Question

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.

Answer 1

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.

Complete step by step answer:
According to the problem, we are given that a number consists of two digits whose sum is 9. If 27 is added to the number, its digits are interchanged. We need to which of the given steps are correct.
Let us verify the given steps by assuming the digits in unit value as the obtained answer and substitute that in each step.
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
Step3: Adding 27 to the number 36, we get 63.
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 $ .
We can see that the answers obtained in step 1-5 resemble with steps 6 and 7.

Note:
 Whenever we get this type of problem, we try to prove the converse of the given steps in the problem. We should keep in mind that 0-9 are called digits and others are not considered as them which is important while solving this type of problem. We can also solve the given problem in our methods to check whether the obtained answer is true or not.