Problem 67 - Algebra 1: Concepts and Skills

Question

Write the numbers in order from least to greatest. $$ \frac{1}{3}, \frac{4}{15}, \frac{2}{5} $$

Step-by-Step Solution

Answer

The numbers in order from least to greatest are \( \frac{4}{15}, \frac{1}{3}, \frac{2}{5} \)

Step 1: Convert to Decimals

Transform the fractions into decimals for easier comparison. Do this by dividing the numerator by the denominator.\( \frac{1}{3} = 0.333 \),\( \frac{4}{15} = 0.266 \), \( \frac{2}{5} = 0.4 \)

Step 2: Compare Decimals

Once the fractions are converted into decimals, it is easier to compare them. Since 0.266 < 0.333 < 0.4, it follows that \( \frac{4}{15} < \frac{1}{3} < \frac{2}{5} \)

Step 3: Verify the Order

It is always a good step to recheck your answer by cross multiplication which is yet another method to compare fractions.

Key Concepts

Converting Fractions to Decimals

Understanding how to convert fractions to decimals is essential for comparing and ordering different numbers. You simply divide the numerator (the top number) by the denominator (the bottom number). This can be done on a calculator or by long division if you're practicing your arithmetic skills.

For instance, in the given exercise, the fraction \( \frac{1}{3} \) is converted to a decimal by dividing 1 by 3, which gives us 0.333... It's important to note that some fractions, like \( \frac{1}{3} \) in this case, produce a repeating decimal. This means that the digit (or group of digits) after the decimal point will continue indefinitely. While it is not possible to write down an infinite number, we can use a few digits after the decimal point for a reasonable approximation in most practical situations.

Ordering Fractions

Having the ability to order fractions from least to greatest or vice versa is a powerful skill in mathematics. The most straightforward way to order fractions is to convert them into decimals, as it allows for easy comparison. However, when the fractions have different denominators, as in our exercise with \( \frac{4}{15} \) and \( \frac{2}{5} \) as examples, simply comparing the numerators won't work.

If you cannot easily convert the fractions into decimals, another method is finding a common denominator. Then you can compare the equivalent fractions with the same denominator to determine their order. It requires a good understanding of multiples and may take a bit longer, but it's another reliable approach to ensure you can always order any set of fractions.

Decimal Comparison

Understanding how to compare decimals is useful in many facets of life, from financial transactions to scientific measurements. When comparing decimals, start by looking at the digit in the first (leftmost) decimal place after the decimal point. In our exercise, the decimals (0.266, 0.333, and 0.4) were easy to compare since they are all less than 1 and have different tenths values.

When the tenths are the same, you would then move on to compare the hundredths, and so forth, until a difference is found. Remember that a decimal with a higher value in the leftmost place is greater; for example, 0.4 is greater than 0.333, which is greater than 0.266. Keeping the numbers organized and proceeding systematically will make decimal comparison a breeze.

Cross Multiplication

Cross multiplication is a very handy technique to compare fractions without converting them to decimals. It involves multiplying the numerator of one fraction by the denominator of the other fraction. For the exercise at hand, if we wanted to compare \( \frac{1}{3} \) and \( \frac{4}{15} \) without converting them to decimals, we could cross multiply: \(1 \times 15 = 15\) and \(4 \times 3 = 12\).

Since 12 is less than 15, we know that \( \frac{4}{15} < \frac{1}{3} \). This quick method is particularly useful when dealing with larger numbers where long division might be cumbersome. Practicing cross multiplication can help reduce errors when comparing fractions and increase your efficiency in solving problems.