Problem 1 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 4 goals in 10 attempts

Step-by-Step Solution

Answer

The simplest form of the ratio is \(\frac{2}{5}\).

Step 1: Understanding Ratios

A ratio is a way to compare two quantities. In this exercise, the ratio given is '4 goals in 10 attempts.' This can be expressed as the fraction \(\frac{4}{10}\), with the numerator representing goals and the denominator representing attempts.

Step 2: Reducing Fractions to Simplest Form

To write the fraction \(\frac{4}{10}\) in its simplest form, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 4 and 10 is 2.

Step 3: Simplifying the Fraction

Divide both the numerator and the denominator by their GCD. \(\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}\). Thus, the fraction \(\frac{4}{10}\) simplifies to \(\frac{2}{5}\).

Key Concepts

Understanding Ratios

Ratios are a fundamental concept in mathematics used to compare two quantities. Imagine you're observing a sports game and note down that a player scores 4 goals in 10 attempts. This is an example of a ratio, which is represented as the relationship between the number of goals and the number of attempts made.
To express this in mathematical terms, we write it as a fraction: \( \frac{4}{10} \). Here, the numerator (the top number) is 4, indicating the number of goals, and the denominator (the bottom number) is 10, indicating the number of attempts.
Ratios help us understand proportions and relationships involving changes or comparisons between different quantities. It is a concise way to display how one quantity is relative to another.

Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is a key mathematical concept used to simplify fractions. It is the largest positive integer that divides two or more integers without leaving a remainder.
To find the GCD of two numbers, list the factors of each number and determine the largest number that appears in both lists.

• For example, the factors of 4 are 1, 2, and 4.
• The factors of 10 are 1, 2, 5, and 10.

The common factors between these numbers are 1 and 2, but the greatest common factor is 2.
Using the GCD is an important step in fraction reduction, as it allows us to effectively simplify fractions to their simplest form, making them easier to interpret and understand.

Fraction Reduction

Fraction reduction is also known as simplifying a fraction, and it involves using the greatest common divisor to transform a fraction into its simplest form. This makes comparisons and calculations simpler and more intuitive.
Consider the fraction \( \frac{4}{10} \) from our earlier example. As previously determined, the GCD of 4 and 10 is 2. To reduce the fraction:

• Divide the numerator (4) by the GCD (2) to get 2.
• Divide the denominator (10) by the GCD (2) to get 5.

This results in the simplified fraction \( \frac{2}{5} \).
The process of fraction reduction does not alter the value or proportion that the original fraction represents, but it presents the same ratio in a cleaner, reduced form. This is extremely helpful in mathematical calculations and real-life applications, making it a vital skill to master.