Problema Solution
Jessica has an odd number of stamps in her collection. The sum of the digits in the number of stamps she has is 12. The hundreds digit is three times the ones digit. If she has between 1000 and 2000 stamps in her collection, how many stamps does Jessica have.
Answer provided by our tutors
Let 'x' be the number of stamps that Jessica has.
1000 < x < 2000 follows that x has 4 digits and hundred digit equal to 1 thus can be written in the following form
x = a0 + a1*10 + a2*10^2 + 10^3
where a0 can be 1, 3, 5, 7 or 9 (must be odd)
while a1, a2 are elements from the following set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
a0 + a1 + a2 + 1 = 12
a2 = 3*a0
If we substitute a2 with 3*a0 in a0 + a1 + a2 + 1 = 12 we get
a0 + a1 + 3*a0 + 1 = 12
4*a0 + a1 = 12 - 1
a1 = 11 - 4*a0
Also a1>0 that is 11 - 4a0 > 0 equivalent to 11 > 4a0 that is
11/4 > a0
2.75 > a0 and a0 > 0 this we have that a0 =1
For ao = 1 we have
a1 = 11 - 4*a0 = 11 - 4 = 7
a2 - 3*a0 = 3*1 = 3
and the solution is x = a0 + a1*10 + a2*10^2 + 10^3 = 1 + 7*10 + 3*100 + 1000 = 1371.
Jessica has 1371 stamps.