Problem 14 - Elementary and Intermediate Algebra

Question

Write the ratio of 25 to 4 in two other forms.

Step-by-Step Solution

Answer

The ratio of 25 to 4 can be expressed as \( \frac{25}{4} \) or 25:4.

Step 1: Understanding the Problem

We are given a ratio (25 to 4) and need to express it in two different forms. A ratio compares two quantities and can be written in various ways.

Step 1: Expressing the Ratio as a Fraction

One common way to express a ratio is as a fraction. The ratio of 25 to 4 can be written as the fraction \( \frac{25}{4} \). Here, 25 is the numerator and 4 is the denominator.

Step 2: Expressing the Ratio with a Colon

Another way to express the ratio is using a colon. The ratio of 25 to 4 can be written as 25:4. This is a common format used to directly convey the relationship between the two numbers.

Key Concepts

Fraction Representation

Fractions are one of the most common and intuitive ways to express ratios. In mathematical terms, when we have two numbers, the ratio of these can be written as a fraction. For instance, in the problem "Write the ratio of 25 to 4 in two other forms," the fraction form would be \( \frac{25}{4} \). Here, 25 is known as the numerator, and 4 is the denominator.

• The numerator represents how many parts we have.
• The denominator shows how many parts are making up a whole.

Fractions provide a clear and visual way to understand how one quantity relates to another. They are especially useful in simplifying the comparison between two quantities. For example, when dealing with recipes or when trying to find out the scale of a drawing.

Colon Notation

Colon notation is another straightforward way to express ratios. It uses a colon (:) to separate the two numbers. For example, the ratio of 25 to 4 can be expressed as 25:4.

• This notation presents a direct and clear relationship between the two numbers.
• Colon notation is often used in settings like maps and models where you'd easily compare dimensions or sizes.

This form is simple to read and is widely used in everyday language. Unlike fractions, colon notation does not imply any division operation, but purely the relationship in terms of comparison between two quantities.

Mathematical Expressions

Mathematical expressions go beyond the standard numbers and symbols used in ratios. An expression is a group of numbers, variables, and operators (like +, -, *, /) combined to represent a value or relationship. Ratios as mathematical expressions can be versatile.

• They can be incorporated into larger equations to find unknowns.
• Expressions help model and solve real-world problems involving proportional reasoning.

Consider the ratio 25:4. In algebra, this can be written as \( 25x:4x \). Here, 'x' represents a variable, meaning the ratio holds for any value of 'x'. This demonstrates ratios' power within algebraic operations and systems, expanding basic comparison to solve complex mathematical problems.