Dividing Decimals Mastering The First Step In 2.5 Into 10.50
When dividing decimals, it's crucial to understand the order of operations and the specific steps involved to arrive at the correct answer. In the given problem, we are faced with the division of 10.50 by 2.5, which essentially asks: How many times does 2.5 fit into 10.50? This type of problem commonly arises in everyday scenarios, such as calculating the cost per item when given a total cost and the number of items.
The Initial Challenge: 2.5 into 10.50
The problem presented, , sets the stage for a typical decimal division scenario. Yen paid $10.50 for 2.5 pounds of pretzels, and the core question is to determine the price Yen paid for each pound of pretzels. This translates mathematically into dividing the total cost ($10.50) by the total weight (2.5 pounds). The question explicitly asks for the first step in solving this division problem, with the following options provided: Move decimal, Divide, Multiply, Subtract, and Bring down. To accurately solve this, we need to delve into the mechanics of decimal division and understand which step logically precedes the others.
Identifying the First Critical Step: Moving the Decimal
The correct first step in this division problem is to move the decimal. This action is necessary because we cannot directly divide by a decimal number. To facilitate the division process, we need to transform the divisor (2.5) into a whole number. This transformation is achieved by moving the decimal point to the right.
Why Move the Decimal?
Moving the decimal point in the divisor makes it a whole number, which simplifies the division process significantly. However, this action must be balanced by a corresponding move in the dividend (10.50) to maintain the problem's mathematical integrity. When we move the decimal one place to the right in both the divisor and the dividend, we are essentially multiplying both numbers by 10. This does not change the quotient, which is the answer to the division problem, but it does make the division process more manageable.
The Process of Moving the Decimal
In our problem, 2. 5 becomes 25 when the decimal point is moved one place to the right. To balance this, we also move the decimal point one place to the right in 10.50, which becomes 105. Now, the problem is transformed into dividing 105 by 25, a much simpler calculation. This transformation is crucial because dividing by a whole number is far more straightforward than dividing by a decimal.
Implications of Not Moving the Decimal
If we were to attempt to divide 10.50 by 2.5 without moving the decimal, the process would be significantly more complex and prone to errors. The subsequent steps of division, multiplication, and subtraction would all be complicated by the presence of the decimal in the divisor. Therefore, moving the decimal is not just a procedural step; it's a foundational element in correctly executing decimal division.
The Subsequent Steps: Divide, Multiply, Subtract, and Bring Down
Once the decimal has been appropriately moved, the subsequent steps in the long division process fall into a familiar pattern. After transforming the problem to 25 into 105, we proceed with the standard long division algorithm:
Divide
The first step in the actual division process is to determine how many times the divisor (now 25) goes into the initial digits of the dividend (105). In this case, we ask: How many times does 25 go into 105? This step involves estimating and placing the first digit of the quotient above the appropriate digit in the dividend.
Multiply
Once we've estimated how many times 25 goes into 105, we multiply this estimate by the divisor (25). This product is then placed below the corresponding digits of the dividend, setting up the subtraction step.
Subtract
Subtracting the product obtained in the multiplication step from the corresponding digits of the dividend gives us the remainder. This remainder is a crucial value as it determines the next digit to be brought down.
Bring Down
If there are more digits in the dividend, we bring down the next digit and append it to the remainder. This forms a new number to be divided by the divisor, and the process loops back to the division step. If there are no more digits to bring down, the remainder is either zero (indicating a clean division) or a non-zero value, which may necessitate adding zeros to the dividend to continue the division for a more precise answer.
Applying the Steps to the Problem
In our specific problem of dividing 105 by 25, we've already established the need to move the decimal. Let's walk through the subsequent steps:
Divide
25 goes into 105 four times (4 x 25 = 100).
Multiply
Multiply 4 (our estimate) by 25, which equals 100.
Subtract
Subtract 100 from 105, leaving a remainder of 5.
Bring Down
Since we moved the decimal in 10.50, we now consider the implicit zero after the 5, making our new number 50. Now, 25 goes into 50 exactly two times.
Final Calculation
Thus, 105 divided by 25 is 4.2. This means Yen paid $4.20 for each pound of pretzels.
The Importance of Order in Operations
Understanding the sequence of steps in decimal division is paramount. Moving the decimal first simplifies the problem, making the subsequent steps more manageable. Skipping this step can lead to confusion and errors. Each step – division, multiplication, subtraction, and bringing down – plays a critical role in arriving at the correct quotient. Mastering this sequence ensures accuracy and efficiency in solving division problems involving decimals.
In summary, when dividing decimals, the first and foremost step is to move the decimal in both the divisor and the dividend to transform the divisor into a whole number. This foundational step sets the stage for a smooth and accurate division process, ultimately leading to the correct solution. The subsequent steps of dividing, multiplying, subtracting, and bringing down then follow a logical progression, building upon this initial transformation.
