Question

Ramola’s grandmother’s present age is 6 times as Ramola’s present age. Three years before, Ramola’s grandmother’s age was 8 times Ramola’s age. Find their present ages.

Answer 1

Hint:
Here, we need to find the present ages of Ramola and her grandmother. We will assume the present age of Ramola and her grandmother be \[x\] and \[y\] respectively. We will use the given information to form two equations in terms of \[x\] and \[y\]. Then, we will solve these equations to get the values of \[x\] and \[y\], and hence, the present ages of Ramola and her grandmother.

Complete step by step solution:
Let the present age of Ramola and her grandmother be \[x\] and \[y\] respectively.
The ages of Ramola and her grandmother 3 years before will be 3 less than their present ages.
Thus, we get the ages of Ramola and her grandmother 3 years before as \[x - 3\] and \[y - 3\] respectively.
Now, it is given that the present age of Ramola’s grandmother is 6 times Ramola’s present age.
Thus, we get the equation
\[y = 6x\]
The age of Ramola’s grandmother 3 years before, was 8 times Ramola’s age at that time.
Thus, we get the equation
\[y - 3 = 8\left( {x - 3} \right)\]
Now, we will solve the two equations to get the values of \[x\] and \[y\], and hence, the present ages of Ramola and her grandmother.
Substituting \[y = 6x\] in the equation \[y - 3 = 8\left( {x - 3} \right)\], we get
\[ \Rightarrow 6x - 3 = 8\left( {x - 3} \right)\]
Multiplying the terms 8 and \[x - 3\] using the distributive law of multiplication, we get
\[ \Rightarrow 6x - 3 = 8x - 24\]
Subtracting \[6x\] from both sides, we get
\[\begin{array}{l} \Rightarrow 6x - 3 - 6x = 8x - 24 - 6x\\ \Rightarrow - 3 = 2x - 24\end{array}\]
Adding 24 to both sides of the equation, we get
\[\begin{array}{l} \Rightarrow - 3 + 24 = 2x - 24 + 24\\ \Rightarrow 21 = 2x\end{array}\]
Dividing both sides by 2, we get the value of \[x\] as
\[ \Rightarrow x = \dfrac{{21}}{2} = 10.5\]
Therefore, the present age of Ramola is \[10.5\] years.
Substitute \[10.5\] for \[x\] in the expression \[y = 6x\], we get
Present age of Ramola’s grandmother \[ = 6x = 6 \times 10.5 = 63\] years

Therefore, we get the present ages of Ramola and her grandmother as \[10.5\] years and 63 years respectively.

Note:
We have used the distributive law of multiplication to find the product of 8 and \[x - 3\]. The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].
We can verify the answer by using the given information.
Ramola’s age three years before was \[10.5 - 3 = 7.5\] years.
Ramola’s grandmother’s age three years before was \[63 - 3 = 60\] years.
Now, we can observe that \[7.5 \times 8 = 60\].
Therefore, we have verified that three years before, Ramola’s grandmother’s age was 8 times Ramola’s age.