Problem 17 - Elementary and Intermediate Algebra

Question

Translate each ratio into a fraction in simplest form. 4 boxes to 15 boxes

Step-by-Step Solution

Answer

The fraction in simplest form is \(\frac{4}{15}\).

Step 1: Understand the Given Ratio

The ratio given is 4 boxes to 15 boxes, which compares two quantities. In this case, the ratio 4:15 means 4 boxes for every 15 boxes.

Step 2: Convert the Ratio to a Fraction

Ratios can be expressed as fractions. To convert the ratio 4:15 into a fraction, write the first number (4) as the numerator and the second number (15) as the denominator, forming the fraction \(\frac{4}{15}\).

Step 3: Simplify the Fraction

To simplify \(\frac{4}{15}\), check for any common factors between 4 and 15 other than 1. Since 4 and 15 have no common factors, \(\frac{4}{15}\) is already in its simplest form.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole and are expressed with two numbers: the numerator and the denominator.
In the context of ratios, these two parts can also help illustrate a comparison. The numerator is the top part of the fraction and indicates the number of parts you have.
Meanwhile, the denominator is the number at the bottom, showing the total number of equal parts into which the whole is divided.

• For example, with the fraction \(\frac{4}{15}\), "4" is the numerator and "15" is the denominator.
• This fraction indicates four parts out of a total of fifteen.

Ratios, which are often used in fractions, indicate how many times one number contains another. This can be incredibly helpful when trying to compare differences or similarities between two quantities. Understanding the link between fractions and ratios allows students to grasp how different quantities relate to each other in a manageable way.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by this factor.

• In this exercise, with the fraction \(\frac{4}{15}\), you have 4 as the numerator and 15 as the denominator.
• To simplify, look for any common factors between the two numbers.
• For 4 and 15, the only common factor is 1, which means the fraction is already in its simplest form.

Simplification is essential because it makes fractions easier to understand and use in calculations.
Although 4 and 15 have no common factors other than 1, checking for common factors when faced with larger numbers can help in finding the simplest form.

Mathematical Comparisons

Mathematical comparisons involve evaluating the size or quantity of two or more numbers or items. Ratios and fractions are often used for these comparisons.
Using ratios, such as "4 boxes to 15 boxes," lets us group and compare quantities. This ratio can be transformed into the fractional form \(\frac{4}{15}\). Understanding whether things are similar, larger, or smaller is critical for making informed decisions and evaluations in math. To effectively compare:

• Convert ratios to fractions as necessary for direct comparisons.
• Simplify fractions to quickly see greater or lesser quantities.

Fractions represent the relationship between two numbers clearly, enabling more straightforward, precise mathematical reasoning and decision-making.