Question

mother says to her daughter, if you reverse my own age, the figures represent yours father's age.he is senior to me and the difference between our ages is one-eleventh of their sum. the mother's age is​

Answer 1

Let x and y be the ten's and one's digit representing the age of mother.

$\implies$ Age of mother = (10x + y) years

Mother says to her daughter, if you reverse my own age, the figures represent yours father's age.

In case of father, y represents the ten's digit and x represents the one's digit.

$\implies$ Age of father = (10y + x) years

Difference between our (father and mother) ages is one-eleventh of their sum.

According to question,

⇒ (10y + x) - (10x + y) = (10y + x + 10x + y)/11

⇒ 10y + x - 10x - y = (11y + 11x)/11

⇒ 9y - 9x = (11y + 11x)/11

⇒ 9(y - x) = 11 × (y + x)/11

⇒ 9(y - x) = (y + x )

⇒ 9y - 9x = y + x

⇒ 9y - y = x + 9x

⇒ 8y = 10x

⇒ y/x = 10/8

⇒ y/x = 5/4

Therefore,

Age of father = 10y + x

⇒ 10(5) + 4 = 54 years

Age of mother = 10x + y

⇒ 10(4) + 5 = 45 years

Answer 2

Question :-- mother says to her daughter, if you reverse my own age, the figures represent yours father's age.he is senior to me and the difference between our ages is one-eleventh of their sum. the mother's age is ?

Solution :--

→ Let us assume that , Mother as is (10a+b) year old.

→ Than, According to given data , father age will be = (10b+a) years.

Now , it has been said that, difference between Father and mother ages is one-eleventh of their sum.

So, A/q, ,

→ (10b+a) - (10a+b) = 1/11 of [ 10a+b + 10b+a ]

→ 10b + a - 10a - b = 1/11 of [ 11a + 11b ]

→ 9b - 9a = a + b

→ 9b - b = a + 9a

→ 8b = 10a

→ b/a = 10/8

→ b/a = 5/4.

So, Mother age = (10a+b) = 10*4 + 5 = 45 years.

Hence, Mother is 45 years old.

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Lets Try to Solve the Problem with little bit shortcut now,

First of all we know that,

→ Difference b/w any two digits (After reversing ) is always Divisible by 9 .

→ Sum of Two digits (After reversing ) is always Divisible by 11.

Lets Take Few Example :--

Example (1)

→ 74 = its reverse = 47 .

→ Difference = 74-47 = 27 = 9*3 = Divisible by 9.

→ Sum = 74+47 = 121 = 11 * 11 = Divisible by 11.

Example (2) :-

→ 46 = its reverse = 64 .

→ Difference = 64-46 = 18 = 9*2 = Divisible by 9.

→ Sum = 46+64 = 110 = 11 * 10 = Divisible by 11.

we can take any two digit number, they will satisfy both the condition .

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Now, in Question , it has been said that, mother age was a two - digit number, also Father age was Reverse of mother age .

→ So, their differnce will be Multiply of 9 .

And , it has been also said that, sum of both age was 1/11th of Their Difference,

→ And, sum is divisible by 11 . So, 11 will be cancel here .

So, we get,

→ 9 * Difference = (sum of digit )

or,

→ 9(b - a) = (a + b) [ i assume both digits again as a and b]

→ 9b - b = a + 9a

→ 8b = 10a

→ b/a = 5/4 .

So, Mother age = (10a+b) = 10*4 + 5 = 45 years.

Hence, Mother is 45 years old.

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"Maths is all about , playing with Numbers"..