Problem 68 - Algebra 1: Concepts and Skills

Question

Write the numbers in order from least to greatest. $$ \frac{3}{5}, \frac{4}{10}, \frac{5}{15} $$

Step-by-Step Solution

Answer

The numbers in order from least to greatest are \( \frac{5}{15}, \frac{4}{10}, \frac{3}{5} \).

Step 1: Simplifying the Fractions

Simplify each fraction. So, \( \frac{3}{5} \) is already in its simplest form. However, the fractions \( \frac{4}{10} \) and \( \frac{5}{15} \) can be simplified to \( \frac{2}{5} \) and \( \frac{1}{3} \) respectively.

Step 2: Comparing the simplified values

Now, compare the simplified values of the fractions. As \( \frac{1}{3} < \frac{2}{5} < \frac{3}{5} \), the numbers in order from least to greatest is \( \frac{5}{15}, \frac{4}{10}, \frac{3}{5} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is crucial when working with numbers, making them easier to work with and understand. Simplifying, also known as reducing a fraction, means to make the numerator (top number) and denominator (bottom number) as small as possible while still keeping the same value.

For example, take the fraction \( \frac{4}{10} \). This fraction can be simplified because both 4 and 10 are divisible by 2. When you divide the numerator and denominator by 2, you get \( \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \). Similarly, \( \frac{5}{15} \) simplifies to \( \frac{5 \div 5}{15 \div 5} = \frac{1}{3} \). Simplifying fractions helps with comparing fractions because it's much easier to compare smaller, 'friendlier' numbers.

• Divide both the numerator and denominator by their greatest common divisor (GCD).
• If the numerator is 1 after simplifying, it means you have a unit fraction, which is one of the simplest forms of a fraction.
• Simplified fractions have the same value as the original but with smaller whole numbers.

Comparing Fractions

Comparing fractions is a key skill in mathematics. To determine which fractions are larger or smaller, they must be readable on a common scale, that's where simplifying can help. When the denominators are the same, it's straightforward: the fraction with the larger numerator is larger. But when denominators differ, as in \( \frac{3}{5} \) versus \( \frac{2}{5} \) and \( \frac{1}{3} \), you compare by either finding a common denominator or converting to decimals which is a more advanced technique that we'll discuss shortly.

Here are some tips for comparing fractions:

• If the fractions have the same denominator, the fraction with the larger numerator is the larger fraction.
• For fractions with different denominators, you can cross-multiply to compare values without finding a common denominator or convert fractions to decimals.
• A visual method is to draw pie charts or number lines to see which fraction represents a larger portion of the whole.

Fraction to Decimal Conversion

Converting fractions to decimals is another method that can be used for comparing fractions and can also be helpful in a variety of mathematical contexts. To convert a fraction into a decimal, you divide the numerator by the denominator.

For instance, to convert \( \frac{3}{5} \) into a decimal, you would do the division 3 divided by 5 which equals 0.6. Similarly, converting the simplified \( \frac{2}{5} \) and \( \frac{1}{3} \) results in 0.4 and approximately 0.333, respectively. This approach turns the problem of comparing fractions into a simpler problem of comparing decimal numbers, which is often more intuitive.

Benefits of converting fractions to decimals include:

• Easier to compare and order, especially if you lack a common denominator.
• Decimals are often more familiar and can be easily plotted on a number line.
• Decimal representation is often required in scientific, financial, and statistical calculations where precision is important.