Question

while listing the numbers from 1 to 1000, how many times the digit 5 is written:

(a)297
(b)243
(c)300
(d)273

Answer 1

Hint: The easy method is to write the numbers in the range of 10, 100 and then count the number of 5’s in them.

Complete step-by-step answer:

First we will take the numbers in range of 10:

From the range of numbers 0 to 10, the digit 5 appears 1 time.

Now we add the numbers in range of 100

From the range of numbers 0 to 99, the digit 5 appears 20 times.

(They are: 5, 15, 25, 35, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 65, 75, 85, 95)

Observe that in number 55 5 appears two times.

Now from 100 to 199 we have: 105, 115, 125, 135, 145, 150, 151, 152, 153, 154, 155, 156,

157, 158, 159, 165, 175, 185, 195.The digit 5 appears 20 times.

Similarly,

From 200 to 299 we have 205,215…….245,250,…255,…..,295. Again 20 times.

From 300 to 399 we have 305,…355,…,395. Again 20 times

From 400 to 499 we have 405,…455,…,495. Again 20 times

From 600 to 699 we have 605,…655,…,695. Again 20 times

From 700 to 799 we have 705,…755,…,795. Again 20 times

From 800 to 899 we have 805,…855,…,895. Again 20 times

From 900 to 999 we have 905,…955,…,995. Again 20 times

Now the remaining range is 500 to 599:

Now we take numbers in the range of 10.

From 500 to 509 we have 500, 501,….509. The digit 5 appears 11 times.

From 510 to 519 we have 510, 511,….519. The digit 5 appears 11 times.

From 520 to 529 we have 520, 521,….529. The digit 5 appears 11 times.

From 530 to 539 we have 530, 531,….539. The digit 5 appears 11 times.

From 540 to 549 we have 540, 541,….549. The digit 5 appears 11 times.

From 560 to 569 we have 560, 561,….569. The digit 5 appears 11 times.

From 570 to 579 we have 570, 571,….579. The digit 5 appears 11 times.

From 580 to 589 we have 580, 581,….589. The digit 5 appears 11 times.

From 590 to 599 we have 590, 591,….599. The digit 5 appears 11 times.

Now the last remaining numbers are 550-559:

We have: 550, 551,….559. The digit 5 appears 21 times.

By adding all above, we get:

Total number of times = (20).(9) + (11).(9) + 21

= 180 + 99 + 21

=300

Therefore when we list numbers from 1 to 1000 the digit 5 is written 300 times.

Option (c) is correct.


Note: Don’t forget to count the digit 5 twice in numbers like 55, and also do not forget to count the digit 5 in hundredth’s place in the numbers from 500 to 599.