Jami picked three equally spaced integer numbers on the number line. The sum of the first and the second numbers is 40, while the sum of the second and third numbers is 60. What is the sum of all three numbers?
(A) 70 \text{(A)}\;70 (A) 70 (B) 75 \text{(B)}\;75 (B) 75 (C) 80 \text{(C)}\;80 (C) 80 (D) 85 \text{(D)}\;85 (D) 85 (E) 90 \text{(E)}\;90 (E) 90 Solution 1 Hints (5) Hint 1 Let the numbers be a < b < c a<b<c a < b < c . Hint 2 a + b = 40 , b + c = 60 a+b=40,\quad b+c=60 a + b = 40 , b + c = 60 Hint 4 By Hints 2 and 3, a + b = 40 b + c = 60 ⟹ a + c + 2 b = 100 \begin{aligned}a+b&=40\\ b+c&=60\end{aligned}\quad\Longrightarrow\quad a+c+2b=100 a + b b + c = 40 = 60 ⟹ a + c + 2 b = 100 a + c 2 = b ⟹ a + c = 2 b ( 1 ) \frac{a+c}{2}=b\Longrightarrow a+c=2b\qquad (1) 2 a + c = b ⟹ a + c = 2 b ( 1 ) ⟹ 4 b = 100 ⟹ b = 25 ( 2 ) \Longrightarrow 4b=100\Longrightarrow b=25\qquad (2) ⟹ 4 b = 100 ⟹ b = 25 ( 2 ) Hint 5 ( 1 ) , ( 2 ) ⟹ a + b + c = a + c + b = 2 b + b = 3 b = 3 ( 25 ) = 75 (1),(2)\Longrightarrow a+b+c=a+c+b=2b+b=3b=3(25)=75 ( 1 ) , ( 2 ) ⟹ a + b + c = a + c + b = 2 b + b = 3 b = 3 ( 25 ) = 75 Solution 2 Hints (5) Hint 1 Let the numbers be a < b = a + d < c = a + 2 d a<b=a+d<c=a+2d a < b = a + d < c = a + 2 d . Hint 2 a + b = 40 ⟹ a + a + d = 40 ⟹ 2 a + d = 40 a+b=40\Longrightarrow a+a+d=40\Longrightarrow 2a+d=40 a + b = 40 ⟹ a + a + d = 40 ⟹ 2 a + d = 40 b + c = 60 ⟹ a + d + a + 2 d = 60 ⟹ 2 a + 3 d = 60 b+c=60\Longrightarrow a+d+a+2d=60\Longrightarrow 2a+3d=60 b + c = 60 ⟹ a + d + a + 2 d = 60 ⟹ 2 a + 3 d = 60 Hint 3 a + b + c = a + a + d + a + 2 d = 3 a + 3 d = 3 ( a + d ) a+b+c=a+a+d+a+2d=3a+3d=3(a+d) a + b + c = a + a + d + a + 2 d = 3 a + 3 d = 3 ( a + d ) Hint 4 By Hint 2, 2 a + d = 40 2 a + 3 d = 60 ⟹ 4 a + 4 d = 100 ⟹ a + d = 25 \begin{aligned}2a+d&=40\\ 2a+3d&=60\end{aligned}\quad\Longrightarrow\quad 4a+4d=100\Longrightarrow a+d=25 2 a + d 2 a + 3 d = 40 = 60 ⟹ 4 a + 4 d = 100 ⟹ a + d = 25 Hint 5 By Hints 3 and 4, a + b + c = 3 ( a + d ) = 3 ( 25 ) = 75 a+b+c=3(a+d)=3(25)=75 a + b + c = 3 ( a + d ) = 3 ( 25 ) = 75