Question

Three times Diana’s age is 17 years more than twice Jim’s age. The sum of their ages is 13 years less than their father’s age, which is three times Jim’s age. What are the children’s ages?
A). Diana 21 years, Jim 16 years
B). Diana 15 years, Jim 14 years
C). Diana 15 years, Jim 16 years
D). Diana 20 years, Jim 14 years

Answer 1

Hint: This problem deals with basic algebraic mathematics. Given that there are two children and a father, where the children's names are Diana and Jim, here given the information of their ages. Given that their father’s age is three times Jim's age. Whereas thrice the age of Diana is 17 years more than two times Jim’s age.

Complete step-by-step solution:
Given that three times Diana’s age is 17 years more than twice Jim's age.
Also given that the sum of Diana’s age and Jim’s age is 13 years less than their father’s age.
Given that their father’s age is three times Jim’s age.
Let the age of Diana = $x$
Let the age of Jim = $y$
Let their father’s age = $f$
Now expressing the given information mathematically.
Given that three times Diana’s age is 17 more than twice of Jim’s age, expressing it mathematically:
$ \Rightarrow 3x = 2y + 17$
$ \Rightarrow 3x - 2y = 17$
Now given that the sum of Diana’s age and Jim’s age is 13 years less than their father’s age, as given below:
$ \Rightarrow x + y = f - 13$
Given that their father’s age is three times Jim’s age, as given below:
$ \Rightarrow f = 3y$
Hence substituting the value of $f$ in the equation $x + y = f - 13$, as given below:
$ \Rightarrow x + y = 3y - 13$
$ \Rightarrow x - 2y = - 13$
Now we have 2 equations and 2 variables, hence solving these 2 equations to get the values of $x$and$y$.
Subtracting the 2 equations, as given below:
$ \Rightarrow 3x - 2y = 17$
$ \Rightarrow x - 2y = - 13$
$ \Rightarrow 2x = 30$
$\therefore x = 15$
Now solving the value of $y$, by substituting the value of $x$ in the equation $3x - 2y = 17$:
$ \Rightarrow 3(15) - 2y = 17$
$ \Rightarrow 45 - 17 = 2y$
$ \Rightarrow 28 = 2y$
$\therefore y = 14$
Thus $x = 15,$ and $y = 14$.
Hence the age of Diana is 15 years, and the age of Jim is 14 years.
The children’s ages are, Diana 15 years; Jim 14 years.

Note: Here while solving this problem be careful while writing the expression for their father’s age, the most important thing here is we have to understand the given information in a right way. Here given that the sum of their ages is 13 years less than their father’s age, which is three times Jim’s age, this doesn’t mean that the sum of the children's ages is equal to three times Jim’s age but rather their father’s age is equal to three times Jim’s age. There is a high chance here to go wrong, so please keep this mind.