Problem 48 - Prealgebra

Question

Write each list of numbers in order from least to greatest. $$\frac{2}{7}, 0.027,27 \%$$

Step-by-Step Solution

Answer

0.027, 27%, \(\frac{2}{7}\)

Step 1: Convert to Decimals

First, we convert each number to a decimal to easily compare them.\(\frac{2}{7}\) is approximately 0.2857. By converting the percentage, 27%, to a decimal, we get 0.27.

Step 2: Compare Decimals

Now, compare the decimal values: 0.027, 0.27 (from 27%), and 0.2857 (from \(\frac{2}{7}\)). Determine their order by starting from the lowest value.

Step 3: Write in Ordered List

Arrange the numbers from least to greatest by placing them according to their decimal equivalences: 0.027, 0.27 (27%), \(\frac{2}{7}\).

Key Concepts

Converting Fractions to Decimals

Fractions represent a part of a whole. To compare them with other numbers, it's often useful to convert them into decimals. This can be done by performing a simple division of the numerator by the denominator. For instance, to convert the fraction \( \frac{2}{7} \) into a decimal, simply divide 2 by 7. This results in approximately 0.2857. Remember, the decimal number you obtain may not always be exact, especially for fractions that produce a long or repeating decimal. In these cases, it's best to round them to a manageable number of decimal places, like four or five, for simplicity when comparing or ordering numbers.

Converting Percentages to Decimals

Percentages are another way to express numbers. Since percentages represent a part out of 100, converting them to decimals requires a simple procedure. To convert a percentage to a decimal, divide it by 100. Let's look at 27% as an example. Dividing 27 by 100 results in 0.27.

• Always remember to move the decimal point two places to the left.
• Similarly, percentages like 100% become 1.0 and 0.5% becomes 0.005.

This conversion helps when comparing numbers in different formats, such as fractions and decimals. Converting them all to decimals provides a uniform basis for comparison.

Comparing Decimals

Once all numbers are in decimal form, they can be easily compared. To do this, look at the digits of each decimal number, starting from the left. Here's how:

• Compare the digits in the tenths place. The smaller the number, the smaller the value.
• If they are the same, move to the next digit on the right, and repeat the comparison process.

For example, when comparing 0.027, 0.27, and 0.2857: - Start by checking the tenths digit. Zero is less than two, so 0.027 is the smallest. - Between 0.27 and 0.2857, compare the tenths. They match, so check the next digit. Seven is less than eight, making 0.27 less than 0.2857. Arranging these from least to greatest gives us the sequence: 0.027, 0.27, and 0.2857. When comparing decimals, ensure you're consistent in the number of decimal places considered, which helps avoid errors in order or comparison.