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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?

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  1. freckles
    • one year ago
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    \[26 \text{ is a two digit number } \\ \text{ another way to express this is } 2 \cdot 10 +6 \\ \text{ So say } N=xy \text{ where } x \text{ is in the tens digit } \\ \text{ and } y \text{ is in the one's digit } \\ \text{ So } N=10x+y\] (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
    • one year ago
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    So can you write what K would look like

  3. anonymous
    • one year ago
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    k = 10y + x

  4. freckles
    • one year ago
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    great

  5. freckles
    • one year ago
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    so you have N=10x+y and K=10y+x and you are given K+2N<60

  6. freckles
    • one year ago
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    just plug in your K and your N and combine like terms

  7. freckles
    • one year ago
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    and if you want to find N check a few pairs for (x,y)

  8. anonymous
    • one year ago
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    10y+x + 2(10x+y) <60 10y+x+20x+2y < 60 12y+ 21x < 60 21x<60-12y x= 60/21 - 12/21y

  9. anonymous
    • one year ago
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    Is this correct?

  10. freckles
    • one year ago
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    \[x<\frac{60}{21}-\frac{12}{21} y \text{ though I probably would leave \it like } \\ 12y+21x<60\]

  11. freckles
    • one year ago
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    \[21x+12y<60\]

  12. freckles
    • one year ago
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    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
    • one year ago
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    so (1,0) would be the first pair I look at

  14. freckles
    • one year ago
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    then (1,1) and I would keep doing this to til I bust (you know go over 60)

  15. freckles
    • one year ago
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    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
    • one year ago
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    see if 12 works

  17. anonymous
    • one year ago
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    21(1)+12(2)= 45 < 60 21(1)+12(3)= 57<60 21(1)+12(4)= 69>60

  18. freckles
    • one year ago
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    ok but I don't think that is all

  19. freckles
    • one year ago
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    we can look at N=20,21,22

  20. anonymous
    • one year ago
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    20 and 21 work

  21. freckles
    • one year ago
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    and 3something wouldn;t work because 21(3)=63

  22. freckles
    • one year ago
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    so now we are done

  23. anonymous
    • one year ago
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    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
    • one year ago
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    np i'm just glad my way wasn't too long :p

  25. anonymous
    • one year ago
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    It's a good way. Thank you again!

  26. freckles
    • one year ago
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    np

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