Problem 29 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$100 \mathrm{mg} \text { to } 5 \mathrm{mL}$$

Step-by-Step Solution

Answer

The simplified ratio is \( \frac{20}{1} \).

Step 1: Understand the Units and Terms

Start by identifying the numerical values in the given ratio, which are 100 mg and 5 mL. These represent mass (milligrams) and volume (milliliters), respectively.

Step 2: Set Up the Ratio as a Fraction

Write the given ratio of 100 mg to 5 mL as a fraction. This can be expressed as \( \frac{100}{5} \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{100}{5} \) by finding the greatest common divisor (GCD) of 100 and 5. The GCD is 5, so divide both the numerator and the denominator by 5: \( \frac{100}{5} = \frac{20}{1} \).

Step 4: Express the Simplified Fraction

The simplified form of the fraction \( \frac{100}{5} \) is \( \frac{20}{1} \). This is the simplest form of the fraction, as the numerator and the denominator have no common factors other than 1.

Key Concepts

Fractions

Fractions are a way to represent a part of a whole or to express a ratio between two numbers. They consist of a numerator and a denominator. The numerator is the number on top, representing the number of parts or units, while the denominator is at the bottom, indicating the total number of equal parts. For instance, in a fraction like \( \frac{3}{4} \), the numerator is 3, and the denominator is 4. This means 3 out of 4 equal parts.
Understanding fractions is essential when dealing with ratios. A ratio like 100 mg to 5 mL can be represented as a fraction; in this case, \( \frac{100}{5} \). This shows a direct comparison of the two quantities in a mathematical form, which can then be simplified to make it easier to interpret or compare with other ratios.
To simplify fractions, find a number that divides both the numerator and the denominator without leaving a remainder, thereby expressing the fraction in its simplest form.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide two or more numbers without leaving a remainder. It is an important concept when simplifying fractions because it allows us to reduce the fraction to its simplest form. Finding the GCD helps ensure that the numerator and denominator of a fraction share no common factors other than 1.
For the fraction \( \frac{100}{5} \), we need to determine the GCD of 100 and 5. Begin by listing the divisors of each number:

• Divisors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
• Divisors of 5: 1, 5

The greatest number that appears in both lists is 5. Thus, the GCD of 100 and 5 is 5. This means we can divide both the numerator and the denominator of \( \frac{100}{5} \) by 5, resulting in the simplified fraction \( \frac{20}{1} \).
Using the GCD effectively reduces fractions, making them simpler and easier to work with or compare.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another. It's a crucial skill when dealing with ratios involving different units, such as milligrams (mg) and milliliters (mL) as seen in the exercise.
While converting units might not always be necessary for simplifying a ratio, understanding how to do so is valuable, especially in scientific and mathematical contexts where comparisons of different quantities are common.
To handle unit conversions:

• Identify the units you need to convert.
• Use conversion factors, which are numbers or formulas used to convert from one set of units to another.
• Apply these factors to translate the original measurement into the desired unit.

For instance, converting between mass and volume requires understanding their relationship or, in some contexts, knowing the substance involved to use density as a conversion factor. In this exercise, we did not need to convert units since we focused on simplifying the given ratio without altering units, but mastering conversions can be key in complex problem-solving.