Solving Age Ratio Problems with Variables and Equations

In mathematics, understanding and solving age problems is a classic exercise that helps develop skills in algebra and equation solving. This article provides step-by-step solutions to a series of age ratio problems, using variables and equations. We will explore how to set up and solve these problems, using specific examples and detailed explanations.

Problem 1: The Ages of A and B in the Ratio 3:5

The ages of A and B are in the ratio 3:5. Four years later, the sum of their ages is 48. What are their present ages?

Set up the equation using variables:

Let the present ages of A and B be (a) and (b), respectively. Given the ratio (a : b 3 : 5), we can write:

[a - 4 : b - 4 3 : 5]

Converting this ratio into an equation:

[5(a - 4) 3(b - 4)]

Simplify and solve the equation:

[5a - 20 3b - 12]

Rearranging the equation to isolate terms involving (a) and (b):

[5a - 3b 8]

Use another condition to form a system of equations:

After ten years, their ages will be:

[a 10 : b 10 5 : 6]

Converting this ratio into an equation:

[6(a 10) 5(b 10)]

Rearranging the equation:

[6a 60 5b 50]

Simplify further:

[6a - 5b -10]

Form a system of linear equations:

From step 2, we have:

[-5a 3b 8]

From step 3, we have:

[6a - 5b -10]

Multiply the first equation by 6 and the second by 5:

[-30a 18b 48]

[30a - 25b -50]

Adding the two equations:

[-7b -2]

Solve for (b):

[b 264 / 6 44]

Solve for (a):

[5a - 3(44) 8]

[5a - 132 8]

[5a 140]

[a 28]

The present ages of A and B are 36 years and 44 years, respectively.

Problem 2: Another Approach to the Same Problem

Set up the initial ratio:

[a : b 3 : 5]

Let the common multiplier be (x), then:

[a 3x, b 5x]

Form the equation based on the sum of ages after four years:

[(a - 4) (b - 4) 48]

[3x - 4 5x - 4 48]

[8x - 8 48]

[8x 56]

[x 7]

Calculate the present ages:

[a 3x 3(7) 21, b 5x 5(7) 35]

The present ages of A and B are 21 years and 35 years, respectively.

Common Approach to Solve Age Ratio Problems

Define the variables:

Let the present ages of A and B be (x) and (y), respectively.

Use the given ratio to form an equation:

[x - 4 : y - 4 3 : 5]

Converting this ratio into an equation:

[5(x - 4) 3(y - 4)]

Simplifying:

[5x - 20 3y - 12]

[5x - 3y 8]

Use the next condition to form a second equation:

Ten years later:

[(x 10) : (y 10) 5 : 6]

Converting this ratio into an equation:

[6(x 10) 5(y 10)]

Simplifying:

[6x 60 5y 50]

[6x - 5y -10]

Solve the system of equations:

The system of equations is:

[5x - 3y 8]
[6x - 5y -10]

Using the method of elimination or substitution, solve for (x) and (y).

Conclusion

The ability to solve age ratio problems is crucial in developing algebraic thinking and equation solving skills. By using variables and formulating systems of equations, we can solve complex age-related problems methodically. This article has provided multiple approaches and solutions, ensuring a thorough understanding of the topic.