M and M? No, it's M and N!!!
During a contest, Brenda and her 2 older brothers, Benedict and Bob, were asked to each write down a number. Amazingly, the three numbers form an arithmetic sequence that decreases in the same order as their age (i.e. Benedict's number is the largest of the three, Bob's is in between while Brenda's is the smallest).
Interestingly, Benedict's number can be expressed as and the sum of Benedict and Bob's numbers is . Moreover, Brenda's number can be expressed as while the sum of Bob and Brenda's, . If Bob's number can be expressed as - . where p and q are coprime positive integers, then what is p+q?
The answer is 17.
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1 solution
Let Benedict, Bob and Brenda's numbers be a-d,a, a+d respectively where d is the common difference. We can form 4 equations. a-d = m+n ------(1) (a-d)+a =m-n --------(2) a+d = mn ---------(3) a+(a+d) = n m ----------(4)
(2)-(1): a= -2n Substituting this into (1): (-2n)-d = m+n hence we have d= -m-3n. Now we have (-2n)+(-m-3n) = mn or -m-5n = mn for equation 3 and 2(-2n)+(-m-3n) = n m or -m-7n = n m for equation 4. Rewriting these 2 equations we have -5n = mn+m and -7n = m + n m . Dividing the former by the latter we simply get n= 7 5 . Since a=-2n, a (middle term) is equal to - 7 1 0 . Hence p=10 and q=7 and we have 10 + 7 = 1 7 .