Aptitude Discussion

Q. |
The digits of a three number are in $AP$. If the number is subtracted from the number formed by reversing its digits, the result is 396. What could be the original number? |

✖ A. |
654 |

✔ B. |
135 |

✖ C. |
852 |

✖ D. |
753 |

**Solution:**

Option(**B**) is correct

The difference between 3-digit number and its reverse is 99 times the difference between its extreme (hundred and units) digits.

As the first difference is 396, the second is 4.

Further as the digits are in $AP$ and the hundred's digits is less than the unit's digit, we have following possibilities:

135,246,357,468,579.….

**Edit: **For alternative solution, check comment by **Deepak.**

**Edit 2: **For yet another alternative solution, check comment by **Sravan Reddy.**

**Pooja**

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**Sravan Reddy**

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Killing it from options.

The difference between the last and first term

coming back from options would be quicker here. Also instead of claculating all options, you can directly get to answer by checking first and last terms.

First term minus last term should be 4 or 3 (100th digit of 369). Only possible with 153 and 753.

Last term minus first term should be 6 (Unit digit of 369). Only possible with 153 now

**Deepak**

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Lets say the three digit number is $XYZ$.

Now as per the question:

$(100Z+10Y+X)-(110X+10Y+Z)=396$

$\Rightarrow 99(Z-X)=396$

$\Rightarrow Z-X=\dfrac{396}{99}=4$

So unit's digit($X$) in the original 3-digit number is 4 less than the hundred's digit($Z$). i.e. digit $X$ is 4 less than the digit $Z$.

Also the three digits form A.P.

So,

$Z=X+2d$, where $d$ is the common difference.

Now,

$Z-X=4$

$\Rightarrow (X+2d)-X=4$

$\Rightarrow d=2$

So, the pssible 3 digit number could be:

$XYZ\equiv \{135, 246,357,468,...\}$

Among the possible numbers, 135 is listed in the **Choice B**, so it is the right answer.

Deepak,

Rather It should be

(100X + 10Y + Z ) - (100Z + 10Y +X)

For eg : take 413, how it should be written?

4 x 100 + 1 x 10 + 3 rather than 3 x 100 + 1 x 10 + 4 as you have written...

And according to that your answer should be 531 instead of 135..

While mentioning the digit's, I mistakenly interchanged $X$ and $Z$.

Since reversed number is $ZYX$, you may re-read the sentence as,

'So unit's digit($X$) in the REVERSED 3-digit number is 4 less than the hundred's digit($Z$). i.e. digit $X$ is 4 less than the digit $Z$.'

Other than that, things are fine and the final answer is correct too.

**Ketan**

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please give some clarification about the question

I've tried explaining in detail. See if that's useful.

Deepak,

Rather It should be

(100X + 10Y + Z ) - (100Z + 10Y +X)

For eg : take 413, how it should be written?

4 x 100 + 1 x 10 + 3 rather than 3 x 100 + 1 x 10 + 4 as you have written...

And according to that your answer should be 531 instead of 135.