Problem 9 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 0.9 \\ \times 0.88 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The product of 0.9 and 0.88 is 0.792.

Step 1: Convert Decimals to Fractions

We start by converting the decimals to fractions. The number 0.9 can be written as \( \frac{9}{10} \) and 0.88 as \( \frac{88}{100} \).

Step 2: Multiply the Fractions

We multiply the fractions: \( \frac{9}{10} \times \frac{88}{100} \). This yields \( \frac{792}{1000} \).

Step 3: Convert the Product to a Decimal

Converting \( \frac{792}{1000} \) to a decimal, we shift the decimal point three places to the left, resulting in 0.792.

Key Concepts

Fractions to Decimals

Understanding how to convert fractions to decimals is key when working with numbers in both forms. Fraction conversion is a straightforward process, allowing you to switch from a ratio of two numbers to a simple decimal number easily.
A fraction like \( \frac{9}{10} \) represents nine parts out of ten. To convert this to a decimal, we divide the numerator (9) by the denominator (10). This results in 0.9. Similarly, the fraction \( \frac{88}{100} \) can be converted to a decimal. Divide the numerator (88) by the denominator (100) to get 0.88.

• Write the fraction \(\frac{numerator}{denominator}\).
• Divide the numerator by the denominator.
• The result is your decimal number.

Understanding these conversions helps simplify complex expressions by representing fractions in a decimal format, offering a different perspective and easing calculations.

Multiplying Fractions

Multiplying fractions is an essential skill, especially when dealing with calculations involving decimals and percentages. It's simple: multiply the numerators together and the denominators together.
Let's illustrate this by multiplying \( \frac{9}{10} \) and \( \frac{88}{100} \). You multiply 9 by 88, resulting in 792, and 10 by 100, resulting in 1000. Hence, \( \frac{9}{10} \times \frac{88}{100} = \frac{792}{1000} \).

• Multiply the numerators across: 9 times 88 = 792.
• Multiply the denominators across: 10 times 100 = 1000.
• Your product is the fraction \(\frac{792}{1000}\).

By systematically multiplying across, you can handle any fraction multiplication confidently, laying the groundwork for converting your results back to a decimal.

Converting Decimals to Fractions

Changing decimals back into fractions is just as important as going the other way. It borders on fraction basics but offers a great chance to understand number representation.
To convert 0.792 into a fraction, recognize it as 792 thousandths, written as \( \frac{792}{1000} \). You can simplify this fraction if needed, but in this case, it's already fittingly reduced for the decimal's precision.

• Note the decimal’s place value (tenths, hundredths, etc.). For 0.792, the place value is thousandths.
• Express as a fraction by placing over the respective place value: \(\frac{792}{1000}\).
• Simplify the fraction if necessary.

This conversion is invaluable for interpreting decimals in fractions, useful for both mathematical precision and clarity. It complements problems where you interchange decimal and fractional representations seamlessly.