Coba wong loro \(A\) lan \(B\) sing ora ulang tahun padha, ngendi \(A\) luwih enom saka \(B\) . Tampilake: Ana persis rong rasi lintang umur \(a,b \in \mathbb{N}\) sing ditrapake ing ngisor iki: \(2\cdot a = b\) . Kita pisanan nyetel \(d \in \mathbb{R}^+\) minangka prabédan umur antarane \(A\) 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 wektu sembarang \(x \in \mathbb{R}^+\) sawise lair saka \(A\) karo \(x = x_0 + x_1\) , \(x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[\) .
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.
Ing istilah konkrit, iki tegese, contone: Yen ibumu nglairake sampeyan ing umur \(20\) , dheweke persis kaping pindho saka sampeyan ing \(40\) lan \(42\) . Kasus apa lan nalika ibumu wis \(n\) kaping lawas uga menarik: Kene sampeyan nyetel \(n \in \mathbb{N}\) sewenang-wenang lan njaluk \(x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)\) . Iki bisa digunakake yen lan mung yen prabédan umur integer \( \lfloor d \rfloor = d_0 \) iku kelipatan saka \(n-1\) . Contone, ing kasus ing ndhuwur, ibumu umur \(24\) taun \(6\) kaping umur sampeyan.