In a certain number is divided by the sum of its two digits, the quotient is 6 and remainder is 3. If the digits are interchanged and the r

Question

In a certain number is divided by the sum of its two digits, the quotient is 6 and remainder is 3. If the digits are interchanged and the resulting number is divided by the sum of its digits, then the quotient is 4 and the remainder is 9. Find the number​

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Alice 3 weeks 2021-09-04T17:44:20+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-09-04T17:45:24+00:00

    \bigstar \sf{\red{\underline{\underline{\blue{To\:Find:-}}}}}

    • The Number [According to question] .

    \bigstar \sf{\blue{\underline{\underline{\red{SOLUTION:-}}}}}

    GIVEN :

    • In a certain number is divided by the sum of its two digits, the quotient is 6 and reminder is 3 .
    • If the digits are interchanged and the resulting number is divided by the sum of its digits, then the quotient is 4 and reminder is 9 .

    FORMULA :

    \tt{\green{\boxed{Dividend\:=\:(Divisor\:\times\:Quotient)\:+\:Reminder\:}}}

    CALCULATION :

    (╬) Suppose the original Numbers,

    • unit digit be x
    • tens digit be y
    • The Original Number = (10y + x)

    (╬) According to question,

    • (10y + x) is divided by (x + y) and we get 6 as a quotient and 3 as a reminder .
    1. Dividend = (10y + x)
    2. Divisor = (x + y)
    3. Quotient = 6
    4. Reminder = 3

    \tt{\implies\:(10y\:+\:x)\:=\:{(x\:+\:y)\:\times\:6}\:+\:3\:}

    \tt{\implies\:4y\:-\:5x\:-\:3\:=\:0} ———(1)

    Again,

    • If the digits are interchanged, we get (10x + y) as a number and (10x + y) is divided by (x + y) resulting 4 as a Quotient and 9 as a reminder .
    1. Dividend = (10x + y)
    2. Divisor = (x + y)
    3. Quotient = 4
    4. Reminder = 9

    \tt{\implies\:(10x\:+\:y)\:=\:{(x\:+\:y)\:\times\:4}\:+\:9\:}

    \tt{\implies\:6x\:-\:3y\:-\:9\:=\:0\:} ———(2)

    Now, we solve the Equations (1) and (2) ,

    • see the attachment picture for solvation of Equations (1) and (2)

    we get,

    • x = 5 and y = 7 .

    Therefore, The original Number is ,

    \tt{\blue{Original\:Number\:=\:(10y\:+\:x)\:}}

    \tt{\implies\:Original\:Number\:=\:(10\times7\:+\:5)\:}

    \tt{\implies\:Original\:Number\:=\:75\:}

    \bigstar\:\underline{\boxed{\bf{\red{Required\:Answer\::\:Original\:Number\:=\:75\:}}}}

    0
    2021-09-04T17:46:09+00:00

    \sf\orange{\underbrace{Answer\implies 75}}

    \sf\large\underline{Let:}

    \sf{\implies The\: ones\: digit=y}

    \sf{\implies The\: tens\: digit=x}

    \sf{\implies The\: certain\: number=10x+y}

    \sf{\implies The\: interchanged\: number=10y+x}

    \sf\large\underline{To\: Find:}

    \sf{\implies The\: certain\: number=?}

    \sf\large\underline{Solution:}

    • Here we use formula of checking division to setting up the equation as per the given Question]

    \sf\small\underline{Given\:in\:case\:(i):}

    • A certain number is divided by the sum of its two digits, the quotient is 6 and remainder is 3.]

    =>Dividend=Divisor×Quotient+Remainder:

    \tt{\implies 10x+y=6(x+y)+3}

    \tt{\implies 10x+y=6x+6y+3}

    \tt{\implies 10x-6x+y-6y=3}

    \tt{\implies 4x-5y=3------(i)}

    \sf\small\underline{Given\:in\:case\:(ii):}

    • If the digits are interchanged and the resulting number is divided by the sum of its digits, then the quotient is 4 and the remainder is 9.]

    \tt{\implies 10y+x=4(x+y)+9}

    \tt{\implies 10y+x=4x+4y+9}

    \tt{\implies 4x-x+4y-10y=-9}

    \tt{\implies 3x-6y=-9}

    • Here diving by 3 on both sides]

    \tt{\implies x-2y=-3------(ii)}

    • Now in eq (ii) multiplying by 4 than subtracting from from equation (i) here]

    \tt{\implies 4x-5y=3}

    \tt{\implies 4x-8y=-12}

    • By solving we get, here]

    \tt{\implies 3y=15\implies y=5}

    • Putting the value of y=5 in eq (ii) here]

    \tt{\implies x-2y=-3}

    \tt{\implies x-2(5)=-3}

    \tt{\implies x-10=-3}

    \tt{\implies x=-3+10\implies x=7}

    Hence,

    • The certain number=10x+y
    • The certain number=10×7+5
    • The certain number=70+5
    • The certain number=75

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