Problem 18 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$\frac{1}{2} \operatorname{to} 3 \frac{1}{2}$$

Step-by-Step Solution

Answer

The ratio \(\frac{1}{2}\) to \(3 \frac{1}{2}\) as a fraction in lowest terms is \(\frac{1}{7}\).

Step 1: Express Mixed Number as an Improper Fraction

First, express the mixed number \(3 \frac{1}{2}\) as an improper fraction. To do this, multiply the whole number part \(3\) by the denominator \(2\), which gives \(6\). Then add the numerator \(1\) to get \(7\). So, \(3 \frac{1}{2} = \frac{7}{2}\).

Step 2: Write the Ratio as a Fraction

The problem gives the ratio as \(\frac{1}{2}\) to \(3 \frac{1}{2}\). We can write this ratio as a fraction: \(\frac{\frac{1}{2}}{\frac{7}{2}}\).

Step 3: Simplify the Complex Fraction

To simplify \(\frac{\frac{1}{2}}{\frac{7}{2}}\), multiply the numerator by the reciprocal of the denominator: \(\frac{1}{2} \times \frac{2}{7}\). The twos cancel out, leaving \(\frac{1}{7}\).

Step 4: Write the Final Simplified Fraction

The fraction \(\frac{1}{7}\) is already in the simplest form, so the ratio \(\frac{1}{2}\) to \(3 \frac{1}{2}\) as a fraction in lowest terms is \(\frac{1}{7}\).

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities by showing how many times one number contains another. If you are comparing two numbers, the ratio is expressed as a fraction. For example, if you have 2 apples and 3 oranges, the ratio of apples to oranges is 2:3, or expressed as a fraction, \( \frac{2}{3} \). Understanding and using ratios can help in various everyday situations, like cooking or dividing expenses. When translating a ratio into a fraction, just treat the ratio values as the numerator and the denominator of the fraction. This gives you a clear way to understand and simplify them.

Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms. This means making the numerator and the denominator as small as possible, but still representing the same value. To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator, and divide both by that number.

• Start with the fraction \( \frac{6}{9} \)
• The GCD of 6 and 9 is 3.
• Divide both the numerator and the denominator by 3 to get \( \frac{2}{3} \).

Breaking numbers down into their simplest form makes them easier to interpret and use in different math problems.

What are Mixed Numbers

Mixed numbers are numbers that combine a whole number with a fraction. For instance, \( 3 \frac{1}{2} \) is a mixed number with the whole number 3 and the fraction \( \frac{1}{2} \). Mixed numbers are used to express amounts greater than one whole, in a simple-to-read format. They are very common in real-world applications like cooking recipes or measuring lengths. Whenever you encounter mixed numbers in mathematical problems, you might need to convert them to improper fractions to perform calculations more effectively.

Improper Fractions Explained

Improper fractions are fractions where the numerator is greater than or equal to the denominator. They represent quantities greater than one. For example, \( \frac{7}{4} \) is an improper fraction.Improper fractions can be converted into mixed numbers for easier reading, but they are more convenient for mathematical operations like multiplication and division. To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator of the fractional part to this product.
• Place the result over the original denominator.

Understanding improper fractions helps you work fluently with diverse types of calculations.