Problem 13 - 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.) 42 inches to 21 inches

Step-by-Step Solution

Answer

The ratio that compares the relative sizes of 42 inches to 21 inches is 2:1.

Step 1: Understand the Quantities to be Compared

The quantities to be compared are 42 inches and 21 inches. These are the same units, which is important for a valid comparison.

Step 2: Formulate the Ratio

The ratio is formulated by dividing the larger quantity by the smaller quantity. In this case, it is \(\frac{42}{21}\).

Step 3: Simplify the Ratio

When the calculation \(\frac{42}{21}\) is performed, it results in 2. This is the simplest form of the ratio and it means that 42 inches is twice as long as 21 inches.

Key Concepts

Simplifying Ratios

When we work with ratios, we're often comparing two or more quantities to find out how they relate to one another. Simplifying ratios allows us to express this relationship in the simplest form, which makes it easier to understand and work with. To simplify a ratio, we divide both terms by their greatest common divisor (GCD), until we can't divide them evenly anymore.

For example, the ratio given in the exercise, 42 inches to 21 inches, when divided by the GCD which is 21, results in 2 to 1. This means for every 2 units of the first quantity, there is 1 unit of the second quantity. Remember, ratios should always be expressed in whole numbers, and they must be as small as those numbers can possibly be while still maintaining the same proportion between quantities.

Units of Measurement

Units of measurement are crucial in understanding the quantities being compared in a ratio. The comparison is meaningful only when both quantities are in the same unit. If the units were different, we'd first need to convert them to the same unit before forming the ratio.

As seen in our exercise, the units were already the same: inches. This consistency allows us to create a ratio without the need for conversion. However, if we had 42 inches and 1 yard (which is equal to 36 inches), we'd convert yards to inches or vice versa to get a valid comparison.

Ratio and Proportion

A ratio is a way to show the relative sizes of two quantities, while a proportion states that two ratios are equal. A proportion can be used to solve for an unknown quantity if the three other numbers in the proportional relationship are known.

To find if two ratios form a proportion, you can cross-multiply and see if the products are equal. However, for a single ratio such as the one in the exercise, 42 inches to 21 inches, we are simply comparing two quantities, not setting up an equation to solve for an unknown. The concept here is fundamental to understanding how ratios represent the 'proportionality' of one quantity to another.

Elementary Algebra

Understanding ratios often leads us into the realm of elementary algebra, especially when we deal with unknown quantities or need to manipulate equations to solve for a variable. For simplifying ratios, we might use algebraic techniques such as finding the GCD or converting units.

In advanced problems, algebra can also allow us to work with ratios that have variables. For instance, if we had a ratio like \(x\) inches to 21 inches, where \(x\) is an unknown length, algebra could help us solve for \(x\) when additional information is provided. Elementary algebra provides the foundational skills necessary to manipulate and solve these kinds of ratio problems.