Problem 10 - Elementary Algebra

Question

For the following problems, perform each indicated operation. \(\frac{14}{15} \cdot \frac{21}{28} \cdot \frac{45}{7}\)

Step-by-Step Solution

Answer

Question: Multiply the decimal numbers 2.97 and 3.15 and provide the answer. Answer: 2.97 * 3.15 = 9.3615.

Step 1: Multiply without decimals

First, multiply the two numbers as if they were whole numbers. For \(2.97 \cdot 3.15\), we will ignore the decimals and multiply \(297 \cdot 315\).

Step 2: Perform the multiplication

Multiply the numbers using the standard multiplication method: ``` 297 x 315 -------- 1485 (297 * 5) 1188 (297 * 10) + 891 (297 * 300) -------- 93615 ``` So, we have \(297 \cdot 315 = 93615\).

Step 3: Place the decimal point

Next, we need to place the decimal point in the result. Count the number of decimal places in both original numbers: - \(2.97\) has two decimal places - \(3.15\) has two decimal places So, there is a total of 2 + 2 = 4 decimal places. Hence, the result will have 4 decimal places. Place the decimal point in the result 4 places from the right side, giving us \(9.3615\). So, the final answer is: \(2.97 \cdot 3.15 = 9.3615\).

Key Concepts

Place Value

Understanding the place value is fundamental to dealing with decimals. Place value refers to the value of each digit in a number, based on its position. In a decimal number, digits to the left of the decimal point represent whole numbers, while to the right, they represent fractions of a whole.

For example, in the number 2.97, the digit '2' is in the ones place, '9' is in the tenths place, and '7' is in the hundredths place. Recognizing these positions is crucial when multiplying decimals because you need to accurately account for the number of decimal places in the final product to ensure correctness of the result.

Decimal Multiplication

The process of decimal multiplication may seem daunting at first, but it's quite similar to multiplying whole numbers. The key difference is accounting for the decimal places. After you perform the multiplication as if there were no decimal points, you then adjust the decimal point in the product.

After multiplying, you determine the correct place for the decimal point by counting the total number of decimal places in both of the factors. If you start with two numbers that both have two decimal places, like in the multiplication of 2.97 and 3.15, the combined number of decimal places will be four. Therefore, you need to place the decimal point in the product so that there are four digits to the right of it, thereby obtaining the accurate decimal product.

Standard Multiplication Method

The standard multiplication method, also known as long multiplication, involves a step-by-step process where each digit of one number is multiplied by each digit of another, and the results are added together.

This method can seem confusing with larger numbers, but breaking it down into smaller, more manageable parts makes it easier. Applying this method to decimals doesn’t differ from multiplying whole numbers; it's just vital to remember to reinsert the decimal point at the correct location in the final answer. Once the decimal has been properly placed, you've successfully converted your whole number result back into a decimal format that reflects the factors' place values.