Problem Solving: how many digits did she type?

[Guest]

Mrs. Marva Drew, a resident of Waterloo, Iowa, typed the numbers 1 to 1,000,000 on a manual typewriter. She worked over a period of six years doing this and used 2,473 sheets of paper.
a. How many digits did Mrs. Drew type? Show work dont count commas only digits.
b. What is the sum of all the digits Mrs. Drew typed? Show how you got your answer?
0 and 999,999
1 and 999,998
2 and 999,997

Need help!

 

[soroban]

Re: Problem Solving-Word Problem

Hello, bbug!

Here's my approach to part (a) . . .

Mrs. Marva Drew, a resident of Waterloo, Iowa, typed the numbers 1 to 1,000,000 on a manual typewriter.
She worked over a period of six years doing this and used 2,473 sheets of paper.

a. How many digits did Mrs. Drew type?

There are 9 one-digit numbers: \(\displaystyle \;\;\;\;\;\;\;\;\;\:9\,\times\,1\:=\;\;\;\;\;\;\,9\)
There are 90 two-digit numbers: \(\displaystyle \;\;\;\;\;\;\;\:90\,\times\,2\:=\;\;\;\;\;180\)
There are 900 three-digit numbers: \(\displaystyle \;\;\;\;\:900\,\times\,3\:=\;\;\;\:2,700\)
There are 9,000 four-digit numbers: \(\displaystyle \;\;\:9,000\,\times\,4\:=\;\;\;36,000\)
There are 90,000 five-digit numbers: \(\displaystyle \;\,90,000\,\times\,5\:=\;\,\,450,000\)
There are 900,000 six-digit numbers: \(\displaystyle \,900,000\,\times\,6\:=\,5,400,000\)
There is one seven-digit number: \(\displaystyle \;\;\;\;\;\;\;\:\,1\,\times\,7\:=\;\;\;\;\;\;\;7\)

I get a total of 5,888,896 digits.

 

[steve_b]

b. What is the sum of all the digits Mrs. Drew typed? Show how you got your answer?

============

Use the formula for the sum of an arithmetic progression:

S = (n/2)(first + last), where n is the number of terms in the progression, here, 1000000

S = (1000000/2)(1 + 1000000) = 500000(1000001) = 500,000,500,000

Steve

 

[galactus]

What you done was to add all the numbers from 1 to 1000000.

That's not the sum of the digits typed.

You calculated 1+2+3+4+5+..............+1000000=500000500000.

See here:

http://www.gomath.com/Questions/questio ... tion=42751

 

[Denis]

part a) : agree with Soroban

part b) : 27,000,001 (would be 27,000,000 if last was 999,999)

1 - 9 : 45
10 - 99 : 855
100 - 999 : 12600
1000 - 9999 : 166500
10000 - 99999 : 2070000
100000 - 999999 : 24750000
1000000 : 1
total: 27000001