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# A number consists of two digits whose sum is 9. If 27 is added to the number, its digits are interchanged. Which of the given steps is CORRECT to find the number?Step1: Let the units digit be x.Step2: Then, ten’s digit = $\left( 9-x \right)$ . $\therefore$ Number = $10\times \left( 9-x \right)+x$ . $\Rightarrow 90-10x+x=90-9x$ .Step3: Adding 27 to the number $90-9x$ , we get $117-9x$ .Step4: Number with digits interchanged is $10x+\left( 9-x \right)=9x+9$ .Step5: $117-9x=9x+9$ .Step6: Therefore, unit’s digit = 6 and ten’s digit = 3.Step7: Hence the number = 36.(a) Only Step 4(b) Both Step 1 and Step 2(c) Step 1, 2, 3 and 4.(d) All steps are correct.

Last updated date: 10th Sep 2024
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Hint:
We start solving the problem by assuming the value of x as the obtained answer to prove the converse of the given steps. We then substitute the value of x in each step and then check the values obtained in each step to prove it. We then check the values obtained in steps 1-5 to check whether they are equal to steps 6 and 7.

So, let us assume $x=6$ .
Step2: ten’s digit = $9-6=3$ .
$\therefore$ Number = $10\times \left( 9-6 \right)+6$ .
$\Rightarrow 90-60+6=36$ (which is the obtained answer). So, step 1 and 2 are correct
Step4: Number with digits interchanged is $10\left( 6 \right)+\left( 9-6 \right)=63$ . Which is the required number after reversal of digits. So, step 3 and 4 are also correct
Step5: $117-9\left( 6 \right)=9\left( 6 \right)+9\Leftrightarrow 63=63$ .