A number theory problem by Ahmed Mahmoud
Miranda is a 10th grade student who is very good in mathematics. In fact she just completed an advanced algebra class and received a grade of A+. Miranda has five sisters, Cathy, Stella, Eva, Lucinda, and Dorothea. Miranda made up a problem involving the ages of the six girls and dared Cathy to solve it. Miranda said: “The sum of our ages is five times my age. (By ’age’ throughout this problem is meant ’age in years’.) When Stella is three times my present age, the sum of my age and Dorothea’s will be equal to the sum of the present ages of the five of us; Eva’s age will be three times her present age; and Lucinda’s age will be twice Stella’s present age, plus one year. How old are Stella and Miranda?” “Well, Miranda, could you tell me something else?” “Sure”, said Miranda, “my age is an odd number”.
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1 solution
Denote the present ages of Miranda, Cathy, Stella, Eva, Lucinda and Dorothea by M;C; S;E;L and D respectively. Let x denote the number of years between future and present ages. Then from Miranda’s speech we gather that M + C + S + E + L + D = 5M; (5) S + x = 3M; (6) M + x + D + x = M + S + E + L + D; (7) E + x = 3E; (8) L + x = 2S + 1: (9) From Equation (6) we see that x = 3M ¡ S, using this we may rewrite the remaining equations as follow: E + L + D = 4M ¡ C ¡ S; (10) 7M ¡ 2S + D = M + S + E + L + D; (11) 2E = (3M ¡ S); (12) L = 3S ¡ 3M + 1: (13)
Substituting (10) into (11) and solving for D gives D = 2S ¡ 2M ¡ C: (14) Multiply (5) by two and substitute Equations (12) through (14) to it gives (after simplification): 15M = 11S + 2: Since we are given that Miranda’s age is old then 15M must end in a 050 which implies Stella’s age must end in a 030. Upon trial and error (and recalling that Miranda is in 10th grade) we found that Miranda’s age is 17 and Stella’s age is 23.