A-shop-keeper-gave-a-bill-to-a-customer-The-bill-was-in-two-digits-but-was-mistakenly-interchanged-thereby-undercharging-the-customer-by-45-00-If-the-sum-of-the-digits-is-9-find-the-actual-value-

Question Number 173332 by pete last updated on 09/Jul/22

$$\mathrm{A}\:\mathrm{shop}\:\mathrm{keeper}\:\mathrm{gave}\:\mathrm{a}\:\mathrm{bill}\:\mathrm{to}\:\mathrm{a}\:\mathrm{customer}. \\ $$$$\mathrm{The}\:\mathrm{bill}\:\mathrm{was}\:\mathrm{in}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{but}\:\mathrm{was}\:\mathrm{mistakenly} \\ $$$$\mathrm{interchanged}\:\mathrm{thereby}\:\mathrm{undercharging}\:\mathrm{the} \\ $$$$\mathrm{customer}\:\mathrm{by}\:\$\mathrm{45}.\mathrm{00}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{is}\: \\ $$$$\mathrm{9},\:\mathrm{find}\:\mathrm{the}\:\mathrm{actual}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bill}. \\ $$

Answered by Rasheed.Sindhi last updated on 09/Jul/22

•(10t+u)_(Correct) −(10u+t)_(Mistaken) =45 ⇒9t−9u=45 •u+t=9⇒u=9−t 9t−9(9−t)=45 18t−81=45 t=(45+81)/18=126/18=7 u=9−t=9−7=2 Original 72 Mistakenly 27

$$\bullet\underset{{Correct}} {\underbrace{\left(\mathrm{10}{t}+{u}\right)}}−\underset{{Mistaken}} {\underbrace{\left(\mathrm{10}{u}+{t}\right)}}=\mathrm{45} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{9}{t}−\mathrm{9}{u}=\mathrm{45} \\ $$$$\bullet{u}+{t}=\mathrm{9}\Rightarrow{u}=\mathrm{9}−{t} \\ $$$$\mathrm{9}{t}−\mathrm{9}\left(\mathrm{9}−{t}\right)=\mathrm{45} \\ $$$$\mathrm{18}{t}−\mathrm{81}=\mathrm{45} \\ $$$${t}=\left(\mathrm{45}+\mathrm{81}\right)/\mathrm{18}=\mathrm{126}/\mathrm{18}=\mathrm{7} \\ $$$${u}=\mathrm{9}−{t}=\mathrm{9}−\mathrm{7}=\mathrm{2} \\ $$$${Original}\:\mathrm{72} \\ $$$${Mistakenly}\:\mathrm{27} \\ $$

Commented by pete last updated on 09/Jul/22