Problem 16 - 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.) 24 pounds to 30 pounds

Step-by-Step Solution

Answer

The ratio of 24 pounds to 30 pounds is 4:5.

Step 1: Understand the Problem

A ratio is a comparison between two numbers showing how many times the first number contains the second number. In this problem, the two quantities to be compared are 24 pounds and 30 pounds.

Step 2: Find the Ratio

To find the ratio, one just needs to write the two given numbers in the form 'first number : second number'. In this case, the ratio would be '24 : 30'.

Step 3: Simplify the Ratio

A ratio is usually expressed in the simplest form. Simplifying a ratio involves dividing both numbers by their greatest common divisor (GCD). The GCD of 24 and 30 is 6. So, when we divide both 24 and 30 by 6, we get 4 and 5 respectively. Thus, the simplified ratio is '4 : 5'.

Key Concepts

Simplifying Ratios

Simplifying ratios is an essential skill in mathematics that helps in making comparisons clearer and easier to understand. A ratio in simplest form is like a fraction reduced to its lowest terms. To simplify a ratio, the key step is to divide both terms by their greatest common divisor (GCD), which means you’re finding the largest number that can exist as a divisor for both terms in the ratio. For instance, in the ratio of 24:30, the largest number that can evenly divide both 24 and 30 is 6.

• Divide both numbers, 24 and 30, by 6.
• This gives you 4 and 5 respectively.
• So, the simplified ratio is 4:5.

By simplifying a ratio, you provide a cleaner and more instantly understandable representation of the relationship between the two numbers. It makes it easier to interpret the scale of comparison or relative size.

Greatest Common Divisor

The greatest common divisor (GCD) is an important concept when simplifying ratios. It is the largest number that can exactly divide the given numbers without leaving a remainder. Understanding and finding the GCD is crucial in various areas of mathematics beyond ratios, such as simplifying fractions and performing algebraic operations. To find the GCD of any two numbers:

• List all the factors of each number.
• Identify the largest factor that appears in both lists.

For the numbers 24 and 30:

• The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
• The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
• The largest number common to both lists is 6, making it the GCD of 24 and 30.

Using the GCD allows you to simplify ratios effectively by ensuring each term of the ratio is as small as possible while maintaining the same proportion.

Units of Measurement

Whenever you are asked to compare two quantities using a ratio, ensuring both are in the same units of measurement is vital. This is because a ratio acts as a direct comparison of size, count, or amount. Having different units would mean you’re comparing apples to oranges, which can lead to inaccurate or meaningless results. However:

• If initially the units differ, convert one set of units so both are the same. For example, convert feet to inches if one quantity is in feet and the other in inches before forming the ratio.
• Check the context to confirm the same units for both quantities; this step ensures the ratio represents an agreed comparison point.

In our example exercise, both quantities were already in pounds, ensuring that the ratio was based on a like-for-like comparison without additional conversion complications. By following this practice, ratios provide meaningful insights into the relationship between quantities.