Problem 69 - 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. $$\frac{1}{10} \cdot \frac{3}{10}$$

Step-by-Step Solution

Answer

The product of \(\frac{1}{10} \cdot \frac{3}{10}\) is \(\frac{3}{100}\).

Step 1: Understanding the Multiplication of Fractions

To multiply two fractions, we multiply the numerators together to get the new numerator, and the denominators together to get the new denominator. So, for the problem \(\frac{1}{10} \cdot \frac{3}{10}\), we need to multiply the numerators (1 and 3) and denominators (10 and 10).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \(1 \cdot 3 = 3\). This will be the numerator of the product.

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \(10 \cdot 10 = 100\). This will be the denominator of the product.

Step 4: Combine the Results to Get the Product

Now that we have the numerator and the denominator for the product, we can combine them to get \(\frac{3}{100}\).

Step 5: Verify and Simplify (if necessary)

Check if the fraction \(\frac{3}{100}\) can be simplified. In this case, the greatest common divisor of 3 and 100 is 1, so the fraction is already in its simplest form.

Key Concepts

Multiplying Decimals

When it comes to multiplying decimals, the process closely mirrors that of whole numbers with one key difference. We follow the steps of regular multiplication, but when it is time to place the decimal point in the final product, we need to account for all the decimal places involved.

Consider two decimal numbers. Start by ignoring the decimal points and multiply them just like whole numbers. After obtaining the product, count the total number of digits to the right of the decimals in both original numbers. This total informs you where to place the decimal in the final answer.

For example, multiplying 0.2 (one decimal place) and 0.3 (one decimal place) gives us the following procedure:

• Ignore decimals and multiply: 2 multiplied by 3 equals 6.
• Count decimal places: 0.2 has one, and 0.3 has one, adding up to two.
• Place decimal point two places from the right in the product: 0.06.

This systematic approach ensures precise computations across varying sizes of decimal values.

Numerators and Denominators

Understanding numerators and denominators is crucial for grasping fraction operations.

A fraction like \(\frac{1}{10}\) is composed of two parts:

• **Numerator**: the top number, representing how many parts we have.
• **Denominator**: the bottom number, signaling how many equal parts the whole is divided into.

In multiplication, both numerators and denominators are handled separately yet similarly.

Using the exercise \(\frac{1}{10} \cdot \frac{3}{10}\), here's what happens:

• **Multiply Numerators**: 1 times 3, resulting in 3.
• **Multiply Denominators**: 10 times 10, coming up as 100.
• This results in a fraction of \(\frac{3}{100}\).

In summary, while numerators indicate the number of parts taken, denominators orchestrate how fractions fit into a whole.

Simplifying Fractions

Simplifying fractions is akin to reducing them to their most efficient forms.

This process involves finding the greatest common divisor (GCD) of the numerator and denominator, and then dividing both by this GCD. The aim is to shrink the fraction's numbers without altering its value.

If you have a fraction like \(\frac{3}{12}\), here’s how you simplify it:

• Find the GCD of 3 and 12, which is 3.
• Divide both 3 and 12 by 3 to get \(\frac{1}{4}\).

For our exercise fraction \(\frac{3}{100}\), we check whether it can be simplified. The greatest common divisor of 3 and 100 is 1. Thus, \(\frac{3}{100}\) is its simplest form.
By simplifying fractions, calculations become easier, and identifying fraction equivalence is more straightforward.