Problem 16 - Prealgebra

Question

Find each of the following products. $$7(0.005)$$

Step-by-Step Solution

Answer

The product of 7 and 0.005 is 0.035.

Step 1: Understand the Problem

We need to find the product of the number 7 and the decimal 0.005.

Step 2: Convert the Decimal to Fraction

First, express 0.005 as a fraction. Since 0.005 is equal to 5 thousandths, it can be written as \( \frac{5}{1000} \).

Step 3: Multiply the Integer by the Fraction

Multiply 7 by \( \frac{5}{1000} \): \[ 7 \times \frac{5}{1000} = \frac{7 \times 5}{1000} \] which equals \( \frac{35}{1000} \).

Step 4: Simplify the Fraction

Now, simplify \( \frac{35}{1000} \). Both 35 and 1000 can be divided by 5: \( \frac{35 \div 5}{1000 \div 5} = \frac{7}{200} \).

Step 5: Convert the Fraction Back to Decimal

To convert \( \frac{7}{200} \) to a decimal, divide 7 by 200: \( 7 \div 200 = 0.035 \).

Step 6: Write the Final Answer

Thus, the product of 7 and 0.005 is 0.035.

Key Concepts

Converting Decimals to Fractions

Converting decimals to fractions is a handy skill, especially when performing calculations or solving math problems. It allows you to work with whole numbers instead of decimals, which can be easier in many cases. To convert a decimal into a fraction, follow these steps:

• Identify the place value of the last digit in the decimal. For example, in 0.005, the digit 5 is in the thousandths place, which means it represents 5/1000.
• Write the decimal as a fraction with the corresponding place value as the denominator. So, 0.005 becomes \( \frac{5}{1000} \).
• Ensure the fraction represents the same value as the decimal by placing the decimal's integer part as the numerator.

Converting decimals to fractions boils down to expressing the numbers in terms of whole numbers and place values. This makes operations like multiplication simpler later on.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). Here's how to simplify fractions:

• Identify the GCD of the numerator and denominator. For instance, in the fraction \( \frac{35}{1000} \), the GCD of 35 and 1000 is 5.
• Divide both the numerator and the denominator by their GCD. In this case, \( \frac{35}{5} = 7 \) and \( \frac{1000}{5} = 200 \). Thus, \( \frac{35}{1000} \) simplifies to \( \frac{7}{200} \).
• Check your result to ensure that it's in its simplest form, meaning there are no common factors left except 1.

Simplifying fractions not only makes them easier to work with but also makes them easier to understand. It's a crucial part of handling fractions because it reduces complexity in further calculations.

Decimal Multiplication

Decimal multiplication involves multiplying numbers where one or both of the numbers are decimals. It's a straightforward process if you handle the decimals carefully:

• Ignore the decimal points and multiply the numbers as if they were whole numbers. For instance, when multiplying 7 by 0.005, consider it as 7 times 5 before placing decimals, which equals 35.
• Count the total number of decimal places in the numbers being multiplied. Here, 0.005 has three decimal places.
• Place the decimal point in the product such that the number of decimal places in the product matches the total number of decimal places counted from the numbers being multiplied. Thus, placing the decimal in 35 gives 0.035, aligning with three decimal places.

Decimal multiplication is mostly about adjusting the decimal point's position correctly after performing the multiplication of whole numbers. This step ensures accuracy and precision in the final result.