Question

the sum of the digits of a two-digit number is 12 if the new number formed by reversing the digits is greater than the original number by 54 find the original number

Answer 1

Let xy be the required two-digit number.

Let x be the required number in unit's digit.

Let y be the required number in ten's digit.

Therefore the decimal expansion is 10x+y.


Given that the sum of digits of a two-digit number = 12.

x + y = 12  ---- (1).


Given that if the new number formed by reversing the digits is > than original number by 54.

10x + y + 54 = 10y + x

10x - x + y - 10y = -54

9x - 9y = -54.

x - y = -6   ------- (2)


On solving (1) & (2), we get

x + y = 12

x - y = -6

--------------

2y = 18

y = 9


Substitute y = 9 in (1), we get

x + y = 12

x + 9 = 12

x = 12 - 9

x = 3.


Therefore the required number is 39.


Verification:

x + y = 12

3 + 9 = 12

12 = 12.


Hope this helps!

Answer 2

Aɴꜱᴡᴇʀ

☞ The original number is 39

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Gɪᴠᴇɴ

➳ The sum of the digits of a two digit number is 12.

➳ When the digits are reversed, the new number so formed is greater than the original number by 54.

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Tᴏ ꜰɪɴᴅ

➠ The original number?

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Sᴛᴇᴘꜱ

❍ Let the digit at ten's place be x and the digit at unit's place be y.

✭ $\underline{\boxed{\sf Original \: Number = 10x + y}}$

As per the question,

❍ The sum of the digits of a two digit number is 12.

$\sf \dashrightarrow x + y = 12 \\\\\dashrightarrow \sf y = 12 - x \: \: \: \dots (i)$

Also, it is given that,

❍ When the digits are reversed, the new number so formed is greater than the original number by 54.

Equation :

$\sf \leadsto10y + x = 10x + y + 54 \\\\\leadsto \sf 10y - y - 10x + x = 54 \\\\\leadsto \sf 9y - 9x = 54\\\\\leadsto \sf 9(12-x) - 9x = 54 \\\\\leadsto \sf 108 - 9x - 9x = 54\\\\\leadsto \sf - 18x = 54 - 108 \\\\\leadsto \sf - 18x = - 54 \\\\\leadsto\sf x = \frac{-54}{-18} \\\\\pink{ \leadsto\sf x = 3 }$

❍ Substituting the value of x in (i),

$\sf \dashrightarrow y = 12 - x\\\\\dashrightarrow \sf y = 12 - 3\\\\\dashrightarrow \sf \pink{y = 9}$

➼ Now, original number = 10x+y

$\longmapsto \sf{}10 \times 3 + 9 \\ \\ \longmapsto \sf30 + 9 \\ \\ \sf\pink{\longmapsto39}$

Hence, the required number is 39.

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