Solving Age Problems with Ratios: A Comprehensive Guide

This article provides a comprehensive guide to solving age problems involving ratios. It outlines step-by-step methods to accurately determine the ages of individuals based on given ratios, timeframes, and conditions. The examples below illustrate how to approach and solve such problems using algebraic equations and reasoning.

Problem 1: A's Age is 2/3 of B's Age

Statement: The age of A is 2/3 that of B. Five years ago, the age of A was 3/5 of B's age. What are their present ages?

Solution:

Let the present age of A be a and the present age of B be b.

From the problem, we have the following conditions:

(a frac{2}{3}b)

(a - 5 frac{3}{5}(b - 5))

Substitute (a frac{2}{3}b) into the second equation:

(frac{2}{3}b - 5 frac{3}{5}(b - 5))

Multiply both sides by 15 to eliminate fractions:

10b - 75 9b - 45

Simplify and solve for b:

10b - 9b 75 - 45

(b 30)

Now, find a using (a frac{2}{3}b):

(a frac{2}{3} times 30 20)

Therefore, the present age of A is 30 years, and the present age of B is 20 years.

Problem 2: A and B's Ages after 10 Years

Statement: 10 years ago, the ratio of A's age to B's age was 3:5. After 10 years, the ratio of their ages will be 2:3. What are their present ages?

Solution:

Let the present age of A be x and the present age of B be y.

From the problem:

(frac{x - 10}{y - 10} frac{3}{5})

(frac{x 10}{y 10} frac{2}{3})

Multiply the first equation by 5(y - 10) and simplify:

5(x - 10) 3(y - 10)

(5x - 50 3y - 30)

(5x - 3y 20)

Multiply the second equation by 3(y 10) and simplify:

3(x 10) 2(y 10)

(3x 30 2y 20)

(3x - 2y -10)

Solve the system of equations:

(begin{align*}5x - 3y 20 3x - 2y -10end{align*})

Multiply the first equation by 2 and the second by 3:

(begin{align*}1 - 6y 40 9x - 6y -30end{align*})

Subtract the second from the first:

(x 70)

Substitute (x 60) into the first equation:

(5(60) - 3y 20)

(300 - 3y 20)

(-3y -280)

(y 80)

Therefore, the present age of A is 60 years, and the present age of B is 40 years.

Problem 3: B's and A's Ages with Given Differences

Statement: B is 2 years older than 2.5 times the age of A. A is 45 years old. What is their present age?

Solution:

Let the present age of A be x and the present age of B be y.

From the problem:

(y 2.5x 2)

Given that (x 45):

(y 2.5 times 45 2)

(y 112.5 2)

(y 114.5)

Correcting for the integer age:

(y 115)

Therefore, the present age of A is 45 years, and the present age of B is 115 years.

Conclusion: This guide has demonstrated various methods to solve age problems involving ratios. By setting up appropriate equations and solving them step-by-step, we can accurately determine the ages of individuals based on given conditions and ratios. These problems can be solved using algebraic techniques, and the key is to express the relationships between the ages in terms of equations.