Solving a Mathematical Puzzle: The String Length Enigma

Mathematics is a powerful tool that helps us solve complex puzzles and determine unknown values through logical reasoning and algebraic manipulation. The following mathematical puzzle involves determining the length of a string owned by Sarah, using relationships with strings owned by Daniel and Chris. This problem not only showcases the beauty of algebra but also highlights the practical application of mathematical concepts.

Introduction to the Puzzle

The problem at hand states: "1/2 of Sarah's string is 2/5 of Daniel's. Chris has 3 times as much as Sarah. The total of all three strings is 6 feet." The goal is to determine the length of Sarah's string in feet. This puzzle involves several algebraic relationships between the lengths of strings owned by three individuals. Let's break it down step by step.

Algebraic Representation

Let's denote the lengths of the strings as follows:

S be the length of Sarah's string. D be the length of Daniel's string. C be the length of Chris's string.

From the problem, we have the following relationships:

1/2 S 2/5 D C 3S S D C 6 feet

Step-by-Step Solution

Express D in terms of S.

From the first equation: 1/2 S 2/5 D. Multiplying both sides by 10 to eliminate the fractions:

5S 4D

Now, we can express D in terms of S:

D 5/4 S

Substitute D and C in the total equation.

Next, we substitute D and C in the total length equation:

S D C 6.

Substituting D and C:

S 5/4 S 3S 6

Combining like terms:

S 5/4 S 3S 4S 5/4 S 21/4 S

This simplifies to:

21/4 S 6

Solve for S.

Multiplying both sides by 4/21:

S 6 * (4/21) 24/21 8/7 feet

Therefore, the length of Sarah's string is:

boxed{8/7} feet, which is approximately 1.14 feet.

Verification

Let's verify the solution:

Calculate D:

Since D 5/4 S:

5/4 * 8/7 40/28 10/7 feet

Calculate C:

C 3S:

3 * 8/7 24/7 feet

Sum the lengths:

C S D 24/7 8/7 10/7 (24 8 10) / 7 42/7 6 feet

The total length matches the given condition, confirming the correctness of the solution.

Conclusion

Sarah's string length of 8/7 feet, or approximately 1.14 feet, Daniel's 10/7 feet, or approximately 1.43 feet, and Chris's 24/7 feet, or approximately 3.43 feet, satisfy all given conditions. This puzzle is a beautiful example of how algebra can be used to solve practical problems and develop analytical skills.