## anonymous one year ago 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?

1. freckles

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

2. freckles

So can you write what K would look like

3. anonymous

k = 10y + x

4. freckles

great

5. freckles

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

6. freckles

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

7. freckles

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

8. anonymous

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

9. anonymous

Is this correct?

10. freckles

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

11. freckles

$21x+12y<60$

12. freckles

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

13. freckles

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

14. freckles

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

15. freckles

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

16. freckles

see if 12 works

17. anonymous

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

18. freckles

ok but I don't think that is all

19. freckles

we can look at N=20,21,22

20. anonymous

20 and 21 work

21. freckles

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

22. freckles

so now we are done

23. anonymous

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.

24. freckles

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

25. anonymous

It's a good way. Thank you again!

26. freckles

np