Problem 20 - Calculate With Confidence

Question

Solve the following. Carry division to the hundredths place where necessary. $$15 \times 2.08=$$

Step-by-Step Solution

Answer

15 times 2.08 equals 31.20.

Step 1: Setting up the Multiplication

To calculate \(15 \times 2.08\), we will first ignore the decimal point and treat it as \(15 \times 208\). This will make it easier to handle the multiplication.

Step 2: Multiplying without the Decimal

Now, multiply 15 by 208:\[15 \times 208 = 3120\]

Step 3: Adjust for Decimal Point

Since the original problem was \(15 \times 2.08\) and 2.08 has two decimal places, place the decimal point two places from the right in the result 3120.

Step 4: Find the Final Answer

By placing the decimal point two places from the right in 3120, we get:\[31.20\] Therefore, \(15 \times 2.08 = 31.20\).

Key Concepts

Understanding Decimal Places

Decimal places refer to the number of digits that appear after the decimal point in a number. These are crucial in determining the precision of a number. For instance, in the number 2.08, there are two decimal places, the "0" and the "8." Decimal places matter in multiplication as well because they influence where the decimal point will be placed in your final answer.

When multiplying decimals, count all the decimal places in the numbers being multiplied. After completing the multiplication without considering the decimals, you use this count to correctly place the decimal point in your final result. This ensures that the result maintains the correct level of precision. For example, if both numbers you're multiplying have one decimal place, the result should have two decimal places.

Understanding Multiplication Steps

Multiplication, especially involving decimals, can initially be confusing, but becomes simple when broken down into clear steps. Let's consider the multiplication performed in the exercise.

• Step 1: Ignore the Decimal: Start by ignoring the decimal in the number you are multiplying, treating it like a simple integer operation. For example, multiply 15 by 2.08 by first treating 2.08 as 208, giving you an easier multiplication task like 15 × 208.
• Step 2: Perform the Multiplication: Next, carry out the multiplication just as you would with whole numbers. The exercise shows that multiplying 15 by 208 gives 3120.
• Step 3: Reintroduce the Decimal Point: Finally, adjust your result by placing the decimal point back, shifting according to the total count of decimal places of the original decimals. Here since 2.08 has two decimal places, the calculation involves moving the decimal in 3120 two places to the left, ending at 31.20.

Breaking tasks into smaller, manageable steps often makes difficult operations much easier.

Exploring Numerical Operations

Numerical operations form the backbone of problem-solving in mathematics. Multiplication, one of these key operations, is essential when dealing with decimals. The important thing to note when dealing with any numerical operations, including decimals, is maintaining the integrity of the values involved. This you achieve by accurately managing the decimal places. Another critical aspect is understanding that numerical operations apply rules consistently. Each operation with numbers, whether it's addition, subtraction, multiplication, or division, follows a specific set of rules. For multiplication involving decimals, the rule about decimal places ensures that the multiplication reflects the decimals' order. For example, when you multiply numbers with decimals, the rule about counting the total number of decimal places helps to find where the decimal should be placed in the final answer. This practice ensures you maintain precision, reinforcing the reliability of your mathematical operations, ultimately leading to accurate results.