Problem 81 - 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. $$305(436)$$

Step-by-Step Solution

Answer

The product of 305 and 436 is 132,980.

Step 1: Arrange the Numbers for Multiplication

Write the number 305 on top and 436 below it, aligning the digits by place value (units under units, tens under tens, etc.).

Step 2: Multiply by Each Digit of the Bottom Number

Start with the rightmost digit of the bottom number (6 in 436). Multiply 305 by 6. Write down the result below the line. Repeat this for each digit of 436.

Step 3: Adding Resultant Rows

For each digit multiplied, shift the results one place to the left. Add up all the resulting products. For example, after multiplying by 3 (from 436), shift the result one place to the left before adding to the column.

Step 4: Calculate the Final Sum

Review and ensure all rows are correctly aligned by place value. Add up all the rows to find the final product of the multiplication.

Key Concepts

Whole Numbers Multiplication

Multiplying whole numbers is a fundamental math skill that is essential for understanding more complex operations. When multiplying whole numbers, like 305 and 436, the process involves using place value and aligning digits appropriately.

• Start by writing the larger number, 305, at the top, and the smaller, 436, below it. Make sure the digits are aligned by place value.
• Multiply each digit of the bottom number, starting from the rightmost digit (6 in the case of 436) with the top number, 305.
• For each multiplication, write the result below, keeping in mind to align it correctly with the corresponding columns.

The entire calculation involves a few iterations of multiplication followed by addition of these products. This helps in obtaining the final sum or product of the two numbers.

Keep practicing to become familiar with this process. It lays a solid foundation for learning to multiply decimals and more advanced number types.

Fractions Multiplication

Multiplying fractions involves a straightforward procedure, but it's important to understand each step clearly. Unlike whole numbers, where we multiply by using place values, fractions require multiplying the numerators and denominators separately.

• Take the first fraction and multiply its numerator by the numerator of the second fraction. This gives the new numerator for your product.
• Similarly, multiply the denominator of the first fraction by the denominator of the second. This gives the new denominator.
• For example, multiplying \( \frac{2}{3} \) by \( \frac{4}{5} \) involves multiplying 2 by 4 for the numerator and 3 by 5 for the denominator, resulting in \( \frac{8}{15} \).

Always simplify the resulting fraction if possible by finding the greatest common divisor. This reduces the fraction to its simplest form and is considered best practice, making calculations easier.

Mixed Numbers Multiplication

Mixed numbers are combinations of whole numbers and fractions. When multiplying mixed numbers, it's easiest to convert them into improper fractions first.

• Start by changing the mixed number into an improper fraction. Multiply the whole number part by the fraction's denominator and add the numerator. This becomes the new numerator over the original denominator.
• For example, for \(2\frac{1}{3}\), multiply 2 by 3, then add 1, resulting in \(\frac{7}{3}\).
• Perform the multiplication of improper fractions as you would with normal fractions: multiply the numerators together and the denominators together.

Once you have the multiplied result in the form of a fraction, you can convert it back if necessary into a mixed number by dividing the numerator by the denominator. The quotient becomes the whole number, and the remainder is the new numerator over the original denominator.

This method ensures accuracy and is useful for more challenging calculations involving mixed numbers.