Problem 285 - Prealgebra

Question

In the following exercises, multiply. $$348(705)$$

Step-by-Step Solution

Answer

245940

Step 1: Set Up the Multiplication

Write the numbers in a vertical format to prepare for long multiplication. The number 348 should be on top and 705 should be directly below it, aligned by their place values: \[ \begin{array}{r} 348 \ \times 705 \ \hline \end{array}\]

Step 2: Multiply by the Units Place

Multiply 348 by the 5 in the units place of 705: \[ 348 \ \times 5 \ \hline 1740\]

Step 3: Multiply by the Tens Place

Multiply 348 by the 0 in the tens place of 705. As 0 multiplied by any number is 0, the intermediate product is 0. Shift this result one place to the left: \[ 348 \ \times 0 \ \hline 0000 \]

Step 4: Multiply by the Hundreds Place

Multiply 348 by the 7 in the hundreds place of 705. Shift this result two places to the left: \[ 348 \ \times 7 \ \hline 243600 \]

Step 5: Add the Intermediate Products

Add the results from Step 2, Step 3, and Step 4 to get the final product: \[ 1740 \ +00000 \ +243600 \ \hline 245940 \]

Key Concepts

Multiplication Steps

Understanding long multiplication involves following systematic steps. Let's break it down to grasp the process fully:

First, we need to arrange the numbers vertically, ensuring each digit is aligned by its place value. This setup is crucial for organizing our work.

For our example, place 348 on top and 705 beneath it. Align the units, tens, and hundreds correctly:
\[\begin{array}{r}348 \times 705 \hline\end{array}\]
Now, we multiply the numbers digit by digit, starting from the rightmost digit.

This methodical approach can eventually tackle even more complex multiplications.

Place Value Alignment

Proper place value alignment is fundamental in long multiplication.

Let's see why it's important: when writing numbers one below the other, align the digits based on their value (units below units, tens below tens, and so forth).

• For instance, in our example, place 348 on top.
• Underneath, align 705 such that the units digit (5) is directly under the 8 of 348, the tens place (0) under the 4, and the hundreds place (7) under the 3.

This alignment ensures that when we multiply and add, each digit is in its correct column, making it easier to keep track of values.

Remember, maintaining strict alignment prevents errors and aids in correctly shifting intermediate products.

Intermediate Products

The role of intermediate products is crucial in the long multiplication process.

We generate these products by multiplying each digit of the bottom number by each digit of the top number.

For example:

• Multiplying 348 by 5 (units place of 705) gives 1740.
• Next, we multiply 348 by 0 (tens place of 705). Since any number times 0 is 0, the product is 0. We write 0000, shifting one place to the left to account for the tens place.
• Multiplying 348 by 7 (hundreds place of 705), the product is 243600. Shift this two places to the left because it’s the hundreds place.

Understanding intermediate products and shifting them appropriately ensures accuracy.

Finally, we sum these intermediate products, giving us the final result, which is 245940 in this scenario.

This breakdown clarifies how intermediate steps collectively lead to the final multiplication result.