Problem 31 - Prealgebra

Question

Divide. Write your answers as decimals. $$5.44 \div 0.04$$

Step-by-Step Solution

Answer

The answer is 136.

Step 1: Understand the Problem

We need to divide 5.44 by 0.04, which involves finding out how many times 0.04 fits into 5.44.

Step 2: Set Up the Division as a Fraction

We express the division problem as a fraction: \( \frac{5.44}{0.04} \). This step helps us in visualizing the problem as a division of decimals.

Step 3: Eliminate the Decimal in the Divisor

To simplify the calculation, we eliminate the decimal from the divisor, 0.04, by multiplying both the numerator and the denominator by 100 (or move the decimal point two places to the right). This gives \( \frac{544}{4} \).

Step 4: Perform the Division

Now perform the division 544 divided by 4. Divide 544 by 4 to get 136. This can be done by long division or using a calculator.

Key Concepts

Decimal Division

Decimal division is the process of dividing numbers that include decimals. Imagine you have some amount of something you want to share, and you need to figure out how many smaller portions of another size can fit into it. For example, in this problem, you want to see how many 0.04s can fit into 5.44. When we divide decimals, it's important to keep track of the decimal points. These tiny dots help decide where the numbers should be split when divided. Therefore, accurately aligning them is crucial. Begin by writing the division as a fraction. This visualization might help you see the problem differently, making it easier to solve. For instance:

• Write it out as 5.44 divided by 0.04 or as the fraction \( \frac{5.44}{0.04} \).

This fraction form is just another way of expressing decimal division, maintaining the balance between numbers while simplifying the solution.

Fraction Representation

Fraction representation is a pivotal step in understanding division, especially with decimals. Turning a division problem into a fraction can make the process less intimidating. A fraction consists of two parts:

• Numerator: This is the number on top, representing the number you are dividing (in this case, 5.44).
• Denominator: This stands for the number by which you are dividing (in this instance, 0.04).

So, writing the division of \(5.44\) by \(0.04\) into fraction form gives us \( \frac{5.44}{0.04} \). This form not only prepares us for the next steps but also helps in visualizing and potentially simplifying the problem. Turning the division problem into a fraction is akin to setting up the puzzle pieces before solving it. In many cases, especially with decimals, having the problem outlined in this manner can clarify the path to the solution.

Eliminating Decimals

To make division simpler, eliminating decimals is often necessary. This involves converting the divisor and sometimes the dividend into whole numbers. Think of it as removing any hindrances that could cause confusion. Here's how you can eliminate decimals:

• Take the divisor (0.04) and determine how many places you need to move the decimal point to turn it into a whole number. Here, you shift it two places to the right to become 4.
• Move the decimal in the dividend, 5.44, the same number of places to keep the equation balanced. So, it also shifts two places to the right, becoming 544.

Now, you have effectively turned the problem \( \frac{5.44}{0.04} \) into \( \frac{544}{4} \), where both the numerator and the divisor are whole numbers. This transformation makes the calculation straightforward and less error-prone.

Long Division with Decimals

Long division with decimals involves several steps, but don't worry, it's all about being organized and careful. Once decimals are eliminated as in previous steps, you proceed as you would with whole numbers. Here's a simple guide to long division:

• Set up your problem, as with pen and paper. Write 544 inside the division box and 4 as the divisor outside.
• Divide the first digit of the dividend by 4. If it doesn't go, move to the first two digits. This will be your first quotient digit.
• Subtract this from the dividend's portion you used, then bring down the next digit.
• Repeat the process until you've worked through all the numbers.

With 544 divided by 4: - 4 goes into 5 one time (the first digit of the quotient). - Subtract 4 from 5 to get 1, bring down the next number to make 14. - 4 goes into 14 three times. - The process continues until you complete the division, resulting in a quotient of 136. Long division with decimals, therefore, turns into a series of manageable steps, especially after eliminating decimals. With practice, it becomes an intuitive process.