Problem 1 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 0.7 \\ \times 0.4 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The product of 0.7 and 0.4 is 0.28.

Step 1: Multiply Without Decimals

First, ignore the decimals and multiply the numbers as if they were whole numbers. So, multiply 7 by 4.\( 7 \times 4 = 28 \)

Step 2: Count Decimal Places

Count the total number of decimal places in both numbers that are being multiplied. In this case, 0.7 has one decimal place, and 0.4 has one decimal place. Total decimal places = 1 + 1 = 2.

Step 3: Adjust the Product for Decimals

Take the result from Step 1, which is 28, and adjust it to have the same number of decimal places as totaled in Step 2. Move the decimal point two places to the left. So, the adjusted product becomes 0.28.

Key Concepts

Decimals

Decimals are fractions written in a special form. Instead of using a fraction, like \( \frac{7}{10} \), we can write it as 0.7.
The number to the right of the decimal point represents parts of a whole, based on place value. Decimal numbers are often used when more precision is needed than whole numbers can offer.
Understanding decimals is crucial because they are a common part of everyday measurements, financial calculations, and more. A key trait of decimals is that their digits hold decreasing value from left to right. For example, in 0.7, the 7 is in the tenths place, meaning 7 parts out of 10.

Place Value

Place value is a foundational concept in math that helps us understand the value of each digit in a number. Each digit in a decimal number represents a power of ten.
For example, in the number 0.4, the 4 is in the tenths place, meaning it represents four tenths. The digits to the right of the decimal point have place values that are fractions of ten, such as:

• The first place right of the decimal is the tenths.
• The second place is the hundredths.
• The third is the thousandths, and so on.

Being able to identify where each digit sits in a decimal number helps in determining the number of decimal places to shift when multiplying. If you encounter a number like 0.04, remember that the 4 is in the hundredths place, which is quite different from a 4 in the tenths place.

Multiplication Steps

Multiplying decimals might seem tricky, but it gets easier when you break down the steps. Let's see how it's done:

**Step 1:** Ignore the decimals initially. Imagine your decimals as whole numbers temporarily. For example, with 0.7 and 0.4, just multiply 7 by 4 to get 28.

**Step 2:** Count the decimal places. Each original number has its own place value. Count how many digits are to the right of the decimal point in your original numbers. Here, 0.7 has one decimal place; 0.4 also has one.

**Step 3:** Adjust your product by the total number of decimal places counted. Once you have your raw product (28 in our case), use your place value count (2 total decimal places) to adjust the placement of the decimal point. Shift the decimal point in your product two places to the left, transforming 28 into 0.28. Voilà, you've successfully multiplied the decimals!