Question

The sum of the ages of a mother and her
daughter is 50 years. Also 5 years ago, the
mother's age was 7 times the age of the daugh-
ter. The present age of the mother is:​

Answer 1

❍ Let the present age of Mother and her daughter be x and y years respectively.⠀

A/Q,

• The sum of the ages of a mother and her daughter is 50 years.

$:\implies\sf x + y = 50 \\\\\\:\implies\sf x = 50 - y \qquad\quad\bigg\lgroup\bf Equation\;(I)\bigg\rgroup$⠀

Also

• Five years ago, the mother's age was seven times the age of the daughter.⠀

Therefore, ⠀

$:\implies\sf x - 5 = 7(y - 5) \\\\\\:\implies\sf x - 5 = 7y - 35 \\\\\\:\implies\sf x = 7y - 30 \qquad\quad\bigg\lgroup\bf Equation\;(II)\bigg\rgroup$⠀⠀⠀⠀

⠀⠀⠀⠀⠀

F R O M E Qn ( I ) : –

$:\implies\sf 50 - y = 7y - 30 \\\\\\:\implies\sf 8y = 80 \\\\\\:\implies\sf y = \cancel\dfrac{80}{8} \\\\\\:\implies{\underline{\boxed{\frak{\pink{y = 10}}}}}\;\bigstar$⠀

⠀⠀⠀⠀⠀

M O T H E R's A G E :

$:\implies\sf x = 50 - y \\\\\\:\implies\sf x = 50 - 10 \\\\\\:\implies{\underline{\boxed{\frak{\pink{x = 40}}}}}\;\bigstar$ ⠀

$\therefore{\underline{\sf{Hence,\; mother's\;age\;is\; \bf{40\;years }.}}}$

$\rule{250px}{.3ex}$

V E R I F I C A T I O N :

• As per given condition, the sum of the ages of Mother and her daughter is 50 years.

Therefore,

$\dashrightarrow\sf x + y = 50 \\\\\\\dashrightarrow\sf 40 + 10 = 50 \\\\\\\dashrightarrow\underline{\boxed{\frak{50 = 50}}}$

⠀⠀⠀⠀⠀

$\qquad\quad\therefore{\underline{\textsf{\textbf{Hence, Verified!}}}}$

Answer 2

Question:

• The sum of the ages of a mother and her daughter is 50 years. Also 5 years ago, the mother's age was 7 times the age of the daughter. The present age of the mother is:

Answer:

• Present age of mother is 40 years.

Explanation:

Given that:

• Sum of ages of mother and her daughter is 50 years.
• 5 years ago mother's age was 7 times the age of daughter.

To Find:

• Present age of mother?

Solution:

• Let their present age of mother be m years and present age of her daughter be n years.
• So, their ages before 5 years are (m - 5) years and (n - 5) years.

According to the question,

• Sum of their ages is 50 years.

Therefore,

➻ $\sf m + n = 50$

➻ $\bf\red{m = 50 - n}\quad - - - (1)$

Also,

• 5 years ago mother's age was 7 times the age of daughter.

Therefore,

➻ $\sf (m - 5) = 7(n - 5)$

➻ $\sf m - 5 = 7n - 35$

➻ $\sf m = 7n - 35 + 5$

➻ $\bf \purple{m = 7n - 30}\quad - - - (2)$

From (1) and (2) we get,

➻ $\sf 50 - n = 7n - 30$

➻ $\sf 7n + n = 50 + 30$

➻ $\sf 8n = 80$

➻ $\sf n = {\cancel{\dfrac{80}{8}}}$

➻ $\bf\pink{n = 10}\quad - - - (3)$

From (3) put in (1) we get,

➻ $\sf m = 50 - 10$

➻ $\bf\blue{m = 40}$

∴ Present age of mother is 40 years.

Let's Verify:

We know that,

➻ $\sf (m - 5) = 7(n - 5)$

Put m = 40 and n = 10 in above equation we get,

➻ $\sf 40 - 5 = 7(10 - 5)$

➻ $\sf 35 = 7\:\times\:5$

➻ $\sf 35 = 35$

➻ $\bf\green{LHS = RHS}$

Hence, Verified ✔

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