Page numbers and digits

Hello, and welcome to Myriora Maths! A printer numbers the pages of a book, from page one to page two hundred. How many digits does the printer use altogether?

Split the pages into groups, by how many digits they have. Pages one to nine have one digit. Pages ten to ninety-nine have two digits. And pages one hundred to two hundred have three digits.

Now count how many pages are in each group. One to nine is nine pages. For ten to ninety-nine, do ninety-nine take away ten, plus one. That's ninety pages. And one hundred to two hundred is two hundred take away one hundred, plus one: a hundred and one pages.

Now multiply each group by its digits. Nine times one is nine. Ninety times two is a hundred and eighty. A hundred and one times three is three hundred and three. Add them up: four hundred and ninety-two digits!

Exam questions often go backwards. A book uses two hundred and fifty-two digits. How many pages does it have? Use the groups again. The first ninety-nine pages use a hundred and eighty-nine digits. That leaves sixty-three digits. Each page from one hundred on uses three digits, so that's twenty-one more pages. Ninety-nine plus twenty-one is a hundred and twenty pages!

A common mistake is forgetting the plus one. From ten to ninety-nine, some people do ninety-nine take away ten, and get eighty-nine pages. But that counts the gaps between the pages, not the pages! Like fence posts, add one: ninety pages.

Your turn! How many digits do you need to number the pages from one to a hundred and fifty? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! Pages one to nine use nine digits. Pages ten to ninety-nine use a hundred and eighty. Pages one hundred to a hundred and fifty are fifty-one pages, using a hundred and fifty-three digits. Altogether, that's three hundred and forty-two digits!

Quick review. Split the pages into one-digit, two-digit and three-digit groups. Count the pages in each group: last take away first, plus one. Then multiply by the digits, and add them up. See you next time!