Problem 32 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$450 \mathrm{mg} \text { to } 20 \mathrm{mL}$$

Step-by-Step Solution

Answer

The fraction in lowest terms is \( \frac{45}{2} \).

Step 1: Convert Ratio to Fraction

First, set up the given ratio as a fraction. The ratio is given in units of milligrams to milliliters, which is \[ \frac{450 \text{ mg}}{20 \text{ mL}} \]

Step 2: Simplify the Fraction

To simplify \( \frac{450}{20} \), we need to find the greatest common divisor (GCD) of 450 and 20. Divide both numbers by their GCD, which is 10. \[ \frac{450 \div 10}{20 \div 10} = \frac{45}{2} \]

Step 3: Verify if Simplified Fraction is in Lowest Terms

Check if \( \frac{45}{2} \) is in its simplest form. Since there are no common divisors between 45 and 2 other than 1, the fraction \( \frac{45}{2} \) is in its lowest terms.

Key Concepts

Ratios

Ratios are a way to compare two quantities, showing how many times one value contains or is contained within the other. It's like a relationship between two numbers. For example, if you have 450 milligrams of a substance and want to compare it to 20 milliliters of another, you create a ratio like so:

• Amount in milligrams: 450
• Amount in milliliters: 20

This can be expressed in the form of a fraction: \( \frac{450}{20} \). Ratios are extremely useful in real-life applications, from cooking to chemistry, helping us understand proportions and balance in various contexts.
When expressing a ratio as a fraction, you usually want to find a simpler form to make it easier to understand and use. This leads us to the next concept: simplifying fractions.

Simplifying Fractions

Simplifying a fraction means making it as simple as possible without changing its value. This involves dividing the numerator (the top number) and the denominator (the bottom number) by a common factor.

• For the fraction \( \frac{450}{20} \), we look for a number that can divide both 450 and 20 without leaving a remainder.
• This common number, known as a common divisor, helps reduce the fraction to its simplest form.

For example, dividing both the numerator and the denominator by 10 reduces the fraction to \( \frac{45}{2} \). When simplified, fractions are easier to interpret and use in further calculations. They become more intuitive, reducing complexity while retaining mathematical accuracy.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that can divide two or more numbers without leaving a remainder. Identifying the GCD is essential when simplifying fractions.

• To find the GCD of 450 and 20, list out the factors of each number.
• For 450, factors are 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, and 450.
• For 20, factors are 1, 2, 4, 5, 10, and 20.
• The largest common factor is 10. Therefore, the GCD of 450 and 20 is 10.

By dividing both the numerator and the denominator of \( \frac{450}{20} \) by their GCD, we simplify the fraction to \( \frac{45}{2} \). Understanding how to find and use the GCD is a crucial skill for anyone working with ratios and fractions regularly. It allows for the simplification of expressions, making calculations more manageable and solutions clearer.