Problem 9 - Elementary Algebra

Question

For the following problems, perform each indicated operation. \(\frac{3}{7} \cdot \frac{14}{18} \cdot \frac{6}{2}\)

Step-by-Step Solution

Answer

Answer: The product of 4.26 and 3.2 is 13.632.

Step 1: Multiply Without Decimal Point

First, we will ignore the decimal points and simply multiply the numbers as if they were whole numbers. So we'll multiply \(426\) by \(32\): \(426 × 32 = 13632\)

Step 2: Count Decimal Places

Now, we count the decimal places in both original numbers. In \(4.26\), there are two decimal places. In \(3.2\), there is one decimal place. In total, we have \(2+1=3\) decimal places.

Step 3: Place the Decimal Point

Since we have \(3\) decimal places in total, we will move the decimal point three places to the left in the result from step 1: \(13632 → 13.632\) So, \(4.26 \cdot 3.2 = 13.632\).

Key Concepts

Place Value

Understanding the place value is crucial when multiplying decimals, as it helps to keep track of the 'weight' of each digit depending on its position. In the decimal number system, the value of a digit is determined by its position relative to the decimal point. For instance, in the number 4.26, the digit 4 represents 4 units, 2 represents two-tenths (0.2), and 6 signifies six hundredths (0.06).

When multiplying, if we ignore the decimal points temporarily, we treat 4.26 as 426 and 3.2 as 32, counting each digit's place as if it were a whole number. Calculating as if working with whole numbers simplifies the initial multiplication process. However, the most important step is to keep track of the number of decimal places in the original numbers so that we can correctly place the decimal in the final answer. In the exercise given, knowing that 4.26 has two places after the decimal and 3.2 has one, ensures accuracy when we reintroduce the decimal point for the final result.

Decimal Arithmetic

When performing decimal arithmetic, such as multiplication, it is essential to align numbers by their place value, especially when multiplying by hand. However, a convenient method is treating the decimals like whole numbers initially, then adjusting the decimal place afterward. This trick simplifies calculation and reduces errors.

After multiplying the numbers as if they were integers, we adjust the decimal point's placement based on the combined count of decimal places from the original numbers. So for our exercise, the product of 426 and 32 gives us 13632. As there are a total of three decimal places in the original decimals, we move the decimal point three places to the left. It is pivotal to ensure that the total number of decimal places in the result accurately reflects the sum of decimal places from the factors involved.

Elementary Algebra

Though our given problem focuses primarily on decimal arithmetic, the underlying principles of elementary algebra still apply. Algebra teaches us methods for dealing with unknown values and complex calculations in a systematic way. For this exercise, the algebraic thinking involves an understanding that the decimal point represents a 'placeholder' that determines the value of digits and that we can manipulate this placeholder following algebraic rules.

In algebra, each step in a calculation follows logically from the previous one, much like moving the decimal place after the multiplication of numbers. Seeing this relationship helps us understand the importance of each step in solving a problem and reinforces why each step must be precisely carried out to achieve the correct result. In summary, elementary algebra instills a strong sense of the order of operations and the correct handling of numbers in various formats, like decimals.