Problem 71 - Prealgebra

Question

If there are 3 girls in a family with 5 children, then we say that \(\frac{3}{5}\) of the children are girls. If there are 4 girls in a family with 5 children, what fraction of the children are girls?

Step-by-Step Solution

Answer

\(\frac{4}{5}\) of the children are girls.

Step 1: Identify Total and Part

First, identify the total number of children, which is 5, as well as the number of girls, which is 4.

Step 2: Create the Fraction

To find the fraction of children who are girls, use the number of girls as the numerator and the total number of children as the denominator.

Step 3: Setup the Fraction

Substitute the values into the fraction: \[\frac{4}{5}\]

Step 4: Simplify if Necessary

Since 4 and 5 have no common factors other than 1, the fraction \(\frac{4}{5}\) is already in its simplest form and does not need to be simplified further.

Key Concepts

Numerator

The numerator is the top part of a fraction, and it represents a specific part or number that you are interested in. When articulating fractions, the focus often centers on the numerator because it tells how many parts of the whole are being considered. For instance, in the fraction \(\frac{4}{5}\), the numerator is 4, which indicates that 4 parts of a total of 5 are being focused on. This particular setup is crucial when trying to answer questions like "What fraction of the children in this scenario are girls?" because it directly reflects the count of girls in relation to the full group. It's like having a cake cut into several pieces: the numerator shows how many pieces you have, while the whole cake represents the total.

Denominator

The denominator, on the other hand, forms the lower part of a fraction and denotes the total number of equal parts that make up a whole. It provides the context for the numerator, allowing us to understand what is being measured or considered. In the fraction \(\frac{4}{5}\), the number 5 is the denominator. It tells us that there are 5 total parts or units (in this case, children in the family) which the numerator's parts belong to. Without the denominator, it would be impossible to grasp the proportion each part represents of the total. Thus, the denominator plays a crucial role in shaping our understanding of the relationship between the part and the whole. When learning about fractions, always remember the denominator is the foundation for knowing how many total units you’re working with.

Simplest Form

Simplifying fractions involves reducing them to their simplest form. This means ensuring that the numerator and denominator have no common factors other than 1. When a fraction is already simplified, you can be confident it is as reduced as possible, making it easier to understand at a glance. For example, in the fraction \(\frac{4}{5}\), since the numbers 4 and 5 have no common divisors other than 1, the fraction is already in its simplest form. Simplifying helps in comparing fractions and in performing operations like addition or subtraction. It's a critical step in problem-solving because simplified fractions provide clarity and are most efficiently used in further calculations and mathematical discussions. Always check if and how you can simplify a fraction to make your work more straightforward.