In this chapter, we will discuss the basic concepts of problems on ages involving fractions. We will start by defining what is meant by "ages" and "fractions". We will also discuss the different types of problems that can be solved using these concepts.

Table of Contents

1. Introduction to Age Problems
2. Basic Concepts and Formulas for Problems on Ages
3. Problems on Ages Involving Fractions
4. Problems on Ages with Multiple Persons
5. Problems on Ages with Reversed Ages
6. Problems on Ages with Time Gaps
7. Problems on Ages Involving Death of a Person
8. Problems on Ages with Current Age and Age at a Future Date
9. Problems on Ages with Average Ages
10. Problems on Ages with Ages in Geometric Progression
11. Problems on Ages with Ages in Harmonic Progression
12. Problems on Ages with Ages and Income
13. Problems on Ages with Ages and Discounts
14. Problems on Ages with Ages and Partnership
    

Formulae for Solving Problems on Ages Involving Fractions

• Ratio formula: If the ratio of the present age of a person to another person is given, and the difference between their ages at a certain point in the past or future is known, we can use the following formula:
  
  

(Present age of the first person / Present age of the second person) = (Age difference at a certain point / Age difference in years)

For example, if the ratio of A's age to B's age is 3:4 and the difference between their ages 5 years ago was 6 years, we can use the formula:

(3/4) = (6 / 5)

3 x 5 = 4 x 6

A's present age = 15 years

B's present age = 20 years

• Average formula: If the average age of a group of people is given, and the fraction of the group that belongs to a certain age group is known, we can use the following formula:

(Average age of the group) = (Sum of ages of the group / Number of people in the group)

For example,  For example, if the average age of a group of 5 people is 12 years, and 2 of them are 10 years old, we can use the formula:

12 = (Sum of ages of the group) / 5

Sum of ages of the group = 60 years

Age of the remaining 3 people = (60 - 2 x 10) / 3 = 13.33 years

• Time formula: If the ratio of the ages of two people at different points in time is given, we can use the following formula:
  
  

(Present age of the first person) = (Age at the earlier point in time) x (Ratio) / (Sum of the ratio - 1)

For example,   if the ratio of A's age to B's age 5 years ago was 2:3, we can use the formula:

(Present age of A) = (Age of A 5 years ago) x (2/3) / (2 + 3 - 1)

(Present age of A) = (A's age 5 years ago) x (2/4)

(Present age of A) = (A's age 5 years ago) x (1/2)

By using these formulae and applying the appropriate algebraic operations, we can solve problems on ages only involving fractions more efficiently.

• Proportion Formula: The proportion formula can be used to solve problems that involve comparing the ages of two or more people at different points in time. The formula is as follows:
  
  

(a / b) = (c / d)

where, a = age of the first person at a certain point in time

b = age of the second person at the same point in time

c = age of the first person at another point in time

d = age of the second person at the same point in time

For example, The age of a father is 42 years and that of his son is 18 years. In how many years will the father be twice as old as his son?

Solution: Let the number of years required be x. Then, we have:

(42 + x) / (18 + x) = 2

Simplifying the equation, we get:

42 + x = 36 + 2x

Hence, x = 6 years.

Therefore, the father will be twice as old as his son after 6 years.

Examples on Ages Involving Fractions

Question 1: A person's age was one-fourth of their father's age 12 years ago. The person's father will be twice the person's age in 10 years. Find the present age of the person and their father.

Concept used: Let the person's present age be x and the father's present age be y. Then, using the given information, we can write two equations: x-12 = y/4 and y+10 = 2(x+10).

Solution: From the first equation, y = 4(x-12). Substituting this into the second equation, we get 4(x-12)+10 = 2(x+10), which simplifies to 2x = 44, or x = 22.
Therefore, the person's present age is 22 years old. Substituting x into the first equation,
we can find that the father's present age is y = 4(22-12) = 40 years old.

Question 2: A mother is three times as old as her daughter. Twelve years ago, the mother was four times as old as her daughter. Find the present age of the mother and her daughter.

Concept used: Let the daughter's present age be x and the mother's present age be y. Then, using the given information, we can write two equations: y = 3x and y-12 = 4(x-12).

Solution: Substituting the first equation into the second equation, we get 3x-12 = 4(x-12), which simplifies to x = 36. Therefore, the daughter's present age is 36 years old. Substituting x into the first equation, we can find that the mother's present age is y = 3(36) = 108 years old.

Question 3: The sum of two people's ages is 40. Five years ago, the ratio of their ages was 3:2. Find their present ages.

Concept used: Let one person's present age be x and the other person's present age be y. Then, using the given information, we can write two equations: x+y = 40 and (x-5)/(y-5) = 3/2.

Solution: Rearranging the second equation, we get 2x-15 = 3y-15, which simplifies to 2x = 3y. Substituting this into the first equation, we get 5y = 120, or y = 24. Therefore, one person's present age is 24 years old. Substituting y into the first equation,
we can find that the other person's present age is x = 40-24 = 16 years old.

Question 4: The difference between two people's ages is 10. Three years ago, the ratio of their ages was 3:2. Find their present ages.

Concept used: Let one person's present age be x and the other person's present age be y. Then, using the given information, we can write two equations: x-y = 10 and (x-3)/(y-3) = 3/2.

Solution: Rearranging the first equation, we get x = y+10. Substituting this into the second equation, we get (y+7)/(y-3) = 3/2, which simplifies to y = 19. Therefore, one person's present age is 19+10 = 29 years old. Substituting y into the first equation, we can find that the other person's present age is 19 years old.

Question 5:  A father is twice as old as his son. In 10 years, the father will be three times as old as his son. Find the present age of the father and his son.

Concept used:  Let the son's present age be x and the father's present age be y. Then, using the given information, we can write two equations: y = 2x and y+10 = 3(x+10).

Solution: From the first equation, we get x = y/2. Substituting this into the second equation, we get y+10 = 3(y/2+10), which simplifies to y = 40. Therefore, the father's present age is 40 years old. Substituting y into the first equation,

we can find that the son's present age is x = 40/2 = 20 years old.

Question 6:  A grandfather is five times as old as his granddaughter. In 15 years, he will be four times as old as his granddaughter. Find the present age of the grandfather and his granddaughter.

Concept used:  Let the granddaughter's present age be x and the grandfather's present age be y. Then, using the given information, we can write two equations: y = 5x and y+15 = 4(x+15).

Solution: From the first equation, we get x = y/5. Substituting this into the second equation, we get y+15 = 4(y/5+15), which simplifies to y = 75. Therefore, the grandfather's present age is 75 years old. Substituting y into the first equation,
we can find that the granddaughter's present age is x = 75/5 = 15 years old.

Question 7:  The sum of two people's ages is 50. Four years ago, the ratio of their ages was 2:3. Find their present ages.

Concept used:  Let one person's present age be x and the other person's present age be y. Then, using the given information, we can write two equations: x+y = 50 and (x-4)/(y-4) = 2/3.

Solution: Rearranging the first equation, we get x = 50-y. Substituting this into the second equation, we get (46-y)/(y-4) = 2/3, which simplifies to y = 28. Therefore, one person's present age is 28+22 = 50 years old. Substituting y into the first equation,
we can find that the other person's present age is 22 years old.

Question 8:  The difference between two people's ages is 18. Six years ago, the ratio of their ages was 2:3. Find their present ages.

Concept used:  Let one person's present age be x and the other person's present age be y. Then, using the given information, we can write two equations: x-y = 18 and (x-6)/(y-6) = 2/3.

Solution: Rearranging the first equation, we get x = y+18. Substituting this into the second equation, we get (y+12)/(y-6) = 2/3, which simplifies to y = 30. Therefore, one person's present age is 30+18 = 48 years old. Substituting y into the first equation,
we can find that the other person's present age is 30 years old.