Question

138. 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:
a) 30 years
b) 35 years
c) 40 years
d) 45 years​

Answer 1

Given :-

• The sum of the ages of a mother and daughter = 50
• 5 years ago, mother's age was 7 times the age of daughter

To find :-

The present age of the mother.

Let the daughter's age be x.

5 years ago,

Daughter = x - 5

Mother = 7(x-5) =≥ 7x - 35

Hence, after 5 years/present age of mother = 7x - 35 + 5 =≥ 7x - 30

As we know that the sum of their present ages is 50, by substituting the values,

Mother's age + Daughter's age = 50

7x - 30 + x = 50

8x - 30 = 50

Transposing (-30) to the RHS(Right Hand Side),

8x = 50 + 30

8x = 80

$x = \dfrac{80}{8}$

$x = 10$ Therefore the mother's age is (7x - 30) =≥ 7(10) - 30

Mother's age = 40 (option c)

Answer 2

${\large{\frak{\pmb{\underline{Correct \; Question}}}}}$

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 -

• a) 30 years
• b) 35 years
• c) 40 years
• d) 45 years

${\large{\frak{\pmb{\underline{Given \; that}}}}}$

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

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

${\large{\frak{\pmb{\underline{To \; find}}}}}$

★ The present age of mother.

${\large{\frak{\pmb{\underline{Solution}}}}}$

★ The present age of mother = 40 year's Option c)

${\large{\frak{\pmb{\underline{Assumptions}}}}}$

★ The present age of mother be a

★ The present age of daughter be b

${\large{\frak{\pmb{\underline{Full \; Solution}}}}}$

• As it's given that the sum of the ages of a mother and her daughter is 50 years. Henceforth,

$\begin{gathered} \sf :\implies a + b = 50 \\ \\ \sf :\implies \: \: a = 50 - b\end{gathered}$

• It is also given that, 5 years ago, the mother's age was 7 times the age of the daughter.

$\begin{gathered}\sf :\implies a - 5 = 7(b - 5) \\ \\ \sf :\implies a - 5 = 7b - 35 \\ \\ \sf :\implies a = 7b - 35 + 5 \\ \\ \sf :\implies a = 7b - 30 \\ \\ \bf \: Now \: we \: have \: to \: use \: a = 7b - 30 \: and \: 50 - b \\ \\ \sf :\implies 50 - b \: = 7b - 30 \\ \\ \sf :\implies 50 + 30 = 7b + b \\ \\ \sf :\implies 50 - b \: = 7b - 30 \\ \\ \sf :\implies 50 + 30 = 7b + 1b \\ \\ \sf :\implies 80 = 8b \\ \\ \sf :\implies 80 \div 8 = b \\ \\ \sf :\implies 10 = b \\ \\ \sf :\implies b = 10 \end{gathered}$

Henceforth, b = 10 means age of daughter is 10 years.

• Now let's find the age mother

$\begin{gathered}\sf :\implies a = 50 - b \\ \\ \sf :\implies a = 50 - 10 \\ \\ \sf :\implies a = 40\end{gathered}$

Henceforth, value of a is 40, mother's age = 40 years means Option c) is correct