Understanding Growth Rates: A Mathematical Puzzle and Its Solution

Today, we dive into a fascinating problem related to the growth rates of children. Let's explore a puzzle involving the growth of two individuals over a period of ten years. Through this exploration, we will not only solve the problem but also delve into the broader concept of growth rates.

The Problem Statement

Let’s consider two individuals: a boy and a girl. At the start, the girl is 10 units taller than the boy. In ten years, the boy will have grown 50 units taller, while the girl will have grown 10 units shorter. We are asked to determine how much faster the boy will be growing compared to the girl over this ten-year period.

Solving the Puzzle

Let's begin with the initial conditions:

tLet the initial height of the boy be X units. tThe girl's initial height is then 1.1X units.

After ten years:

tThe boy's height will be such that he is 50 units taller, making his height 1.5X units. tThe girl's height will be 10 units less than the boy's, making it 1.5X - 10 units. Since we are comparing her initial height, we set her new height to be 1.35 times her initial height, giving us 1.35X units.

Now, let’s calculate the growth rate for both:

Boy's Rate of Growth:

t( text{Boy's initial height} X ) t( text{Boy's height after 10 years} 1.5X ) t( text{Boy's growth over 10 years} 1.5X - X 0.5X ) t( text{Boy's annual growth} frac{0.5X}{10} frac{X}{20} )

Girl's Rate of Growth:

t( text{Girl's initial height} 1.1X ) t( text{Girl's height after 10 years} 1.35X ) t( text{Girl's growth over 10 years} 1.35X - 1.1X 0.25X ) t( text{Girl's annual growth} frac{0.25X}{10} frac{X}{40} )

Therefore, the boy grows at a rate of 2.2 times faster than the girl over the ten-year period:

t( frac{text{Boy's annual growth}}{text{Girl's annual growth}} frac{frac{X}{20}}{frac{X}{40}} 2.2 )

Conclusion

Through this problem, we learned that the boy grows 2.2 times faster than the girl over a span of ten years. This puzzle highlights the importance of understanding growth rates and provides insightful practice for those who enjoy mathematical challenges.

For those interested in honing their skills further, explore similar puzzles in our Quora Space. Engage with a community of learners, join discussions, and share your insights. Delve into the beauty and complexity of mathematical problems, and enjoy the journey of discovery!