Problem 64 - Elementary Algebra Within Reach

Question

Find a ratio that compares the relative sizes of the quantities. (Use the same units of measurement for both quantities.) 4 days to 30 hours

Step-by-Step Solution

Answer

The ratio that compares the relative sizes of the quantities 4 days and 30 hours is 16:5.

Step 1: Convert Days to Hours

Firstly, convert days into hours, knowing that one day contains 24 hours. Therefore, 4 days are equivalent to \( 4 \times 24 = 96 \) hours.

Step 2: Find the Ratio

Next, write the ratio comparing the number of hours. Here we have 96 hours (from the converted 4 days) to 30 hours. The ratio then becomes \( 96:30 \).

Step 3: Simplify the Ratio

Finally, simplify the ratio by dividing both quantities by their greatest common divisor (GCD). The GCD of 96 and 30 is 6. After dividing both 96 and 30 by 6, the ratio is simplified to \( 16:5 \).

Key Concepts

Convert Days to Hours

Understanding how to convert from one unit of time to another is indispensable in mathematics, especially when comparing quantities with different units. To compare quantities with ratios, we must first express them in the same unit.

One day is made up of 24 hours. So when converting days to hours, multiply the number of days by 24. For instance, to convert 4 days into hours, calculate as follows: \[\begin{equation} 4 \text{ days} \times 24 \frac{\text{hours}}{\text{day}} = 96 \text{ hours}. \end{equation}\] This conversion allows you to align the units of measure, thereby making your calculations more accurate and your comparisons consistent.

Simplify Ratios

Ratios are used to compare two or more quantities, and simplifying them helps to understand the comparison easily. Simplifying a ratio involves dividing both terms of the ratio by a common number until they cannot be divided any further by the same number except one. This process is akin to simplifying fractions.

To apply this to our initial example where we have a ratio of 96 hours to 30 hours, or \[\begin{equation} 96:30, \end{equation}\] we need to find a common divisor, which in many cases is the greatest common divisor to make the simplification process faster. When simplified, our ratio effectively represents the same relationship in its most reduced form.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that divides two or more integers without leaving a remainder. Finding the GCD is a crucial step in simplifying ratios because it lets you reduce the numbers to their smallest equivalent ratio.

To find the GCD of two numbers, like 96 and 30, you can list out the factors of each number or use prime factorization. Then, identify the largest factor that appears in both lists. However, a more effective method for larger numbers is the Euclidean algorithm, which involves repeated division. In our example, the GCD is 6, so you divide both numbers by 6 to simplify the ratio to \[\begin{equation} 16:5. \end{equation}\] Familiarity with the concept of GCD not only assists in simplifying ratios but also in solving problems involving fractions and integer properties.