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Question Number 170305 by MathsFan last updated on 20/May/22

A number with two digits is equal to four times the sum of its digits. The number formed by reversing the order of the digit is 27 greater than the given number. Find the number

$$\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{digits}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{equal}}\:\boldsymbol{\mathrm{to}} \\ $$$$\boldsymbol{\mathrm{four}}\:\boldsymbol{\mathrm{times}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{its}}\:\boldsymbol{\mathrm{digits}}.\:\boldsymbol{\mathrm{The}} \\ $$$$\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{formed}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{reversing}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{order}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{digit}}\:\boldsymbol{\mathrm{is}}\:\mathrm{27}\:\boldsymbol{\mathrm{greater}}\:\boldsymbol{\mathrm{than}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{given}} \\ $$$$\boldsymbol{\mathrm{number}}.\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{number}} \\ $$

Answered by Rasheed.Sindhi last updated on 20/May/22

Let the 2-digit number is x and its reversed number is x′ ∴ sum of digits=((x+x′)/(11)) (1): x=4(((x+x′)/(11)))⇒11x=4x+4x′⇒x′=((7x)/4) (2): x′=x+27 ((7x)/4)=x+27 7x=4x+108 3x=108 x=36

$${Let}\:{the}\:\mathrm{2}-{digit}\:{number}\:{is}\:{x}\: \\ $$$$\:\:\:\:{and}\:{its}\:{reversed}\:{number}\:{is}\:{x}' \\ $$$$\therefore\:{sum}\:{of}\:{digits}=\frac{{x}+{x}'}{\mathrm{11}} \\ $$$$\left(\mathrm{1}\right): \\ $$$${x}=\mathrm{4}\left(\frac{{x}+{x}'}{\mathrm{11}}\right)\Rightarrow\mathrm{11}{x}=\mathrm{4}{x}+\mathrm{4}{x}'\Rightarrow{x}'=\frac{\mathrm{7}{x}}{\mathrm{4}} \\ $$$$\left(\mathrm{2}\right): \\ $$$${x}'={x}+\mathrm{27} \\ $$$$\frac{\mathrm{7}{x}}{\mathrm{4}}={x}+\mathrm{27} \\ $$$$\mathrm{7}{x}=\mathrm{4}{x}+\mathrm{108} \\ $$$$\mathrm{3}{x}=\mathrm{108} \\ $$$${x}=\mathrm{36} \\ $$

Commented by otchereabdullai@gmail.com last updated on 21/May/22

nice!

$$\mathrm{nice}! \\ $$

Commented by peter frank last updated on 21/May/22

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

Answered by floor(10²Eta[1]) last updated on 20/May/22

10a+b=4(a+b) ⇒6a=3b⇒2a=b 10b+a=27+10a+b⇒9b=9a+27 ⇒b=a+3⇒2a=a+3⇒a=3⇒b=6 the number is 36

$$\mathrm{10a}+\mathrm{b}=\mathrm{4}\left(\mathrm{a}+\mathrm{b}\right) \\ $$$$\Rightarrow\mathrm{6a}=\mathrm{3b}\Rightarrow\mathrm{2a}=\mathrm{b} \\ $$$$\mathrm{10b}+\mathrm{a}=\mathrm{27}+\mathrm{10a}+\mathrm{b}\Rightarrow\mathrm{9b}=\mathrm{9a}+\mathrm{27} \\ $$$$\Rightarrow\mathrm{b}=\mathrm{a}+\mathrm{3}\Rightarrow\mathrm{2a}=\mathrm{a}+\mathrm{3}\Rightarrow\mathrm{a}=\mathrm{3}\Rightarrow\mathrm{b}=\mathrm{6} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{36} \\ $$

Commented by MathsFan last updated on 20/May/22

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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