Problem 32 - Prealgebra

Question

Multiply. Round to nearest hundredth if necessary. $$10,302.25(0.06)\left(\frac{1}{4}\right)$$

Step-by-Step Solution

Answer

The product rounded to the nearest hundredth is 154.53.

Step 1: Break Down the Expression

The given expression is \[ 10,302.25 \times 0.06 \times \frac{1}{4} \]It involves three numbers and operations of multiplication. We'll solve this step-by-step starting from left to right.

Step 2: Multiply the First Two Numbers

Calculate the product of the first two numbers:\[ 10,302.25 \times 0.06 \]Executing this multiplication, we get:\[ 618.135 \]

Step 3: Multiply by the Fraction

Next, multiply the result from Step 2 by the fraction:\[ 618.135 \times \frac{1}{4} \]To multiply by \( \frac{1}{4} \), divide by 4:\[ \frac{618.135}{4} = 154.53375 \]

Step 4: Round to the Nearest Hundredth

Round the result from Step 3 to the nearest hundredth. The number is 154.53375, and looking at the thousandth digit (which is 3), determines that rounding down:\[ 154.53 \]

Key Concepts

Rounding to nearest hundredth

Rounding a number to the nearest hundredth involves looking at the third digit to the right of the decimal point. This position is called the thousandth place. If this digit is 5 or higher, we round the number up. If it is 4 or lower, we round the number down.
For example, in our case, the number 154.53375 requires rounding. Look at the thousandth digit, which is 3. Since 3 is less than 5, round down. This means the hundredth digit, which is 3, remains unchanged. As a result, the rounded number is 154.53.
Rounding is a common method used in many calculations to simplify numbers, especially when achieving precision is not critical or when simplifying communication is the goal.

Decimals multiplication

Multiplying decimal numbers is straightforward once you understand the concept of place value. When multiplying, don't worry about the decimal points initially. Just consider the numbers as whole numbers.
Perform the multiplication as usual. Once you have the product, count the total number of decimal places in the original numbers. This is how many decimal places the final product should have.

• For instance, in multiplying 10,302.25 by 0.06, treat them initially as 1030225 and 6.
• The multiplication yields 6181350. Now, adjust by placing the decimal point so that there are as many decimal places as in the original numbers combined, where 10,302.25 has two decimal places and 0.06 has two, totaling four in the result.
• Thus, 6181350 becomes 618.135, reflecting the proper decimal placement.

It's important to understand how many decimal places affect the product, ensuring accuracy in the result.

Fractions in multiplication

When involving fractions in multiplication, the process is usually straightforward. The rule is to multiply the numerator (top number) of the fraction by the whole number or the result you currently have, and then divide by the denominator (bottom number).
To multiply by a fraction, follow these steps:

• Multiply the current value by the numerator. In our exercise, this involved multiplying 618.135 by 1.
• Next, divide that result by the denominator, which is 4 in this case. So, take 618.135 and divide by 4, resulting in 154.53375.

This approach simplifies what could otherwise be a complex operation and makes handling fractions in multiplication accessible and clear.