Problem 82 - Prealgebra

Question

Determine which of the two given statements is true. 8.5934 < 8.554 or 8.5934 > 8.554

Step-by-Step Solution

Answer

The statement 8.5934 > 8.554 is true.

Step 1: Identify the place value

Consider the first number 8.5934. Each digit has a place value. Here, 8 is in the ones position, 5 is in the tenths, 9 in the hundredths, 3 in the thousandths, and 4 in the ten-thousandths. Do the same for 8.554.

Step 2: Compare digit by digit

Start comparing the two numbers from the leftmost non-zero digit. - Compare the hundreds place: both have 8. - Compare the tenths place: 5 in both. - Compare the hundredths place: 9 in the first number, 5 in the second. Since 9 is greater than 5, the number 8.5934 is greater.

Step 3: Determine the conclusion

Since 8.5934 has a greater digit at the hundredths place than 8.554, the inequality that holds true is 8.5934 > 8.554.

Key Concepts

Decimal Numbers

Decimal numbers are numbers that have a whole number part and a fractional part, separated by a decimal point. For example, in the number 8.5934, "8" is the whole number part, while the numbers "5934" after the decimal point represent the fractional part.

• The whole number part is composed of integers, just like in non-decimal numbers.
• The fractional part breaks down into tenths, hundredths, thousandths, and so on. This allows us to represent parts of a whole number.

Decimals are especially useful for expressing numbers that are not whole, which is common in measurements or calculations involving money. When comparing decimal numbers, focusing on the value of each digit based on its position (like tenths, hundredths) helps determine which number is larger or smaller.

Place Value

Place value is essential for understanding the value of each digit in a decimal number. Each position to the right of the decimal point has a specific value:

• The first position is the tenths place.
• The second is the hundredths place.
• The third is the thousandths place.
• And the fourth is the ten-thousandths place.

Place value dictates that the farther to the right a digit is, the smaller its value relative to the whole number. For example, in 8.5934, the "5" is in the tenths place, giving it a value of 0.5. In contrast, "4" is in the ten-thousandths place, giving it a much smaller value of 0.0004, even though both are whole numbers "5" and "4" respectively. This hierarchical structure aids in accurate number interpretation and comparison.

Inequality

Inequality involves comparing two values to understand the relationship between them. When we say one number is smaller or greater than another, we are exploring these inequalities. The symbols "<" and ">" help represent a smaller than or greater than relationship between two numbers.
For example, in comparing 8.5934 and 8.554:

• Start by comparing numbers from the left, moving to the right digit by digit.
• If digits are the same, move to the next place value (hundredths, thousandths).
• A larger digit at any point between two numbers indicates that number is greater.

In the given example, by identifying the unequal digits in the hundredths place (9 > 5), we conclude that 8.5934 is greater than 8.554, leading to the inequality 8.5934 > 8.554. Understanding inequalities is crucial in math to compare and order numbers or find ranges.