Problem 83 - Prealgebra

Question

To understand how to multiply decimals, we need to understand multiplication with whole numbers, fractions, and mixed numbers. The following problems review these concepts. $$5(420+3)$$

Step-by-Step Solution

Answer

The result of the expression \(5(420 + 3)\) is \(2115\).

Step 1: Understand the Expression

The expression given is \(5(420 + 3)\). This is a multiplication of a number with a sum. To simplify, you'll perform the operation inside the parenthesis first.

Step 2: Evaluate the Parenthesis

Inside the parenthesis, we have \(420 + 3\). Calculate this addition: \(420 + 3 = 423\). Now, the expression becomes \(5 \times 423\).

Step 3: Multiply the Numbers

Now, we need to multiply \(5\) and \(423\). Start by breaking \(423\) into hundreds, tens, and units: \(423 = 400 + 20 + 3\).Calculate each:1. \(5 \times 400 = 2000\)2. \(5 \times 20 = 100\)3. \(5 \times 3 = 15\)Finally, sum these results: \(2000 + 100 + 15 = 2115\).

Step 4: Write the Final Solution

The product of \(5(420 + 3)\) is \(2115\).

Key Concepts

Whole Numbers

Whole numbers are the simplest form of numbers you encounter. They include zero and all positive numbers without fractions or decimals. Multiplying whole numbers, like in the exercise with the numbers 5 and 423, follows a straightforward process of repeated addition. For instance, when multiplying 5 by 423, you essentially add 5 to itself 423 times. Given their simplicity, operations with whole numbers serve as a foundational step before dealing with more complex number types.

• Whole numbers: 0, 1, 2, 3, ...
• In the exercise, multipling 5 by 423 requires understanding that 5 is added repeatedly.

Breaking down 423 into smaller components such as hundreds, tens, and units allows for easier handling of large numbers in multiplication by addressing each part separately before summing up the results.

Fractions

Fractions represent parts of a whole and are used in multiplication to find parts of quantities. While the original exercise didn't explicitly involve fractions, understanding them complements the multiplication of decimals. When multiplying fractions, you multiply the numerators together and the denominators together. For example, \[\text{If multiplying } \frac{1}{2} \times \frac{3}{4}, \text{ compute } \frac{1 \times 3}{2 \times 4} = \frac{3}{8}. \]Fractions are everywhere, even in operations with mixed numbers. They extend the understanding of multiplication beyond whole numbers and decimals.

• Fractions are parts of whole numbers, like \( \frac{1}{2} \).
• Multiplying fractions requires handling the numerators and denominators separately.

Mixed Numbers

Mixed numbers consist of a whole number and a fraction, such as 3 1/2. This concept combines the multiplication skills from both whole numbers and fractions. When multiplying mixed numbers, you first convert them to improper fractions. This conversion allows for operations similar to simple fractions. For example,

• Convert 3 1/2 to an improper fraction: 3 1/2 = \( \frac{7}{2} \)
• Multiply with another mixed number converted to a fraction: 2 1/3 becomes \( \frac{7}{3} \).
• Perform the multiplication: \( \frac{7}{2} \times \frac{7}{3} = \frac{49}{6} \).

Finally, convert back to a mixed number: \( \frac{49}{6} = 8 \frac{1}{6} \). Mastering these conversions ensures accuracy in calculating real-world scenarios involving mixed quantities.