Problem Solving

How many positive even integers less then 100 contain digits 4 or 7.

A.) 16
B.)17
C.)18
D.) 19
E.)20

I tried something as follows but am unable to get the correct answer.

first i started with 4

so 4, 40, 42, 44, 46, 48, 24, 64, 84 giving me 9 numbers that have a 4 and are even.


now using 7

70,72,74,76,78, since 7 is not even number it can not be in the second place so we have only 5 numbers

9 number + 5 numbers is only 14... that is not even answer choice... what numbers am i missing?

14,34,54,94

damn it i was fixated on number starting with 4... and 7 only...

Is there any formula i could use here to avoid hicups as such?

thanks for the answer..

I'd suggest being more systematic. Choose something to list and then stick with it. It ensures you don't forget any, and it makes counting easier. If you have to find a lot more numbers, then you can write out enough to see the pattern, and then just go based on the pattern. There isn't a formula exactly, but if you've done a few of these you're less likely to forget a large group.

In this problem, one way to write the list would be:

First, try the numbers that have 4 at the end:

4,14, 24,34, 44, 54, 64, 74, 84, 94 = 10 ways
Then the numbers that have 4 at the beginning

40, 42, (44), 46, 48, = 4 ways

Then the numbers that have 7 at the end, well none of those are even.

Then the numbers that have 7 at the beginning:
70, 72, (74), 76, 78 = 4 ways

My approach:

Find all integers ending in 4 [they re all even]

04[same as 4.This is logic],14,24,...94 [This is like counting from 0 to 9 [in tens digit]. Therefore no of integers both inclusive =9-0=9 +1 [for inclusive]=10

Now 40 42 46 48 [Do not count 44 - already included in first list].Total =4

Now 70 72 76 78 =4 [ 74 would be repeated.]

I didn't include any integers that have 7 as their units digit ,for they are all odd.

total integers = 10+4+4 =18

Very first thing is to read the question very carefully!

It says 4 or 7...not starting with 4 or 7, and also not 4 and 7!

Then I did the manual listing of nos. and counting. Since the range is 100, this is the least complicated way for me and also to not get confused.

For bigger ranges, I am not too sure of the approach.