SOLUTION: A number is between 20 and 30 and is three times the sum of its digits. What is the number?

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Question 829757: A number is between 20 and 30 and is three times the sum of its digits. What is the number?
Found 3 solutions by josgarithmetic, edjones, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Two-digit number, x%2A10%2By%2A1.
Expect the number 10x%2By%3E20 and 10x%2By%3C30.
We must expect only Natural Numbers for x and y, so if graphing, these must be only in quadrant 1.

The other part of the description:
highlight_green%283%28x%2By%29=10x%2By%29.
Simplify.
3x+3y=10x+y
3y=7x+y
2y=7x
highlight_green%28-7x%2B2y=0%29

Next plan is that we can plot that line. Intercepts are (0,0) only. The equation can be shown as y=%287%2F2%29x; and because of the inequalities expected above, y%3E-10x%2B20 and y%3C-10x%2B30.

The easiest path seems to be find all the Natural Number coordinate pairs on the line y=%287%2F2%29x and stay at y%3C10.
-
Best guess it the point, (2,7).
The number, highlight_green%28highlight%2827%29%29 ?
No other natural number combination on that line for y%3C10.

CHECK THIS:
3%282%2B7%29=27?
3%289%29=27 yes.

Answer by edjones(8007) About Me  (Show Source):
Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

A number is between 20 and 30 and is three times the sum of its digits. What is the number?
Let tens and units digits be T and U, respectively

Then number is: 10T + U

So, 20 < 10T + U < 30 ------- inequality (i)

Also, 10T + U = 3(T + U)____10T + U = 3T + 3U___7T = 2U___U+=+7T%2F2 ----- inequality (ii)

20 < 10T + 7T%2F2< 30 ------ Substituting 7T%2F2 for U in inequality(i) 

40 < 20T + 7T < 60 ----- Multiplying by LCD, 2

40 <    27T   < 60

40%2F27+%3C+T+%3C+60%2F27 ------ Dividing by 27

1%2613%2F27+%3C+T+%3C+2%266%2F27



Since tens digit, or T+%3E+1%2613%2F27, but T+%3C+2%266%2F27, then T or tens digit MUST BE 2

U = 7(2)/2 ------- Substituting 2 for T in inequality (ii)

U+=+14%2F2, or 7



With T being 2 and U being 7, number is: highlight_green%2827%29

You can do the check!! 



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