Problem 29 - Prealgebra

Question

Multiply each of the following. Be sure all answers are written in lowest terms. $$\frac{2}{5} \cdot 20$$

Step-by-Step Solution

Answer

The product is 8.

Step 1: Understand the Problem

We need to multiply a fraction, \( \frac{2}{5} \), by a whole number, 20. Our goal is to find the product and express it in its simplest form.

Step 2: Write Whole Number as a Fraction

Convert the whole number into a fraction. The number 20 can be expressed as \( \frac{20}{1} \). This helps us multiply it with the given fraction.

Step 3: Multiply the Fractions

To multiply fractions, multiply their numerators and denominators. Here, \( \frac{2}{5} \times \frac{20}{1} = \frac{2 \times 20}{5 \times 1} = \frac{40}{5} \).

Step 4: Simplify the Fraction

Simplify \( \frac{40}{5} \). Divide 40 by 5 to get the simplest form. \( \frac{40}{5} = 8 \).

Step 5: Conclusion

The product of \( \frac{2}{5} \) and 20 in its simplest form is 8.

Key Concepts

Simplifying Fractions

Understanding how to simplify fractions is an essential skill in mathematics, making complex calculations much easier to handle. Simplifying a fraction involves reducing it to its lowest terms so that the numerator and denominator share no common factors other than 1. For example, once we reach the fraction \( \frac{40}{5} \), we can simplify it by recognizing that both numbers can be divided by 5, which is their greatest common divisor (GCD).

• The numerator here is 40, which can be divided by 5 as \( 40 \div 5 = 8 \).
• The denominator is 5, and dividing 5 by itself gives us 1, like \( 5 \div 5 = 1 \).

Putting these together, \( \frac{40}{5} \) simplifies to 8, which is expressed as a whole number since any number divided by 1 remains the same.

Fraction Multiplication

When it comes to multiplying fractions, the process is straightforward. You multiply the numerators (top numbers) to find the new numerator and multiply the denominators (bottom numbers) to find the new denominator. This process ensures that the multiplication remains consistent with the properties of fractions. Let's consider the exercise:

• We have \( \frac{2}{5} \) and need to multiply it by \( \frac{20}{1} \).
• Multiply the numerators: 2 and 20, giving us \( 2 \times 20 = 40 \).
• Multiply the denominators: 5 and 1, giving us \( 5 \times 1 = 5 \).

Thus, \( \frac{2}{5} \times \frac{20}{1} = \frac{40}{5} \). The product is \( \frac{40}{5} \), which we can later simplify.

Whole Numbers as Fractions

Any whole number can be transformed into a fraction, which often simplifies the process of operations involving fractions, like multiplication. To express a whole number as a fraction, you can place the number over 1. For instance, the whole number 20 can be represented as the fraction \( \frac{20}{1} \). This step makes the arithmetic with fractions intuitive and follows the same rules as working with regular fractions.

• It allows us to use the fraction multiplication rule by multiplying across numerators and denominators efficiently.
• This approach unifies the method, whether you're multiplying two fractions or a fraction and a whole number.

In summary, converting whole numbers to fractions helps maintain consistency and simplifies computations involving different types of numbers.