Problem 15 - Prealgebra

Question

Multiply the decimals. (9.9)(6.7)

Step-by-Step Solution

Answer

The product of 9.9 and 6.7 is 66.27.

Step 1: Understanding the Problem

We need to multiply the two decimals: 9.9 and 6.7.

Step 2: Removing Decimal Points

First, remove the decimal points to make the multiplication easier. Convert 9.9 to 99 by shifting the decimal one place to the right. Similarly, convert 6.7 to 67.

Step 3: Performing the Multiplication

Now, multiply the whole numbers as if there were no decimals: calculate 99 times 67.

Step 4: Calculating 67 times 9

First, multiply 67 by 9. We find that 67 times 9 equals 603.

Step 5: Calculating 67 times 90

Next, multiply 67 by 90. We break 90 into 9 times 10, so this means 603 (from step 4) times 10, which equals 6030.

Step 6: Adding Products

Now, add up the results from the multiplications. We add 603 and 6030 to get 6627.

Step 7: Restoring the Decimal Places

Because we initially moved the decimal one place to the right for each number, the combined effect is two places total. So, in the final answer, shift the decimal point two places to the left in 6627, giving us 66.27.

Key Concepts

Step-by-Step Solution

When dealing with decimal multiplication, following a step-by-step solution makes the process straightforward. Initially, identify the decimals to be multiplied; in this case, these are 9.9 and 6.7. The first step is to simplify the process by removing the decimal points.

• Convert 9.9 to 99 by shifting its decimal one place to the right.
• Similarly, convert 6.7 to 67 by moving the decimal point one place to the right.

Next, you treat these adjusted numbers as whole numbers. Perform the multiplication as you would with any whole numbers. Specifically, find the product of 99 multiplied by 67.

• Firstly, calculate 67 multiplied by 9, which gives you 603.
• Then, calculate 67 times 90 by multiplying 603 by 10, resulting in 6030.

Finally, combine these calculated products: 603 and 6030, to achieve the result of 6627.

Decimal Point Adjustment

Adjusting decimal points is an essential strategy in simplifying decimal multiplication. When you move the decimal points before multiplying, you shift them back in the final step. For our exercise:

• Initially, we shifted the decimal one place to the right in each number: from 9.9 to 99 and from 6.7 to 67.

This method of adjusting involves temporarily treating the decimals as whole numbers, minimizing complications during multiplication. Once the multiplication is complete and the product found, it is crucial to correct the decimal position. By shifting each original decimal one place, their combined effect equals two places. Hence, after obtaining the whole number result of 6627 from our multiplication, the final step is shifting the decimal point two places to the left. Thus, the end result is recalibrated correctly: 66.27.

Place Value in Multiplication

Understanding place value is key when multiplying decimals. Each digit in a number has a specific place value depending on its position. When decimals are involved, placement dictates their worth more intricately, affecting both interim calculations and the final result. In the case of (9.9)(6.7):

• Shifting the decimal points converts each decimal into whole numbers, considering them as units of tens or hundreds temporarily.
• By resolving to whole numbers, like 99 and 67 here, multiplication becomes straightforward because the place values behave predictably.

However, once the raw multiplication is done, remember the decimal's impact. In converted numbers, the dig: 603 and 6030. Once placed accurately back into their decimal form, 66.27 ensures accuracy respective to the original decimal place values and the overall measure of multiplying two decimals. Recognizing this vital aspect guarantees precision in future calculations.