problem 260 mathcounts

Mathematics
Tutor: None Selected Time limit: 1 Day

If RATS*4=STAR, and each letter represents a different digit from 0 to 9, inclusive, what is the value of S+T+A+R?

Oct 19th, 2014

For RATS x 4 = STAR 
1) R can only be either 1 or 2, if more than 2 then result will be a 5 digit number 
2) R can only be 2, because there's no way for 4 to multiply any number with a result of 1 
 4S cannot be ?1 
3) 4S will then be ?2, which means S can only be 3 or 8 
4) S cannot be 3, if has to be 8, since R = 2, 4R = 8 
5) A cannot be 3 or more, because 4A needs to stay as single digit, or else if it is more than 10 
the ten will goes over to the value of S which will make S more than 8 and which will be wrong 
6) Since R = 2 already, A cannot be 2, therefore A = 1 
7) Since S = 8, 4S = 32, the 3 carry over to the left, therefore 4T + 3 =?A = ?1 
which means 4T = ?1 - 3 = ?8 
8) For 4 multiply a number and get a 8, it can only be 2 or 7, two is taken, therefore T can only be 7 
which makes 4T = 28, and 28 + 3 = 31, the 3 carry over to the left 
9) Since A = 1, 4 x 1 = 4; 4 + 3 = 7 (T ) , T=7 checks out 
10) therefore the number RATS = 2178 
11) 2178 x 4 = 8712 
12) 8+7+1+2 = 18

Oct 19th, 2014

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