Problem 19 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 11 gallons to 11 quarts

Step-by-Step Solution

Answer

The ratio is \(\frac{4}{1}\) in simplest form.

Step 1: Understand the Units

In this exercise, we need to express the ratio of gallons to quarts as a fraction. We know that one gallon is equal to 4 quarts.

Step 2: Convert Gallons to Quarts

Since 1 gallon equals 4 quarts, 11 gallons will be \(11 \times 4 = 44\) quarts.

Step 3: Write the Ratio as a Fraction

The ratio of 11 gallons to 11 quarts can now be written in terms of quarts: \(\frac{44}{11}\).

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{44}{11}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 11. This gives \(\frac{44 \div 11}{11 \div 11} = \frac{4}{1}\).

Key Concepts

Fractions

Fractions are a way to express a part of a whole. They consist of two parts: the numerator, which represents how many parts you have; and the denominator, which indicates how many equal parts the whole is divided into. For instance, the fraction \( \frac{4}{1} \) tells us that we have 4 parts, with each part being the whole itself. When dealing with fractions, it's essential to understand these components to manipulate and interpret them effectively.
Fractions aren't just about dividing something physical, like a pizza. They are also used to express ratios, which compare sizes of two quantities. For example, the exercise about gallons and quarts uses a fraction to describe a ratio. This is a powerful way to understand relationships between quantities in mathematics.

Simplifying Fractions

Simplifying fractions involves making the fraction as simple as possible. This means reducing it so that the numerator and the denominator have no common factors other than 1. To simplify effectively, you divide both the numerator and the denominator by their greatest common divisor (GCD).

• Start by finding the largest number that can divide both parts of the fraction evenly.
• Once you've identified this number, divide both the numerator and the denominator by it.

In our earlier example, the fraction \( \frac{44}{11} \) was simplified to \( \frac{4}{1} \) by recognizing that 11 was the greatest common divisor. This step is crucial for comparing and working with fractions efficiently.

Converting Units

Converting units is about changing a measurement from one unit to another. It requires an understanding of how units relate to each other. For liquid volumes, we often see conversions from gallons to quarts and vice versa. Knowing that 1 gallon equals 4 quarts is fundamental.
For example, if you have 11 gallons, multiplying by 4 converts it to quarts: 11 gallons \( \times \) 4 quarts/gallon = 44 quarts. It’s important to ensure unit consistency when solving problems, particularly in ratios. This ensures you’re comparing or calculating with compatible measurements.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide two or more numbers without leaving a remainder. It's a key concept when simplifying fractions.

• To find the GCD, list the factors of each number.
• Identify the largest factor common to both lists.

For \( 44 \) and \( 11 \), the GCD is \( 11 \), as it is the highest number dividing both completely. Using the GCD to simplify fractions involves dividing the numerator and denominator by this number, thereby creating the simplest form of the fraction. This process is efficient for making fractions more intuitive and easier to understand in further calculations.