Problem 38 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\frac{2.99}{\frac{1}{2}}$$

Step-by-Step Solution

Answer

The simplified expression is 5.98.

Step 1: Understanding the Problem

We need to simplify the expression \( \frac{2.99}{\frac{1}{2}} \) which involves dividing a decimal by a fraction. Our goal is to express this in the simplest decimal form.

Step 2: Convert Division into Multiplication

To divide by a fraction, we multiply by its reciprocal. Thus, \( \frac{2.99}{\frac{1}{2}} \) becomes \( 2.99 \times 2 \).

Step 3: Calculate the Product

Multiply 2.99 by 2: \( 2.99 \times 2 = 5.98 \).

Step 4: Final Expression

The simplified expression of \( \frac{2.99}{\frac{1}{2}} \) in decimal form is \( 5.98 \).

Key Concepts

Division by Fraction

When you see a division problem involving a fraction, it might seem a bit intimidating at first, especially with decimals. However, there's a simple way to handle it. Dividing by a fraction means you're essentially finding out how many of those fraction parts fit into the number you're dividing. For instance, in the problem \(\frac{2.99}{\frac{1}{2}} \), you're determining how many halves fit into 2.99.
To make this easier, you employ the strategy of converting this division into a multiplication problem, which is much easier to solve. This leads us to our next concept of multiplication by reciprocal. But always remember, dividing by a fraction is the same as multiplying by its inverse.

Multiplication by Reciprocal

One powerful tool when dividing by fractions is using the reciprocal. A reciprocal of a number is simply flipping the numerator and denominator. For example, the reciprocal of \( \frac{1}{2} \) is \( 2 \).
When you're asked to divide by a fraction, you can instead multiply by its reciprocal. So in our example, \( \frac{2.99}{\frac{1}{2}} \) turns into \( 2.99 \times 2 \). This technique simplifies your calculations significantly, as multiplying numbers is often more straightforward than dividing by fractions.
Remember, the key is always to identify the fraction you're dividing by and then flip it to find the reciprocal. With that, you're ready to turn any division into a comfortable multiplication problem.

Decimal Multiplication

Once we've converted division to multiplication using the reciprocal, we find ourselves with a decimal multiplication problem. Multiplying decimals might seem tricky, but it's not too different from multiplying whole numbers.
To multiply decimals like \( 2.99 \times 2 \), you ignore the decimal points and multiply the numbers as if they were whole numbers: \( 299 \times 2 = 598 \).
Next, count the total number of decimal places in the factors you multiplied. Since \( 2.99 \) has 2 decimal places, place the decimal in your product so that it has the same number of decimal places: \( 5.98 \).
Following these steps ensures accuracy in your decimal multiplication, wrapping up the simplification process.