Coba pikirake wong loro \(A\) lan \(B\) sing ora lair dina sing padha lan \(A\) luwih enom tinimbang \(B\) . Tampilake: Ana persis loro konstelasi umur \(a,b \in \mathbb{N}\) , sing ditrapake: \(2\cdot a = b\) . Kaping pisanan nyetel \(d \in \mathbb{R}^+\) minangka bedane umur antara \(A\) lan \(B\) nalika lair \(A\) karo \( d = d_0 + d_1 \) , \( d_0 \in \mathbb{N}_0, d_1 \in \mathbb{R}, d_1 \in [0;1[\) . Saiki kita nimbang titik sewenang-wenang wektu \(x \in \mathbb{R}^+\) sawise lair \(A\) kanthi \(x = x_0 + x_1\) , \(x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[\) .
Ing wektu iki, miturut definisi, \(a = \lfloor x \rfloor \) lan \(b = \lfloor x+d \rfloor\) . Saiki kita nemtokake kabeh \(x\) sing nyekel:
$$2 \lfloor x \rfloor = \lfloor x+d \rfloor \Leftrightarrow 2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor$$
Kasus kaping 1: \(0 \leq x_1 + d_1 < 1\):
Banjur $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 \Leftrightarrow x_0 = d_0.$$
Iki tegese $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + x_1 \rfloor = d_0$$ lan $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ umur pisanan sing $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ .
Kasus kaping 2: \( 1 \leq x_1 + d_1 < 2 \):
Banjur $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 + 1 \Leftrightarrow x_0 = d_0 + 1.$$
Iki tegese $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + 1 + x_1 \rfloor = d_0 + 1$$ lan $$b = \lfloor x+d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + 1 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor = 2 d_0 + 2$$ konstelasi umur kaping loro sing dikarepake.
Khusus, tegese, kayata: Yen ibumu nglairake sampeyan nalika umur \(20\) taun, umure umure kaping pindho kaping pindho umure \(40\) lan \(42\) taun. Sampeyan uga bakal narik kawigaten yen lan nalika dheweke wis \(n\) umure luwih tuwa: Ing kene \(n \in \mathbb{N}\) sewenang-wenang lan kita entuk \(x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)\) . Iki bisa digunakake persis nalika bedane umur ongko ongko \( \lfloor d \rfloor = d_0 \) pirang-pirang \(n-1\) , kayata ing kasus ing ndhuwur ibune \(24\) taun \(6\) umur sampeyan.