Exploring Age Ratios and Proportional Growth in Mathematics

This article delves into the fascinating world of age ratios and proportional growth, focusing on a specific mathematical problem involving Steve and Mark. We’ll walk through the problem step by step, ensuring a thorough understanding of the concepts involved. By the end, you’ll be able to solve similar problems with ease. Let’s begin with the given information and explore the solution in detail.

Initial Problem Context

Given that Steve is older than Mark by 4 years, and the current ratio of their ages is 7:9, we need to determine the new ratio of their ages when Mark is twice as old as he is now.

Step 1: Finding Current Ages

Let Mark’s current age be M, and Steve’s current age be S.

From the problem, we know:

Steve is 4 years older than Mark: S M 4 The ratio of their current ages is 7:9: S/M 7/9

Substituting S M 4 into the ratio equation:

(M 4)/M 7/9

Multiplying both sides by 9M:

9(M 4) 7M

9M 36 7M

36 -2M

M -36 / -2 18

Since age cannot be negative, let’s re-evaluate with correct algebraic manipulation:

9(M 4) 7M

9M 36 7M

36 -2M 7M

36 2M

M 18

Therefore, Mark’s current age is 18, and Steve’s current age is 18 4 22.

Step 2: Future Ages when Mark is Twice as Old

When Mark is twice as old as he is now, he will be:

2 * 18 36

Let’s denote Steve’s future age as S'. Since Steve is always 4 years older than Mark:

S' 36 4 40

Now, we need to find the new ratio S' / 36:

S'/36 40/36

This simplifies to:

10/9

Thus, the new ratio when Mark is twice as old as he is now is 10:9.

Alternative Solutions and Verifications

Let’s explore the alternative solutions given in the problem statement and verify them step by step.

Alternative 1: Ratio 7:9 as 21:27

Using a different approach, we can multiply the current ratio 7:9 by 3 (since 2*3 6, which is the difference in their ages):

Mark: 7 * 3 21 Steve: 9 * 3 27

When Mark is twice as old as he is now (42):

Steve will be 42 6 (since the difference remains 6): 42 6 48

Thus, the new ratio is:

42:48

Which simplifies to:

7:8

Alternative 2: Mark: 7x, Steve: 9x

Let Mark’s current age be 7x, and Steve’s current age be 9x.

From the problem, 9x - 7x 6, so x 3 Mark’s current age: 7 * 3 21 Steve’s current age: 9 * 3 27

When Mark is 42:

Steve will be 42 6, so Steve will be 48

Thus, the new ratio is:

42:48

Which simplifies to:

7:8

Conclusion

Through careful analysis and multiple approaches, we have determined that the new ratio of Steve and Mark’s ages when Mark is twice as old as he is now is 7:8. Understanding these steps and alternative solutions will help you solve similar age and ratio problems more effectively.

Remember, age ratios and proportional growth problems require a clear understanding of the given information and careful manipulation of equations. Practice these skills regularly to enhance your problem-solving abilities in mathematics.