capital city.

1. Nobody walks all day and all night. Say you walk 8 hours a day: 8 × 5 = 40 km a day.
2. Suppose the neighbouring state's capital is about 250 km away.
3. Days needed ≈ 250 ÷ 40 = 6.25, so about 6 to 7 days.

In shortAbout 6 to 7 days of walking for a capital 250 km away. The distance, and so the answer, depends on where you live.

(c) The southernmost point in India to the northernmost point in India.

1. India's southernmost point (Indira Point) is on an island, so for a walk start from the southern tip of the mainland, Kanyakumari.
2. From there to the northernmost point India measures about 3,214 km in a straight line.
3. At 40 km a day: 3,214 ÷ 40 ≈ 80 days.
4. Roads bend and climb mountains, so the real walk is longer than the straight line.

In shortAt least about 80 days (nearly 3 months) of walking 8 hours every day, and longer in practice because roads are longer than the straight line.

Watch this explained “Distance from a map, and testing the method on India's length”, 8:05 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 71One way to think about it

1. A good estimation question cannot be answered by direct counting.
2. It should break into one small measurement plus a multiplication (like counting words on a page, or steps in ten metres).
3. It should leave room for different reasonable assumptions, so classmates can disagree and compare methods.

In shortFor example: 'How many grains of rice are in a 1 kg packet?', 'How many steps do you take in a whole day?', or 'How many litres of water does your school use in a week?' — each needs a small measurement multiplied up, not a direct count.

Watch this explained “Write an estimation question of your own”, 9:12 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 72Matches NCERT’s answer

1. Neighbours are the cells directly to the left, right, above and below.
2. Only the centre, 62,871, is bigger than all its neighbours, so it is the only supercell.
3. Look at the four cells touching the centre. 39,344 is bigger than 16,200 and 29,765. 23,609 is bigger than 16,200 and 19,381. 45,306 is bigger than 29,765 and 38,408. 50,319 is bigger than 19,381 and 38,408.
4. So each of these four beats all its neighbours except 62,871. The big centre is the only thing stopping them.
5. Make the centre small: swap the 6 and the 1 in 62,871 to get 12,876.
6. 12,876 is smaller than 39,344, 23,609, 45,306 and 50,319. Now those four are supercells and the centre is not: 4 supercells.
7. Checking every possible two-digit swap in the grid shows this is the only one that gives 4 supercells.

AnswerSwap the 6 and the 1 in 62,871 to make 12,876. Then 39,344, 23,609, 45,306 and 50,319 become the 4 supercells.

Watch this explained “One digit swap takes the count from one to four”, 9:13 into Supercells: a number's status comes from its neighbours · हिंदी में देखें

Open NCERT p. 72One way to think about it

1. Write your birth year as a 4-digit number (it must have at least two different digits).
2. Round: arrange its digits to make the largest number (A) and the smallest number (B), keeping leading zeros if the smallest number needs them to stay 4 digits.
3. Subtract: C = A − B. This C is the input to the next round.
4. Repeat rounds 2, 3, ... using the same two steps on each new C, until C = 6174. Count how many rounds it took.
5. Example, for the birth year 2012: Round 1: 2210 − 0122 = 2088. Round 2: 8820 − 0288 = 8532. Round 3: 8532 − 2358 = 6174. So 2012 takes 3 rounds.

In shortThe number of rounds depends on the actual birth year typed in, but it is always somewhere between 1 and 7 rounds. For example, the year 2012 reaches 6174 in exactly 3 rounds.

Watch this explained “Count your own rounds: your year of birth”, 8:33 into Kaprekar's 6174: a process that always ends in the same place · हिंदी में देखें

Open NCERT p. 72Matches NCERT’s answer

1. The odd digits are 1, 3, 5, 7 and 9. Every digit of our numbers must be one of these.
2. Largest: the number must be less than 75,000, so the first digit is at most 7. With first digit 7, the second digit must be less than 5, so the biggest odd choice is 3. Every other digit can be 9. This gives 73,999.
3. Smallest: the number must be at least 35,000, so the first digit is at least 3. With first digit 3, the second digit must be at least 5, so the smallest odd choice is 5. Every other digit can be 1. This gives 35,111.
4. Closest to 50,000: 4 is even, so no number in the group starts with 4. Below 50,000 the biggest member is 39,999, which is 50,000 − 39,999 = 10,001 away.
5. Above 50,000 the smallest member is 51,111, which is 51,111 − 50,000 = 1,111 away.
6. 1,111 is less than 10,001, so 51,111 is the closest.

AnswerLargest: 73,999. Smallest: 35,111. Closest to 50,000: 51,111.

Watch this explained “The largest 5-digit one — and why there is no largest at all”, 6:16 into How many numbers have a given number of digits, and why digit sums behave · हिंदी में देखें

Open NCERT p. 72One way to think about it

1. Weekends: a year has 52 weeks. If your school is closed on Saturdays and Sundays, that is 52 × 2 = 104 days. (If it closes only on Sundays, use 52.)
2. Vacations: summer, festival and winter breaks together are about 45 days. About 2 out of every 7 of those days are Saturdays and Sundays, already counted, so add only about 32 new days.
3. Festival and national holidays: about 15, but some fall on a weekend or inside a vacation, so add about 12.
4. Estimate: 104 + 32 + 12 = 148 holidays.
5. Cross-check: schools are open about 220 days a year, and 365 − 220 = 145, close to the estimate.
6. For the exact number, count the holidays on your school calendar and compare.

In shortDifferent schools give different answers. One estimate: 104 + 32 + 12 = 148 holidays in a year (with Saturdays and Sundays off). Do not count a weekend twice when it falls inside a vacation. Then count on your school calendar to see how close you were.

Watch this explained “Write an estimation question of your own”, 9:12 into Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 72One way to think about it

1. Use a 1-litre water bottle as the measuring unit: how many bottles would fill each container?
2. Mug: a drinking mug holds about a third of a bottle, about 0.3 litres (300 ml). A bathroom mug is bigger, about 1 litre.
3. Bucket: a household bucket takes about 15 to 20 bottles, so about 15 to 20 litres.
4. Overhead tank: a common household tank takes about 50 buckets of 20 litres: 50 × 20 = 1,000 litres. Tanks of 500 and 2,000 litres are also common.

In shortRoughly: a mug holds about 0.3 litres (a bathroom mug about 1 litre), a bucket about 15 to 20 litres, and an overhead tank about 1,000 litres. Sizes vary; many tanks have their capacity printed on the side, which lets you check.

Watch the lesson Estimation: when an approximate answer is the right answer · हिंदी में देखें

Open NCERT p. 72Checked by computerAnswers can differ: one example

1. Pick a convenient 5-digit number that leaves a manageable remainder: 18,000.
2. Remainder to make up with two 3-digit numbers: 18,670 − 18,000 = 670.
3. Split 670 into two 3-digit numbers: 400 and 270.
4. Check: 18,000 + 400 + 270 = 18,670.

Answer18,000 + 400 + 270 = 18,670.

Open NCERT p. 72Checked by computerAnswers can differ: one example

1. Choose 340, which is between 210 and 390.
2. Build a pattern like figure (a) of Section 3.9, where long and short rows take turns:

3. Row	Numbers
   1	20 20 20 20
   2	10 10 10 10 10
   3	20 20 20 20
   4	10 10 10 10 10
   5	20 20 20 20

4. Quick way: rows 1, 3 and 5 hold 3 × 4 = 12 twenties, which is 12 × 20 = 240.
5. Rows 2 and 4 hold 2 × 5 = 10 tens, which is 10 × 10 = 100.
6. Total = 240 + 100 = 340.

AnswerMany patterns work. One: choose 340 and arrange five rows — four 20s, five 10s, four 20s, five 10s, four 20s. That is 12 × 20 + 10 × 10 = 240 + 100 = 340.

Watch this explained “Twelve 40s and ten 50s: count the rows, not the squares”, 7:23 into Rearranging a sum so it can be done in your head · हिंदी में देखें

Open NCERT p. 73One way to think about it

1. The Collatz rule is: if the number is even, halve it; if it is odd, multiply by 3 and add 1.
2. Every power of 2 (2, 4, 8, 16, 32, ...) is even, so the rule always just halves it.
3. Halving a power of 2 gives the next smaller power of 2 (for example, 32 → 16 → 8 → 4 → 2 → 1), which is still even, so the rule keeps halving.
4. The sequence never becomes odd until it reaches 1, so the 'multiply by 3 and add 1' branch is never even needed — the sequence walks straight down to 1.

In shortEvery power of 2 is even, so the Collatz rule just keeps halving it, walking straight down the chain 2ⁿ → 2ⁿ⁻¹ → ... → 2 → 1 without ever hitting an odd number along the way. So the conjecture holds immediately, and obviously, for every power of 2.

Watch this explained “The one family we can settle: every power of 2”, 7:35 into The Collatz conjecture: a question a child can ask and nobody can settle · हिंदी में देखें

Open NCERT p. 73Checked by computer

1. Start at 100 (even): halve it to get 50.
2. Keep applying the rule (halve if even, ×3+1 if odd): 50→25→76→38→19→58→29→88→44→22→11→34→17→52→26→13→40→20→10→5→16→8→4→2→1.
3. The sequence reaches 1 after 25 steps, so the conjecture holds for 100.

AnswerYes. Starting from 100, the sequence is 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 — it reaches 1 after 25 steps.

Watch this explained “100 never exceeds itself. 27 climbs to 9,232.”, 8:29 into The Collatz conjecture: a question a child can ask and nobody can settle · हिंदी में देखें

Open NCERT p. 73One way to think about it

1. Work backwards from the target, 22. Whoever says 22 wins, so a player wants to force their opponent into a spot they cannot escape from.
2. Since each move adds at most 3, if a player can say a number 4 less than a 'safe' number, they can always add enough (1, 2 or 3) to reach it whatever their opponent does — so the safe numbers are every 4th one counting down from 22: 22, 18, 14, 10, 6, 2.
3. 2 is reachable from the very start (0), since a first move can add 1, 2, or 3 — so the first player should open by adding 2.
4. After that, whatever the opponent adds (1, 2 or 3), the first player replies with 4 minus that amount, landing exactly on 6, then 10, then 14, then 18, then 22.

In shortThe first player wins by adding 2 first (landing on 2), and after that always adding whatever brings the total up by exactly 4 from their last move (if the opponent adds 1, add 3; if they add 2, add 2; if they add 3, add 1). That forces the running total through 2, 6, 10, 14, 18, 22 — so the first player always says 22.

Watch this explained “Move the target to 22 — and the winner does NOT change”, 7:09 into Winning strategies: working a game backwards from its end · हिंदी में देखें