Problem 74 - Prealgebra

Question

To understand how to multiply decimals, we need to understand multiplication with whole numbers, fractions, and mixed numbers. The following problems review these concepts. $$7 \cdot \frac{7}{10}$$

Step-by-Step Solution

Answer

The result of multiplying 7 by \( \frac{7}{10} \) is 4.9.

Step 1: Understanding the Problem

We need to multiply an integer, 7, by a fraction, \( \frac{7}{10} \). This requires us to use basic multiplication concepts involving fractions.

Step 2: Multiply the Integer by the Numerator

To multiply an integer by a fraction, multiply the integer by the numerator of the fraction. Here, we multiply 7 (the integer) by 7 (the numerator of the fraction). Thus, \( 7 \times 7 = 49 \).

Step 3: Divide by the Denominator

Now, we take the result from the previous step and divide it by the denominator of the fraction. So, we need to divide 49 by 10. This is \( \frac{49}{10} \).

Step 4: Convert to Decimal (if needed)

To express \( \frac{49}{10} \) as a decimal, divide 49 by 10, which gives 4.9. Thus, the final result of the multiplication in decimal form is 4.9.

Key Concepts

Whole Numbers

Whole numbers are numbers that do not have any fractional or decimal parts. They include zero and all positive integers like 1, 2, 3, and so on. When multiplying a whole number by another whole number, the process is straightforward. It's simply repeated addition. For example, 3 times 2 means adding 3 two times, which results in 6. Let's break this down further:

- **Multiplication is commutative**: This means that the order of the numbers doesn't matter. For example, 4 times 5 is the same as 5 times 4, both equaling 20.
- **Multiplication is associative**: Grouping of numbers doesn’t affect the result. For instance, (2 times 3) times 4 equals 2 times (3 times 4). Understanding whole numbers lays the foundation for more complex multiplication, like with fractions and mixed numbers.

Fractions

Fractions represent parts of a whole and are written as two numbers separated by a line: the numerator and the denominator. The numerator indicates how many parts we are considering, and the denominator shows the total number of equal parts.
When it comes to multiplying a whole number by a fraction, you follow these steps:

• Multiply the whole number by the numerator of the fraction.
• Divide the result by the denominator.

For example, when we multiply 7 by the fraction \( \frac{7}{10} \), we first multiply 7 by 7, resulting in 49. Then, we divide 49 by 10, which gives us 4.9. This shows that multiplying by a fraction effectively scales down the whole number, as fractions represent values less than one.

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction, like 3\( \frac{1}{2} \). They are useful when expressing quantities that are between two whole numbers. To multiply mixed numbers, it’s often easiest to convert them into improper fractions first.
Here's a simple approach:

• Convert the mixed number to an improper fraction.
• Multiply the improper fractions as you would multiply regular fractions.

To convert a mixed number into an improper fraction, multiply the whole number by the denominator, add the numerator, and place it over the original denominator. For example, with 2\( \frac{2}{3} \), you multiply 2 by 3 to get 6, add 2 to get 8, and write it over 3, making \( \frac{8}{3} \).
Once converted, proceed with multiplication as usual, making sure to simplify the resulting fraction when possible. Mixed numbers can seem tricky at first, but with practice, they become just another tool in your mathematical toolbox.