Problem 79 - Prealgebra

Question

To understand how to multiply decimals, we need to understand multiplication with whole numbers, fractions, and mixed numbers. The following problems review these concepts. $$2 \frac{1}{10} \times \frac{7}{100}$$

Step-by-Step Solution

Answer

\(2 \frac{1}{10} \times \frac{7}{100} = \frac{147}{1000}\)

Step 1: Convert Mixed Number to Improper Fraction

Start by converting the mixed number \(2 \frac{1}{10}\) into an improper fraction. Multiply the whole number (2) by the denominator (10) and add the numerator (1) to get the new numerator. So, \(2 \times 10 + 1 = 21\). The fraction becomes \(\frac{21}{10}\).

Step 2: Multiply the Fractions

Multiply the two fractions: \(\frac{21}{10} \times \frac{7}{100}\). To do this, multiply the numerators and multiply the denominators separately.The numerator is \(21 \times 7 = 147\) and the denominator is \(10 \times 100 = 1000\). Thus, the product is \(\frac{147}{1000}\).

Step 3: Simplify the Fraction

The fraction \(\frac{147}{1000}\) cannot be simplified further because 147 and 1000 have no common factors other than 1. Therefore, \(\frac{147}{1000}\) is already in its simplest form.

Key Concepts

Multiplying Fractions

Multiplying fractions is easier than it seems once you understand the steps. The main rule to remember is that when multiplying fractions, you need to multiply the numerators together and the denominators together. Let’s break it down further.

• Numerator: The top part of a fraction. In our problem, the numerators are 21 and 7.
• Denominator: The bottom part of a fraction. In the example, the denominators are 10 and 100.

To find the product of two fractions, \ \( \frac{a}{b} \times \frac{c}{d} \ \), multiply \( a \times c \) to get the new numerator and \( b \times d \) to get the new denominator.

In the given exercise, 21 (from the improper fraction derived from the mixed number) is multiplied by 7, giving 147 for the numerator. Likewise, 10 is multiplied by 100, yielding a denominator of 1000. So, \( \frac{21}{10} \times \frac{7}{100} = \frac{147}{1000} \).

This is now the complete product of the fractions, and the last step is simplification, if needed. In this case, since \( \frac{147}{1000} \) was found to already be in its simplest form, no further steps are required.

Mixed Numbers

A mixed number consists of two parts: a whole number and a fraction. When faced with operations involving mixed numbers, it’s helpful to convert them into improper fractions first. The reason is that improper fractions are easier to handle in multiplication and division operations. Let’s see how this conversion is done.

• Whole Number: The number before the fraction, such as 2 in the mixed number \( 2 \frac{1}{10} \).
• Fraction: The part of the mixed number that comes after the whole number, in this case, \( \frac{1}{10} \).

To convert a mixed number into an improper fraction:

• Multiply the whole number by the fraction's denominator.
• Add the numerator to the result.
• The total becomes the new numerator with the original denominator remaining the same.

For \( 2 \frac{1}{10} \), multiply 2 by 10 (denominator) to get \( 20 \), then add 1 (numerator) to get \( 21 \). The resulting improper fraction is \( \frac{21}{10} \). Using the improper fraction simplifies calculations and ensures accuracy when multiplying with other fractions.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This might seem strange at first, but they are quite useful, especially in multiplication and division.

An improper fraction like \( \frac{21}{10} \) is what you get when you convert a mixed number. To be comfortable working with improper fractions, remember these key points:

• An improper fraction is easy to use in calculations. You multiply or divide as you do with regular fractions.
• After operations, you might need to convert the result back to a mixed number for interpretation, especially with larger numbers.

To multiply an improper fraction by another fraction, such as in the exercise, simply follow the basic fraction multiplication rules. Here, \( \frac{21}{10} \times \frac{7}{100} \) leads directly to a product of \( \frac{147}{1000} \), demonstrating the straightforward nature of using improper fractions.

Once you master improper fractions, handling mixed numbers becomes easier, allowing for seamless calculations in various math problems.