Problem 54 - Prealgebra

Question

Perform the indicated operations. \(165 \div 2.20\)

Step-by-Step Solution

Answer

The result of dividing 165 by 2.20 is 75.

Step 1: Understand the Problem

You need to perform the division of two numbers: 165 and 2.20.

Step 2: Set Up for Long Division

Since we're dividing a whole number by a decimal, convert the division into one involving whole numbers by multiplying both the dividend (165) and the divisor (2.20) by 100. This removes the decimal: 165 becomes 16500 and 2.20 becomes 220.

Step 3: Perform the Long Division

Now divide 16500 by 220 using long division. Start by seeing how many times 220 fits into the leading part of 16500. Repeat this process until you've worked through the entire number.

Step 4: Simplify the Calculation

220 fits into 1650 (the leading digits) 7 times (220 x 7 = 1540). Subtract 1540 from 1650 to get 110. Combine this with the next digit '0' from 16500, making it 1100.

Step 5: Continue the Division

Next, see how many times 220 fits into 1100. It fits 5 times (220 x 5 = 1100) exactly. Hence, 16500 divided by 220 results in 75.

Step 6: Conclude the Long Division

As there are no remaining digits in 16500 to bring down, the division completes with no remainder.

Key Concepts

Long Division

Long division is a method used to divide large numbers easily. It breaks down a division problem into smaller, manageable parts. In this exercise, we performed long division by first removing the decimal from the divisor. We did this by multiplying both the dividend and the divisor by 100. This step is crucial because it simplifies the problem, converting it into a division of whole numbers.

Here's a simple way to understand long division:

• Write down the larger number (the dividend) inside the long division bracket.
• Place the smaller number (the divisor) outside the bracket.
• Determine how many times the divisor fits into the number formed by the initial digits of the dividend.
• Record this number above the dividend, multiply, then subtract from the dividend.
• Bring down the next digit from the dividend and repeat these steps until complete.

Long division may seem complex at first, but practicing it with numbers of varying lengths will build confidence and skill.

Decimal Conversion

Decimal conversion is key when dealing with division involving decimals. The main idea is to transform the problem into a division of whole numbers, which is easier to handle. By multiplying both the dividend and divisor by the same factor, the decimal point is removed.

Consider the original problem of dividing 165 by 2.20. By turning 2.20 into 220 through multiplication by 100, we changed the equation. It's essential to apply the same multiplication to the dividend, making it 16500 instead of 165. This process allows us to use straightforward long division.

Transformation of decimals to whole numbers helps simplify calculations, ensuring accuracy and making division problems less intimidating. This technique avoids dealing with messy decimal division directly, reducing possible errors.

Basic Arithmetic Operations

Basic arithmetic operations form the foundation of all math problems. They include addition, subtraction, multiplication, and division. The process of division, particularly long division, brings all these operations together.

In long division:

• Multiplication is used to find how many times the divisor fits into parts of the dividend.
• Subtraction occurs when removing the product of the divisor and quotient parts from the current portion of the dividend.
• Addition might be used when carrying digits down to form a new number to continue the division process.

Understanding these basic operations helps in efficiently solving division problems. Each operation plays a critical role in breaking down complex calculations into simple steps, aiding in finding the correct answer.