Problem 34 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$-8 \div \frac{4}{5}$$

Step-by-Step Solution

Answer

The quotient is -10.

Step 1: Understand the Division Problem

The problem presented is a division operation involving a negative integer and a fraction: \(-8 \div \frac{4}{5}\). We need to find out how many times \(\frac{4}{5}\) fits into \(-8\).

Step 2: Convert the Division into Multiplication

To divide by a fraction, you multiply by its reciprocal. The reciprocal of \(\frac{4}{5}\) is \(\frac{5}{4}\). Thus, \(-8 \div \frac{4}{5}\) becomes \(-8 \times \frac{5}{4}\).

Step 3: Simplify the Multiplication

Multiply \(-8\) by \(\frac{5}{4}\). This results in \(-8 \times \frac{5}{4} = \frac{-8 \times 5}{4} = \frac{-40}{4}\).

Step 4: Calculate the Final Quotient

Simplify \(\frac{-40}{4}\) by dividing the numerator by the denominator. \(-40 \div 4 = -10\). Thus, the quotient is \(-10\).

Key Concepts

Division by Fractions

When dividing by a fraction, the process may initially seem a bit confusing. But it becomes quite simple when you know the trick: to divide by a fraction, you multiply by its reciprocal. This means that instead of tackling the division head-on, you flip the fraction and change the division to multiplication.

For example, if we need to compute \(-8 \div \frac{4}{5}\), we will first invert the fraction \(\frac{4}{5}\) to its reciprocal, which is \(\frac{5}{4}\). After this step, we simply change the division into multiplication, turning the problem into \(-8 \times \frac{5}{4}\). This change simplifies our task significantly and allows us to find the answer more comfortably. So anytime you encounter a division by fractions, remember to multiply by the reciprocal!

Negative Numbers

Negative numbers can appear challenging, but they play a crucial role in mathematics. They are numbers less than zero and are usually represented with a minus sign \(-\). Understanding how they work in operations like addition, subtraction, multiplication, and division is essential.

When dealing with negative numbers in division, remember these simple rules:

• A negative number divided by a positive number results in a negative quotient.
• A negative number divided by another negative number results in a positive quotient.
• A positive number divided by a negative number results in a negative quotient.

Applying these principles to our example of \(-8 \div \frac{4}{5}\), we see that since \(-8\) is negative and \(\frac{4}{5}\) is positive, our quotient will be negative. This forms the basis of why the answer becomes \(-10\).

Reciprocals

Reciprocals are an essential concept when dealing with fractions, especially in division. A reciprocal of a number is simply exchanging the numerator (top number) and the denominator (bottom number) of a fraction. By doing this, you effectively switch their positions.

For example, the reciprocal of \(\frac{4}{5}\) is \(\frac{5}{4}\). It's important to remember that any number multiplied by its reciprocal equals 1. This property makes reciprocals valuable in division because it lets you transform a division problem into a multiplication problem.

Using reciprocals streamlines calculations and is useful, particularly in problems where direct division would otherwise be quite complex. Always keep the reciprocal trick handy whenever you encounter fractions in division!

Multiplying Fractions

Once the division problem is converted into multiplication by using the reciprocal, the next step is to multiply the fractions. This step is straightforward when you know the basic rules.

To multiply fractions, simply multiply the numerators across to get the new numerator and the denominators across to get the new denominator. For a whole number, such as \(-8\), think of it as \(\frac{-8}{1}\).

• Multiply the numerators: \(-8 \times 5 = -40\).
• Multiply the denominators: \(1 \times 4 = 4\).
• Combine to get \(\frac{-40}{4}\).

Now, simplify if necessary. In this case, dividing \(-40\) by 4 gives us \(-10\). That’s how we get the final answer in our division problem.