91Ó°ÊÓ

Fractions are a handy way to represent quantities smaller than a whole number, using two parts: the numerator (top part) and the denominator (bottom part). In this exercise, when Abigail cuts off a piece of string that is one-tenth the length of the entire mile, it can be written as a fraction \( \frac{1}{10} \). This fraction tells us she takes one part out of ten equal parts of the mile.
Understanding fractions helps in determining proportions and parts of a whole, which are crucial in everyday calculations. During the exercise, each subsequent piece of string cut follows the rule of dividing the previous string's length by ten. Thus, Barbara's cut is \( \frac{0.1}{10} = \frac{1}{100} \), and so on. This quick division with fractions and understanding how they can represent parts smaller than a whole aids in scientific notation conversions and problem-solving scenarios.

Measurement conversion allows one to translate various units into other units for consistent calculations. In this exercise, we start with the length in miles and need to convert it to feet to assess suitability for shoelace length. Knowing that 1 mile equals 5280 feet becomes critical here.
For instance, Abigail's piece of 0.1 miles translates to 528 feet by multiplying: \( 0.1 \times 5280 = 528 \text{ feet} \). This conversion becomes vital when comparing lengths in practical, real-world contexts.

• Barbara's conversion: \( 0.01 \times 5280 = 52.8 \text{ feet} \)
• Cruz's conversion: \( 0.001 \times 5280 = 5.28 \text{ feet} \)
• Damien's conversion: \( 0.0001 \times 5280 = 0.528 \text{ feet} \)

These conversions clearly demonstrate how different small fractions of a mile translate into actual physical lengths in feet, which is much easier to visualize and understand for practical assessments such as shoelace length.

Problem-solving often involves breaking a complex problem into smaller manageable steps, just like in the string exercise. Each question requires a logical step of deduction and application of previous results.
To answer part (b), observe the consistent pattern in the exercise. Each person cuts a length that is one-tenth of the previous cut. This pattern allows for direct prediction without calculation at each step. For Damien's cut, he cuts \( 0.0001 \) miles, continuing the trend.
In part (c), problem-solving involves measuring the adequacy of each piece. Here, converting miles to a more practical measurement such as feet helps assess feasibility—understanding that a shoelace needs to be at least 1.5 feet highlights practical applications of this exercise.

• Check if length exceeds 1.5 feet.
• Convert using the mile-to-feet conversion.
• Evaluate and conclude adequacy against shoelace length criteria.

Such exercises reinforce learning by applying conceptual knowledge to everyday situations, enhancing comprehension and practical skills.