Solving the Age Difference with Ratio: A Step-by-Step Guide
Finding the ages of individuals when given their age ratios and a specific difference is a common mathematical problem. In this guide, we will walk through a detailed step-by-step process to find the age of S given that S is younger than R by 7 years and that their ages are in the ratio of 7:9. This example is particularly suitable for SEO as it includes a variety of keyword mentions relevant to age ratio problems and mathematical problem-solving.
The Problem Statement
The main problem is to find the age of S given the following conditions:
S is younger than R by 7 years. The ratio of their ages is 7:9.We will solve this step-by-step, outlining each calculation and reasoning process to ensure clarity and ease of understanding.
Step 1: Establishing Variables
Let's denote the age of S as ( x ) and the age of R as ( y ).
Since the ratio of their ages is 7:9, we can write:x : y 7 : 9
This can be expressed as:
frac{x}{y} frac{7}{9}
Step 2: Formulating the Equation Based on the Age Difference
According to the problem, S is younger than R by 7 years. Therefore, we can write:
y - x 7
Substituting ( y ) from the ratio equation, we get:
frac{9x}{7} - x 7
Step 3: Solving for x (Age of S)
Let's simplify and solve the above equation:
frac{9x - 7x}{7} 7
Simplifying the numerator:
frac{2x}{7} 7
Multiplying both sides by 7:
2x 49
Dividing both sides by 2:
x frac{49}{2} 24.5
Therefore, the age of S is 24.5 years.
Step 4: Verifying the Ages
To verify, we find the age of R:
y frac{9}{7}x frac{9}{7} times 24.5 31.5
Check the difference:
31.5 - 24.5 7 (which matches the given condition)
Conclusion
We have successfully found that the age of S is 24.5 years and the age of R is 31.5 years. The problem is well-constructed, making it a useful example for teaching and practicing age ratio problems.
Additional Practice Problems
For further practice, consider the following similar problems:
There are two friends, A and B. A is younger than B by 5 years, and their ages are in the ratio 8:11. Find the age of A. Given that C is 10 years younger than D, and their ages are in the ratio 5:7, what is the age of C?