Problem 28 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 331.5 pages in 8.5 weeks

Step-by-Step Solution

Answer

The simplified ratio is 39.

Step 1: Understand the Ratio

The ratio is given as 331.5 pages in 8.5 weeks. We need to express this as a fraction where the number of pages is the numerator and the number of weeks is the denominator.

Step 2: Set Up the Fraction

Write the ratio as a fraction with the numerator being the pages and the denominator being the weeks: \( \frac{331.5}{8.5} \).

Step 3: Simplify the Fraction

To simplify the fraction, divide both the numerator and the denominator by the greatest common divisor. In this case, divide both by 8.5: \( \frac{331.5 \div 8.5}{8.5 \div 8.5} \).

Step 4: Perform the Division

Perform the division for both the numerator and the denominator: \( 331.5 \div 8.5 = 39 \) and \( 8.5 \div 8.5 = 1 \). So the simplified fraction becomes \( \frac{39}{1} \).

Step 5: Finalize the Simplest Form

Since the denominator is 1, the fraction simplifies to just 39. Therefore, the simplest form of the ratio is 39.

Key Concepts

Understanding Fractions

A ratio can be expressed as a fraction, which is a way to describe a part of something relative to the whole. In this context, we have the task of expressing a ratio—331.5 pages in 8.5 weeks—as a fraction. Understanding fractions is crucial, as they are used to represent divisions between quantities. A fraction consists of a numerator, which is the top part, and a denominator, which is the bottom part. In the original exercise, the number of pages (331.5) is the numerator, and the number of weeks (8.5) is the denominator. This formation allows us to communicate how many pages are read per week.

• The numerator indicates how many parts we have.
• The denominator tells us into how many parts the whole is divided.
• Expressing ratios as fractions helps simplify comparisons.

If you master fractions, you will find it easier to solve problems that involve splitting or grouping things into portions.

Simplifying Fractions for Clarity

Simplifying fractions means reducing them to their simplest form where both the numerator and the denominator have no common factors other than 1. This makes the fraction easier to understand and work with. In many mathematical situations, simplified fractions can provide a clearer picture of the relationship between numbers. To simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• If the fraction reduces to a whole number, ensure it is shown as such.

For the exercise at hand, dividing both 331.5 and 8.5 by the common divisor 8.5 yields 39/1. Since the denominator becomes 1, the fraction is equivalent to the whole number 39. Simplifying fractions not only helps mathematically but also enhances comprehension when dealing with ratios in daily life.

The Role of Mathematical Division

Mathematical division is a foundational operation that involves splitting a number into defined parts. When expressing ratios as simplified fractions, division plays a key role in simplifying the comparison between two quantities. For instance, when given the fraction \(\frac{331.5}{8.5}\), division is used to find how many times the denominator can "fit" into the numerator, breaking down the complex fraction into a simpler form.Steps involved in performing division in the context of simplifying fractions are:

• Identify the numbers to divide (in this case, 331.5 divided by 8.5).
• Perform the division operation to determine the simplified form.
• Maintain accuracy to ensure that the correct results are achieved.

Using division correctly allows one to reduce fractions quickly and accurately, aiding in better interpretation of data through simplified numbers. Understanding and applying division not only helps in school work but is also beneficial in everyday problem-solving scenarios.