Cabanga ngabantu ababili \(A\) no \(B\) abangazalwanga ngosuku olufanayo futhi \(A\) bancane kuno \(B\) . Khombisa ukuthi kunemilaza emibili yeminyaka ncamashi \(a,b \in \mathbb{N}\) , esebenza kuyo: \(2\cdot a = b\) . Siqale ukusetha \(d \in \mathbb{R}^+\) njengomehluko wobudala phakathi kuka \(A\) no \(B\) ekuzalweni kwe \(A\) nge \( d = d_0 + d_1 \) , \( d_0 \in \mathbb{N}_0, d_1 \in \mathbb{R}, d_1 \in [0;1[\) . Manje sibheka iphuzu lokuphikisana nesikhathi \(x \in \mathbb{R}^+\) ngemuva kokuzalwa kwe- \(A\) nge \(x = x_0 + x_1\) , \(x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[\) .
Ngalesi sikhathi ngesikhathi, ngencazelo, \(a = \lfloor x \rfloor \) kanye \(b = \lfloor x+d \rfloor\) . Manje sinquma konke \(x\) okuphethe:
$$2 \lfloor x \rfloor = \lfloor x+d \rfloor \Leftrightarrow 2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor$$
Icala lokuqala: \(0 \leq x_1 + d_1 < 1\):
Ngemuva kwalokho i- $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 \Leftrightarrow x_0 = d_0.$$
Lokhu kusho ukuthi $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + x_1 \rfloor = d_0$$ futhi $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ lokuqala lobudala $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ .
Icala lesibili: \( 1 \leq x_1 + d_1 < 2 \):
Ngemuva kwalokho i- $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 + 1 \Leftrightarrow x_0 = d_0 + 1.$$
Lokhu kusho ukuthi $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + 1 + x_1 \rfloor = d_0 + 1$$ futhi $$b = \lfloor x+d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + 1 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor = 2 d_0 + 2$$ lesibili leminyaka eliyifunayo.
Ngokuqondile, lokhu kusho, ngokwesibonelo: Uma umama wakho wakubeletha eneminyaka \(20\) , uneminyaka ephindwe kabili yobudala bakho eminyakeni engu - \(40\) kanye \(42\) . Kuyathakazelisa futhi uma ngabe \(n\) izikhathi ezindala kunalezi: Lapha \(n \in \mathbb{N}\) ngokungafanele futhi sithola \(x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)\) . Lokhu kusebenza ngqo lapho umehluko weminyaka ophelele \( \lfloor d \rfloor = d_0 \) kwe- \(n-1\) , isb. Esimweni esingenhla unyoko uneminyaka \(24\) \(6\) iphinda iminyaka yakho.