Problem 44 - Prealgebra

Question

Carry out cach of the following divisions only so far as needed to round the results to the nearest hundredth. $$3.99 \div 0.5$$

Step-by-Step Solution

Answer

The result rounded to the nearest hundredth is 7.98.

Step 1: Set Up the Division

To divide 3.99 by 0.5, set up the equation as a division problem: \( 3.99 \div 0.5 \).

Step 2: Convert Division to Multiplication

Instead of dividing by 0.5, you can multiply by 2 (since \( 0.5 = \frac{1}{2} \)), transforming the problem into \( 3.99 \times 2 \).

Step 3: Perform the Multiplication

Calculate the multiplication: \( 3.99 \times 2 = 7.98 \).

Step 4: Round to the Nearest Hundredth

Since the result, 7.98, is already to the hundredth decimal place, no further rounding is necessary. The hundredth digit is 8 and there are no additional digits that affect rounding.

Key Concepts

Multiplication as Inverse Operation

Understanding division can sometimes be tricky, but did you know that multiplication is its trusty sidekick? They work together in a really cool way. Division essentially asks how many times one number fits into another. Instead of dividing a number, you can flip the operation to multiplication by using the reciprocal of the divisor. This method simplifies the calculation and gives the same result as traditional division.

• In our exercise, instead of dividing by 0.5, we multiplied by 2, which is the inverse or reciprocal of 0.5.
• Mathematically, this is because dividing by a number is the same as multiplying by its reciprocal: \( a \div b = a \times \frac{1}{b} \).
• This is super handy because for many, multiplication feels more intuitive and straightforward than division.

So, every time you face division, remember you can switch to multiplication by using the inverse. Your calculations will become easier and often quicker!

Rounding Numbers

Rounding is like giving a number a nice clean-up when you don’t need all those extra digits. When you're asked to round to the nearest hundredth, you're focusing on the second digit after the decimal point.

• If the next digit (third digit after the decimal) is 5 or more, you round up. If it's less than 5, you round down.
• Consider the number 7.985. Here, since the third digit is 5, 7.98 becomes 7.99 after rounding.
• Rounding helps make numbers easier to work with in real-world situations, like money calculations or when precision is not vital.

In our solution, the multiplication result was 7.98, which was already neatly rounded. Hence, no additional rounding was necessary!

Decimal Division

Dealing with decimals in division might seem daunting initially, but it’s just another step in your mathematical journey. Here’s how you can handle it with confidence:

• Firstly, understand how many decimal places you need in the final answer. In many cases, it will be specified as to the nearest whole number, tenth, hundredth, etc.
• Use multiplication as seen in the inverse operation technique to simplify decimal division. For instance, \(3.99 \div 0.5\) was converted to \(3.99 \times 2\) making it a lot simpler.
• If you must perform traditional division, ensure to line up your decimals properly to avoid any confusion, which can often be a common source of errors.

With these tips, dividing decimals can be as straightforward as dealing with whole numbers, and it's a useful skill that extends to many real-life scenarios.