Problem 10 - Elementary and Intermediate Algebra

Question

b. Fill in the blank: \(\frac{1 \text { day }}{24 \text { hours }}=\)

Step-by-Step Solution

Answer

1

Step 1: Understanding the Units

We need to break down the fraction to understand what each part represents. Here, the numerator represents 1 day and the denominator represents 24 hours.

Step 2: Establish the Relationship

Recognize that 1 day is equivalent to 24 hours. This is a fundamental time conversion that we need to keep in mind.

Step 3: Reformulating the Fraction

By the definition of 1 day being equal to 24 hours, we express the problem as a fraction: \(\frac{1 \text{ day}}{24 \text{ hours}}\).

Step 4: Simplify the Fraction

Since both the day and hour refer to units of time and 1 day equals 24 hours, the fraction simplifies to 1. Hence, \(\frac{1 \text{ day}}{24 \text{ hours}} = \frac{24 \text{ hours}}{24 \text{ hours}} = 1\).

Step 5: Conclusion

Finally, we can fill in the blank with the simplified fraction which evaluates to 1.

Key Concepts

Fractions

Fractions are a way of representing parts of a whole. In mathematics, a fraction consists of a numerator, which is the top number, and a denominator, the bottom number. Fractions can represent quantities that are less than a whole, equal to a whole, or greater than a whole.
Understanding fractions is crucial in various mathematical operations and real-life applications. They are used in situations ranging from splitting groups into parts to calculating ratios and proportions.
Some key points about fractions:

• The numerator defines how many parts we are focusing on.
• The denominator tells us into how many parts the whole is divided.
• If the numerator is smaller than the denominator, the fraction is "proper" indicating less than a whole.
• When the numerator is equal to or greater than the denominator, the fraction is "improper" indicating a whole or more.

In our exercise, the fraction \(\frac{1\,\text{day}}{24\,\text{hours}}\) shows the relation between days and hours, emphasizing that 1 day consists of 24 equal parts of hours.

Time Conversion

Time conversion involves changing the units used to measure time from one unit to another, like hours to minutes or days to hours. This is a helpful skill for solving problems in real life and various academic exercises where time is expressed in different units.
For example:

• 1 minute = 60 seconds
• 1 hour = 60 minutes
• 1 day = 24 hours
• 1 week = 7 days

In the given exercise, time conversion is crucial because we need to convert the unit of a day into hours to create our fraction. We established that 1 day equals 24 hours. Knowing these fundamental conversions allows you to perform operations such as addition, subtraction, and simplification between different time-based units smoothly.
Always double-check your conversions to avoid errors, especially in multi-step problems.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their lowest terms. This means rewriting the fraction so that the numerator and denominator have no common factors other than 1. Simplifying fractions makes them easier to interpret and use in equations.
Here's how you can simplify fractions:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and denominator by the GCD.
• The fraction is in its simplest form when the greatest factor that divides both is 1.

In our exercise, the fraction \(\frac{1\,\text{day}}{24\,\text{hours}}\) simplifies easily because both the units represent time and are undoubtedly equivalent: 1 day is exactly 24 hours. Thus, this fraction simplifies to 1, since both the numerator and the denominator effectively convey the same measure.
Always remember, simplifying helps us redefine fractions to their most understandable and straightforward form, which is especially useful when comparing or adding fractions.