Problem 18 - 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.) 3 inches to 2 feet

Step-by-Step Solution

Answer

The ratio of 3 inches to 2 feet simplifies to 1:8 when both measurements are in the same units (inches).

Step 1: Convert feet to inches

Firstly, convert the measurement in feet to inches. To do this, remember that there are 12 inches in each foot. Therefore, 2 feet equals 2*12 = 24 inches.

Step 2: Get the Ratio

Now that both quantities are in the same unit (inches), we can directly compare them. So, the given ratio of 3 to 24 can be simplified by dividing both numbers by their greatest common divisor, which is 3. This will get the simplest form of ratio. Thus, the ratio 3:24 simplifies to 1:8.

Key Concepts

Unit Conversion

Unit conversion is a process that allows you to express a quantity in different units. When comparing two quantities in a ratio, it is crucial to first ensure they are measured in the same units. This way we can directly compare them without any mismatches. For instance, converting feet to inches is necessary when the given quantities use different units.

• In the original exercise, one quantity was measured in inches and the other in feet.
• Remember that 1 foot equals 12 inches.
• To convert feet into inches, multiply the number of feet by 12.

In the given problem, converting 2 feet to inches means calculating 2 multiplied by 12, which results in 24 inches. This conversion step makes it possible to compare 3 inches directly to 24 inches, establishing a foundational aspect of the ratio problem.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is a key step in simplifying ratios. The GCD is the largest number that can divide two numbers without leaving a remainder. To simplify a ratio, you divide both parts of the ratio by their GCD. This effectively reduces the ratio to its simplest form.

• First, take the numbers in your ratio—in our example, 3 and 24.
• Identify the greatest number that divides both 3 and 24 evenly.
• In this case, 3 is the largest number that can divide both numbers evenly.

By dividing both 3 and 24 by the GCD, 3, we simplify the ratio to 1:8. This reduction makes the ratio easier to understand and interpret, ensuring a more straightforward comparison between the quantities.

Simplest Form of Ratio

A ratio is simplified when it uses the smallest whole number values possible for comparison. Simplifying a ratio involves using the greatest common divisor, but the goal is the end result: clarity and ease of understanding. In simpler terms, it answers how many times the first number contains the second number, using the smallest possible numbers.

• After converting units and finding the GCD, ensure the numbers in the ratio have no common factors other than 1.
• The simplest form is achieved when this condition is met.
• For example, the ratio 3:24 can be simplified to 1:8, which is clearer and more concise.

The simplest form of a ratio is essential for making data more comprehensible, especially when comparing two different sizes or amounts. It gives a direct, uncomplicated view of their relationship.