Student Question

# I am a 3-digit whole number. If you double me, I remain a 3-digit number. If you add 2 to me after I am doubled, I become a 4-digit number. What number am I?

The number doubled is a 3-digit number so 2x<1000 or x<500.

2 more than the number doubled is a 4-digit number so 2x+2>999 or x>498

The solution to 498<x<500 is x=499

(Alternatively `2x+2>=1000 ==> x>=499`

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First, we are told that we are dealing with a three digit number. At this point, this can be any number from 100 to 999. We are told that, if this number is doubled, the number will remain a three digit number. From this, we can deduce that the doubled number cannot be more than 999.

We are also told that, if we add 2 to the doubled number, the number immediately becomes a four digit number. To experiment, we can add 2 to 999 (999+2= 1001).

However, we soon realize that 999 does not divide by 2 without a remainder. So, now we know that we can't use 999 as the doubled number. At this point, the best thing to do is to try 998, which divides by 2 easily, giving us 499. Also, when we add 998 with 2, we get 1000 (a four digit number). So the original three digit number is 499.

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You can do this a couple of ways.  Let's try this. . .

Let x = the number.  Then, we have:

x < 1000

since x is specified to be a 3 digit number.  Then, if we double it, x is still a 3 digit number.  So:

2x < 1000

Then, if you add 2 to it, x is a 4 digit number.  So, we would have:

2x+2 > 1000

We have "greater than or equal to" since 1000 is a 4 digit number.

Solving this for x, we get:

x > 499

But, then, if you consider any other numbers greater than 499 and double them, they would be 4 digit numbers.  So, the number has to be 499.