Problem 74 - Prealgebra

Question

Perform the indicated operations. $$1.5 \times 45$$

Step-by-Step Solution

Answer

The product is 67.5.

Step 1: Set Up the Multiplication

To solve the multiplication of 1.5 and 45, we need to multiply the two numbers directly. Start by writing the multiplication as \(1.5 \times 45\).

Step 2: Convert Decimal to Fraction (Optional)

For easier multiplication, you can convert 1.5 to a fraction. Note that 1.5 is equivalent to \(\frac{3}{2}\). Thus, the multiplication becomes \(\frac{3}{2} \times 45\).

Step 3: Simplify the Multiplication

Next, multiply \(\frac{3}{2}\) by 45. You can multiply the numerator by 45 first: \(3 \times 45 = 135\).

Step 4: Divide by the Denominator

Now, divide the result by the denominator 2: \(\frac{135}{2} = 67.5\). This step gives you the product of the original multiplication.

Key Concepts

Step by Step Multiplication

When performing multiplication with decimals, it's useful to tackle the problem in manageable steps. Start by writing the multiplication equation clearly, as this sets the foundation. In our example, we want to multiply 1.5 by 45. It’s important to recognize that decimals are just another form of numbers, and they behave similarly to whole numbers in multiplication. First, imagine multiplying the numbers as if they are both whole: you could ignore the decimal point temporarily to make the process straightforward. This makes it easier to carry out without getting bogged down by decimal placement. After completing the multiplication as if both are whole numbers, later, you will need to adjust for the decimal to get the accurate result.

Converting Decimals to Fractions

Converting a decimal into a fraction can make multiplication simpler. Many find fractions less cumbersome because multiplying fractions uses whole numbers rather than decimals. To convert the decimal 1.5 into a fraction, consider what 1.5 represents. The number 1.5 can be expressed as \( \frac{15}{10} \) since it consists of 15 tenths. Simplifying \( \frac{15}{10} \), by dividing both the numerator and the denominator by their greatest common divisor, which is 5, yields \( \frac{3}{2} \). This fraction form can now replace the decimal in the multiplication equation: \( \frac{3}{2} \times 45 \). By using these simpler forms, we can make calculations easier and quicker, particularly when multiplying by whole numbers.

Simplifying Fractions in Multiplication

When you multiply fractions, the process involves multiplying the numerators (top numbers) and then the denominators (bottom numbers). However, when one of the numbers is a whole number, like 45 in our equation \( \frac{3}{2} \times 45 \), it's useful to consider the whole number as a fraction itself by placing it over 1 (i.e., \( \frac{45}{1} \)). Now, multiply \( \frac{3}{2} \times \frac{45}{1} \). This requires multiplying the numerators together (i.e., \( 3 \times 45 = 135 \)) and multipliers of the denominators (i.e., \( 2 \times 1 = 2 \)). Once you have your new fraction \( \frac{135}{2} \), the final step is often to convert this improper fraction back into a decimal or mixed number to complete the multiplication process. Since dividing 135 by 2 yields 67.5, our final simplified product of the original multiplication is 67.5. Simplifying fractions not only aids in cleaner calculations but ensures that our final results are both concise and accurate.