Problem 43 - Prealgebra

Question

Carry out cach of the following divisions only so far as needed to round the results to the nearest hundredth. $$2.99 \div 0.5$$

Step-by-Step Solution

Answer

The result of the division \(2.99 \div 0.5\) rounded to the nearest hundredth is 5.98.

Step 1: Set up the division problem

Write down the division problem: \(2.99 \div 0.5\). This means we need to find out how many times 0.5 fits into 2.99.

Step 2: Convert the divisor to a whole number

To make the division easier, multiply both the dividend and the divisor by 10 to eliminate the decimal in the divisor: \(2.99 \times 10 = 29.9\) and \(0.5 \times 10 = 5\). Now divide 29.9 by 5.

Step 3: Perform the division

Divide 29.9 by 5: \(29.9 \div 5 = 5.98\). Calculate the division up to two decimal places to round to the nearest hundredth.

Step 4: Round the result

The division result is 5.98, which is already calculated to the nearest hundredth. Confirm rounding by checking the third decimal place. Since there is no third decimal here, 5.98 is our rounded result.

Key Concepts

Long Division

Long division is a method used to divide larger numbers, breaking the division process into more manageable parts. It involves setting up the dividend inside a division bracket and the divisor outside, to the left.
In the provided exercise, we began by making the numbers more manageable by eliminating the decimal in the divisor. This process helped fit the divisor into the dividend repeatedly, making calculations straightforward. With our division, we aimed to figure out how many complete times the smaller number (or divisor) could "fit into" the larger number (the dividend).
This step-by-step method helped us accurately find the quotient, or the result of the division. Systematically, long division includes:

• Setting up the problem correctly
• Aligning numbers to ensure each part of the division is handled precisely
• Repeating this method until a meaningful result is achieved

Long division can seem daunting, but by following these structured steps, you ensure accuracy and gain a deeper understanding of the division process.

Decimals

Decimals represent fractions or parts of a whole number. In our exercise, we began with decimal numbers. Decimals use a dot to separate the whole numbers from the tenths, hundredths, and so forth.
The initial problem was to divide 2.99 by 0.5, which both represent less than whole numbers but in decimal forms. Decimals are helpful when precise values are needed, like in financial transactions or scientific measurements.
To simplify, we transformed the division by eliminating the decimal in the divisor. By multiplying both numbers in the division by 10, we managed to work with whole numbers, making our calculations easier without changing the value of our quotient. Handling decimals this way ensures precision while allowing us to perform operations using familiar techniques. Remember these key points about decimals:

• They break numbers into smaller measurement units beyond whole numbers
• Position after the decimal (tenths, hundredths, etc.) determines the specific value
• Decimal adjustments (like multiplication) simplify mathematical operations

Utilizing decimals wisely allows for efficiency and accuracy in real-world applications.

Rounding Numbers

Rounding numbers is a method to simplify numbers while keeping them close to their original value. This process is useful when precision is not paramount, and you need a quick, comprehensible result.
The existing solution required the division result to be rounded to the nearest hundredth. This means keeping two digits after the decimal point. If a division shows more numbers than needed, rounding helps achieve your preferred precision, simplifying the outcome.
In rounding rules, the focus is on the number just beyond the desired precision level. If this number is 5 or more, round up. If it's less than 5, maintain the current value. In our exercise, the operation already provided exactness required. However, confirming always safeguards accuracy and understanding.
Through rounding:

• You can simplify mathematical results
• It helps manage large data sets or complex calculations efficiently
• Clarifies numbers for practical use cases, such as financial estimates

Mastering rounding enhances numerical agility, especially when calculations gravitate towards approximations, helping to make sound, informed decisions.