Problem 81 - Prealgebra

Question

Multiply. $$0.25(74)$$

Step-by-Step Solution

Answer

0.25 times 74 equals 18.5.

Step 1: Understand the Problem

We are given the expression \(0.25 \times 74\). This means we need to calculate the product of 0.25 and 74.

Step 2: Recall the Concept of Multiplication by Decimals

When multiplying by 0.25, which is the same as \(\frac{1}{4}\), we are essentially finding one fourth of the number.

Step 3: Multiply Directly

Multiply 74 by 0.25 directly:\[0.25 \times 74 = 18.5\]To perform this multiplication, think of it as multiplying 74 by 25 and then dividing by 100.

Key Concepts

Decimal Operations

Working with decimals can initially seem daunting, but it's surprisingly straightforward once you get the hang of it. Decimals are a way to represent numbers that aren't whole, using a decimal point to divide the whole from the fractional parts. When performing operations with decimals, such as addition, subtraction, and multiplication, their placement and significance differ slightly compared to whole numbers.

When multiplying decimals, the process involves simply multiplying the numbers as if they were whole numbers and then applying the decimal point later. Here's a quick guide:

• Ignore the decimal points and multiply the numbers as you would normally.
• After multiplying, count the total number of decimal places from both factors in the original multiplication problem.
• Place the decimal point in your answer, moving from the right to the left, by the total number of decimal places counted in the previous step.

This method ensures you maintain accuracy throughout the operation. You can clearly see these principles in the calculation of the product stepping from 0.25 and 74 to the final product, 18.5.

Multiplication Strategies

When dealing with multiplication, especially with decimals, having robust strategies can simplify the process significantly. Let's break it down with useful strategies for efficient multiplication.

The first approach involves multiplying by recognizing simpler equivalent forms. For instance, multiplying by 0.25 can be likened to multiplying by \(\frac{1}{4}\). Therefore, finding one quarter of a number might be easier for some, and then multiplying by that smaller number.

• Convert tough decimal multipliers into simpler fractions for easy calculations.
• Utilize shortcuts by breaking apart the number. For instance, in our example calculation, seeing 0.25 as 25/100 provided a clear approach.
• Estimate the product to ensure results are reasonable. Recognize that 0.25 is a quarter, so multiplying by 74 should yield something less than 74.

An understanding of the number's fractions can help determine an initial rough estimate, verifying the final answer's plausibility. With these strategies, multiplying by decimals becomes an intuitive task.

Fraction to Decimal Conversion

Understanding the foundation of decimals requires fluency in converting between fractions and decimals. This skill is particularly useful because it provides additional insight into what the numbers represent and can make calculations more intuitive.

Decimals can often be viewed more straightforwardly as fractions. For example, 0.25 is equivalent to \(\frac{25}{100}\), which simplifies to \(\frac{1}{4}\). Here's how you can make conversions:

• To convert a fraction to a decimal, divide the numerator by the denominator.
• Recognize common equivalences, such as \(\frac{1}{4}\) being equal to 0.25, or \(\frac{1}{2}\) equal to 0.5.
• Ensure the conversion keeps the logic of the size of the number. Adjust the decimal places to maintain the correct order of magnitude in your calculations.

By mastering the conversion process, you'll better understand how decimals reflect proportions and measurements, enriching your capability to handle decimal operations softly and efficiently.