The digits of a positive two-digit integer N are interchanged to form integer K. Find all possibilities for N under the conditions described: The sum of K and twice N is less than 60.
How do I write an equation for this?

Question

The digits of a positive two-digit integer N are interchanged to form integer K. Find all possibilities for N under the conditions described: The sum of K and twice N is less than 60.
How do I write an equation for this?

freckles · Answer

$$26 \text{ is a two digit number } \ \text{ another way to express this is } 2 \cdot 10 +6  \ \text{ So say } N=xy \text{ where } x \text{ is in the tens digit } \ \text{ and } y \text{ is in the one's digit } \ \text{ So } N=10x+y$$
(by the way this xy I have written for N doesn't mean x and y multiplied; it is just the digits of N)

freckles · Answer

So can you write what K would look like

anonymous · Answer

k = 10y + x

freckles · Answer

great

freckles · Answer

so you have N=10x+y and K=10y+x
and you are given K+2N<60

freckles · Answer

just plug in your K and your N
and combine like terms

freckles · Answer

and if you want to find N
check a few pairs for (x,y)

anonymous · Answer

10y+x + 2(10x+y) <60
10y+x+20x+2y < 60
12y+ 21x < 60
21x<60-12y
x= 60/21 - 12/21y

anonymous · Answer

Is this correct?

freckles · Answer

$$x<\frac{60}{21}-\frac{12}{21} y \text{ though I probably would leave \it like } \ 12y+21x<60$$

freckles · Answer

$$21x+12y<60$$

freckles · Answer

so N is two digits
and N in digit form looks like N=xy
we don't want x to be 0 because basically that would mean N is a one digit number

freckles · Answer

so (1,0) would be the first pair I look at

freckles · Answer

then (1,1)
and I would keep doing this to til I bust (you know go over 60)

freckles · Answer

21(1)+12(0)=21<60 so N=10 is good
21(1)+12(1)=21+12=33<60  so N=11 is good

freckles · Answer

see if 12 works

anonymous · Answer

21(1)+12(2)= 45 < 60
21(1)+12(3)= 57<60
21(1)+12(4)= 69>60

freckles · Answer

ok but I don't think that is all

freckles · Answer

we can look at N=20,21,22

anonymous · Answer

20 and 21 work

freckles · Answer

and 3something wouldn;t work because 21(3)=63

freckles · Answer

so now we are done

anonymous · Answer

I see how to do this now. Thank you sooo much! I was stuck on this yesterday because I divided by 21 and kept getting decimals.

freckles · Answer

np
i'm just glad my way wasn't too long :p

anonymous · Answer

It's a good way. Thank you again!

freckles · Answer

np