Age Problems
a) If a person’s age is x,
\( \bullet \) Their age after \( t \) years: \( x \; + \; t \)
\( \bullet \) Their age \( t \) years ago: \( x \; – \; t \) .
b) If the sum of the ages of n people is x,
\( \bullet \) The sum of their ages after \( t \) years: \( x \;+ \;n \cdot t \)
\( \bullet \) The sum of their ages \( t \) years ago: \( x \; – \;n \cdot t \) .
c) The age difference between two people does not change with years; it is constant.
Example:
Fatih is 8 years older than Yavuz. Since the ratio of Fatih’s age to Yavuz’s age 5 years from now will be \( \large \frac{3}{2} \), let’s find Yavuz’s age.
If Yavuz’s age is called \(x \), Fatih’s age becomes \( x+8 \). 5 years from now, since the ratio of Fatih’s age to Yavuz’s age will be
\[ \frac{(x+ 8 ) + 5 }{x+5 } = \frac{3}{2} \] Yavuz’s age is found to be \( x = 11 \).
Example:
Erdem is 4 years older than his sibling. His father’s age is 3 times Erdem’s age. Given that the sum of their ages 5 years ago was 41, let’s find how old the father was when Erdem was born.
If Erdem’s age is called \(x\), his sibling’s age becomes \( x – 4 \), and his father’s age becomes \(3x \). Since the sum of their ages 5 years ago was \( 41 \), the sum of their current ages is: \[ 41 + 3 \cdot 5 = 56 \] From here, \[ 3x + x + x – 4 = 56 \Rightarrow x = 12 \quad \text{and } \quad 3x = 36 \]
Accordingly, the father’s age when Erdem was born is: \[ 36-12 = 24 \] years old.
Example:
A father’s age is 4 times the difference between his two children’s ages. Since the father’s age 5 years from now will be 2 less than 5 times the difference between his children’s ages, let’s find the father’s current age.
If the difference between the two children’s ages is called \(x \), the father’s current age becomes \(4x \) , and his age 5 years from now becomes \(4x + 5\). Writing the equation for the given problem from here:
\[ 4x + 5 = 5x − 2 \Rightarrow x = 7 \]
and the father’s current age is found to be \[ 4x = 28 \quad . \]
Question 7
Alparslan is 10 years older than Engin. 8 years ago, Alparslan’s age was 2 times Engin’s age. Accordingly, what is the sum of Alparslan’s and Engin’s current ages?
\[
\text{A)} 28 \quad
\text{B) } 30\quad
\text{C) } 32 \quad
\text{D) } 36 \quad
\text{E) } 46
\]
Solution:
If Engin’s age is called \(x \), Alparslan’s age becomes \(x + 10 \); 8 years ago, Engin’s and Alparslan’s ages were \( x − 8 \) and \(x + 2 \), respectively. Writing the equation for the given problem from here,
\[x + 2 = 2 \cdot (x − 8) \Rightarrow x = 18 \] is found. Accordingly, the sum of Alparslan’s and Engin’s current ages is:
\[ x + 10 + x = 18 + 10 + 18 = 46 \quad . \]
\(\textbf{Answer: E} \)
Question 8
Yunus’s age is 3 times Selim’s age. When Yunus’s age increases by half of his own age, the sum of their ages becomes 56. Based on this, what is Selim’s current age?
\[
\text{A)} 8 \quad
\text{B) } 9\quad
\text{C) } 10 \quad
\text{D) } 11 \quad
\text{E) } 12
\]
Solution:
If Selim’s age is called \( 2x \), Yunus’s age becomes \( 6x \). When Yunus’s age increases by half of his own age (meaning \( 3x \) years later), Selim’s age becomes \( 5x \) and Yunus’s age becomes \( 9x \). Accordingly, writing the equation for the given problem:
\[ 5x + 9x = 56 \Rightarrow x = 4 \] then Selim’s age is found to be
\[ 2x = 8\]
\(\textbf{Answer: A} \)
Question 9
The sum of the ages of three siblings is 64. The oldest sibling is 12 years older than the youngest one. When the middle sibling reaches the current age of the oldest sibling, the sum of the siblings’ ages becomes 88. Accordingly, how old is the youngest sibling?
\[
\text{A)} 14 \quad
\text{B) } 16\quad
\text{C) } 18 \quad
\text{D) } 20 \quad
\text{E) } 24
\]
Solution:
Let the youngest sibling’s age be \( x \) , the middle sibling’s age be \( y \), and the oldest sibling’s age be \( x + 12 \). The age of the middle sibling becomes equal to the current age of the oldest sibling after \(x \;+ \; 12 \;- \; y \) years. Accordingly, the sum of the ages of the 3 siblings increases by,
\[ x\; + \; 12 \;-\; y \quad \text{years } \quad 88- 64 = 24 \] . Therefore,
\[ 3 \cdot (x\; +\; 12 \;- \;y ) = 24 \Rightarrow y = x+ 4 \] . Since the sum of the ages of the 3 siblings is 64:
\[ x + (x +4 ) + (x + 12 ) = 64 \Rightarrow x = 16\]
\(\textbf{Answer: B} \)
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