Formulas
1. If the current age is x, then n times the age is nx.
2. If the current age is x, then age n years later/hence = x + n.
3. If the current age is x, then age n years ago = x - n.
4. The ages in a ratio a : b will be ax and bx.
5. If the current age is x, then $\dfrac{1}{n}$ of the age is $\dfrac{x}{n}$
Problems on Ages:
Age problems are very similar to number word problems. Problem on ages can be categorized into three types, i.e. Questions based on calculating the present age, Questions to determine the age of person after k years and questions that calculate age of a person before k years. These three types may cover cases of various types with different combination of ratios, fractions etc.
Before moving forth with the above given types of problems keep in mind these quick tips:-
- If the age of the person is x years then,
- Age after k years will be x + k
- Age before k years will be x – k
- The age k times would be kx
- If ages are given in form of ratio P: Q then, then P: Q would be Px and Qx respectively.
Problem 1:
Karen is thrice as old as Hannah. The difference between their ages is 18 years. What are their ages?
Solution:
First, we let x be Hannah’s age. Since Karen is thrice as old as Hannah, we multiply her age by 3. For instance, if Hannah is 10 years old now, then Karen is 30 or 3(10).
Now, since Hannah’s age is x, then Karen’s age is thrice x or 3x.
Now, to set up the equation, we look at the second statement. The difference between their ages is 18 years. This means that
Karen’s Age – Hannah’s age = 18.
In equation form, we have 3x - x = 18.
This gives us 2x = 18 which means that x = 9.
So, Hannah is 9 years old and Karen is 18.
Problem 2:
Mika is twice as old as Ella. Ten years ago, Mika was three times as old as Ella. What are their present ages?
Solution:
Mika is twice as old as Ella, so if Ella is x years old, then Mika is 2x. Ten years, ago Ella’s age is x - 10 and Mika’s age is 2x - 10.
According to the problem, Mika’s age which was 2x - 10 was three times Ella’s age which is x - 10. This implies that if we multiply Ella’s age by 2, their ages will be equal. That is,
3(x - 10) = 2x - 10
Simplifying, we have
3x - 30 = 2x - 10. This gives us x = 20.
Exercise:
let x be Carlo’s age. Twice his age is 2x. One less than twice his age is 2x - 1 which is Adrian’s age. Since the sum of their ages is 50,
we have
x + (2x - 1) = 50.
Simplifying, we have 3x - 1 = 50. This gives us 3x = 51 and x = 17.
Therefore, Carlo is 17 years old and Adrian is 2(17) - 1 = 33 years old.
Note:
One common mistake in solving problems containing the word “less” is the confusion between “a less than b” and “a less b.” The expression a less than b is b - a, while the expression a less b is a - b. In the problem above, the expression 1 less than twice Carlo’s age is 2x - 1 and not 1 - 2x.
Problem 3 :
Pol is 10 years younger than Greg. In 7 years, he will be 10 years more than one half as old as Greg. Find their age at present.
solution:
Let x be Greg’s age and let x - 10 be Pol’s age. In 7 years, their ages will be
Greg’s age: x + 7
Pol’s age: x - 10 + 7 = x - 3.
In the problem, it is said that in 7 years, Paul’s age will be ten years more than half of Greg’s age. Now, in 7 years, half of Greg’s age will be $\dfrac{1}{2}(x + 7).$
Now, since Paul is 10 more than one half Greg’s age, if we add 10 years to half Greg’s age which is $\dfrac{1}{2}(x + 7)$, their ages will be equal.
Therefore, $\dfrac{1}{2}(x + 7) + 10 $= x - 3.
Multiplying both sides by 2 we have x + 7 + 20 = 2x - 6.
Simplifying, we have x + 27 = 2x - 6 which gives us x = 33.
Therefore, Greg is 33 years old and Pol is 23.
Note:
If ages are given in form of ratio P: Q
then, then P: Q would be Px and Qx
respectively.
The example problem for this type is as follows:
problem 4:
The ratio of present ages of A and B is 6 : 7. Five years hence, this ratio would become 7 : 8. Find the present age of A and B.Solution :
Let the common ratio be ‘n’.=> A’s present age = 6 n years
=> B’s present age = 7 n years
So, according to the question
(6 n + 5) / (7 n + 5) = 7 / 8
=> 48 n + 40 = 49 n + 35
=> n = 5
Thus, A’s present age = 6 n = 30 years and B’s present age = 7 n = 35 years
Exercise:
Then, (4x - 8) + (7x - 8) + (9x - 8) = 56
20x = 80
x = 4.
Their present ages are 4x = 16 years, 7x = 28 years and 9x = 36 years respectively.