Solving Ratio Problems Involving Past and Future Ages
Understanding and solving problems that involve the ratio of ages, both in the past and the future, is a common application of algebra in everyday life and academic settings. In this article, we will solve a specific problem that requires us to find the ratio of ages for individuals A and B, at different points in time.
Problem Statement
Let the present ages of A and B be 8x and 15x respectively, where x is a positive constant.
8 years ago, their ages were in the ratio 6:13.
What is the ratio of their ages 8 years from now?
Step-by-Step Solution
Step 1: Set up the equation for ages 8 years ago
Evaluate the ages of A and B 8 years ago:
A's age: 8x - 8
B's age: 15x - 8
Given that their ages in the ratio 6:13 can be expressed as:
frac{8x - 8}{15x - 8} frac{6}{13}
Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
13(8x - 8) 6(15x - 8)
Step 3: Expand and simplify the equation
Expand both sides of the equation:
104x - 104 9 - 48
Step 4: Solve for x
Rearrange the equation to isolate x:
104x - 9 104 - 48
14x 56
x 4
Step 5: Calculate the present ages
Substitute x 4 into the expressions for A and B:
A's present age: 8x 8 times 4 32 years
B's present age: 15x 15 times 4 60 years
Step 6: Find their ages 8 years from now
In 8 years, their ages will be:
A's future age: 32 8 40 years
B's future age: 60 8 68 years
Step 7: Calculate the ratio of their ages 8 years from now
The ratio of their ages in 8 years will be:
frac{40}{68} frac{10}{17}
Conclusion
The ratio of their ages 8 years from now is 10:17.
Additional Examples and Solutions
Example 1
Consider the ratio of present ages of A and B as 5x : 9x.
8 years ago, the ages were in the ratio 1/2.
Equate the given ratio and solve for x:
frac{5x-8}{9x-8} frac{1}{2}
By cross-multiplying and solving, we get:
x 8
Substitute x 8 to find the present ages of A and B:
A's present age: 5(8) 40 years
B's present age: 9(8) 72 years
After 8 years, the ages of A and B will be:
A's age: 40 8 48 years
B's age: 72 8 80 years
The future ratio: frac{48}{80} frac{3}{5}
Example 2
Let the present age of A and B be 5x and 9x respectively.
8 years ago, the ratio of their ages is 1/2.
Set up the equation and solve for x:
frac{5x-8}{9x-8} frac{1}{2}
By cross-multiplying, we get:
5x - 8 frac{9x - 8}{2}
1 - 16 9x - 8
x 8
Substitute x 8 to find the future ages:
A's age: 5(8) 8 48 years
B's age: 9(8) 8 80 years
The future ratio: frac{48}{80} frac{3}{5}
Example 3
Given the present ages of A and B as 8x and 15x, the ratio of their ages 8 years ago was 13/6.
Equate the given ratio and solve for x:
frac{15x - 8}{8x - 8} frac{13}{6}
By solving the equation, we get:
x 4
Substitute x 4 to find the future ratio:
A's age 8 years hence: 8(4) 8 40 years
B's age 8 years hence: 15(4) 8 68 years
The future ratio: frac{40}{68} frac{10}{17}
Conclusion
Solving problems involving the ratio of ages in the past, present, and future requires a careful setup and application of algebraic principles. Understanding these steps will help in tackling a variety of similar problems and enable you to apply the same techniques to real-life scenarios involving age ratios.