Problem 31 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$375 \mathrm{mg} \text { to } 10 \mathrm{mL}$$

Step-by-Step Solution

Answer

The ratio 375 mg to 10 mL as a fraction in lowest terms is \( \frac{75}{2} \).

Step 1: Write the Ratio as a Fraction

First, express the given ratio as a fraction where the first quantity is the numerator and the second quantity is the denominator. So, the ratio of 375 mg to 10 mL can be written as \( \frac{375}{10} \).

Step 2: Simplify the Fraction

Next, simplify the fraction \( \frac{375}{10} \) by finding the greatest common divisor (GCD) of 375 and 10. The GCD of 375 and 10 is 5. Divide both the numerator and the denominator by their GCD: \( \frac{375 \div 5}{10 \div 5} = \frac{75}{2} \).

Step 3: Verify the Simplification

Ensure that \( \frac{75}{2} \) is in its lowest terms by checking that the numerator and denominator share no common divisors other than 1. Since 75 is divisible by factors such as 3, 5, and 15, and 2 is a prime number, the fraction \( \frac{75}{2} \) is indeed in its simplest form.

Key Concepts

Simplifying Fractions

When simplifying fractions, the goal is to reduce the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. This process makes fractions easier to work with and understand. In the original exercise, the fraction \( \frac{375}{10} \) needed to be simplified. The simplification involves the following steps:

• Identify the greatest common divisor (GCD) of both numbers. In this case, it was 5.
• Divide both the numerator (375) and the denominator (10) by the GCD, resulting in \( \frac{75}{2} \).

After simplification, double-check to ensure that no other common factors remain to verify the fraction is indeed in its simplest form. By simplifying fractions, not only do we make calculations easier, but we also enhance clarity when communicating ratios.

Greatest Common Divisor

The greatest common divisor (GCD) is a vital concept in mathematics that aids in simplifying fractions. It is the largest number that can evenly divide two or more numbers. The process involves finding the largest number that will divide both the numerator and the denominator with no remainder.

• For instance, the GCD of 375 and 10 is 5. This was found by looking at the factors of each number and identifying the highest common one:
• Factors of 375: 1, 3, 5, 15, 25, 75, 125, 375
• Factors of 10: 1, 2, 5, 10
• Common factor: 5

Knowing how to find the GCD allows us to simplify fractions effectively. This method ensures that we reduce fractions to their lowest terms, making them neat and manageable.

Writing Ratios as Fractions

Converting ratios to fractions is an essential skill that helps simplify and understand relationships between quantities. Instead of comparing numbers like 375 mg to 10 mL directly, writing them as a fraction \( \frac{375}{10} \) provides a clearer perspective.

• Ratios are expressed as fractions where the first number in the ratio becomes the numerator and the second becomes the denominator.
• In this way, students can visualize and calculate the proportion between two values more effectively.

The process also paves the way for simplification. Once in fraction form, ratios can be precisely simplified using the GCD, as done in this exercise, to become \( \frac{75}{2} \). Writing ratios as fractions is a foundational mathematical skill that facilitates better problem-solving and understanding.