Problem 11 - Algebra 1

Question

Find the quotient. $$10 \div\left(-\frac{1}{5}\right)$$

Step-by-Step Solution

Answer

-50

Step 1: Identify the dividend and the divisor

Here, the dividend (the number to be divided) is 10 and the divisor (the number by which we are dividing) is \(-\frac{1}{5}\).

Step 2: Recognize that dividing by a fraction is the same as multiplying by its reciprocal

Instead of dividing by the fraction \(-\frac{1}{5}\), we can multiply by its reciprocal, which is \(-5\). This converts the operation to multiplication, making it easier to carry out.

Step 3: Perform multiplication

Multiply 10 by \(-5\) to get \(-50\). This is the quotient when 10 is divided by \(-\frac{1}{5}\).

Key Concepts

Reciprocal of a Fraction

When dealing with fractions in division, it's important to understand the concept of a reciprocal. Each fraction has a unique reciprocal. Essentially, the reciprocal of a fraction is created by swapping its numerator and denominator. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). This is a handy tool for simplifying division problems.

In the given exercise, instead of dividing by \(-\frac{1}{5}\), we need its reciprocal: \(-5\). By converting the division problem into a multiplication problem, it usually becomes easier to solve. Remember, finding a reciprocal is a great strategy to change division into multiplication.

Multiplying Integers

Multiplying integers might seem straightforward, but it's essential to be mindful of signs. When multiplying two integers, the rule is:

• If the signs are the same, the result is positive.
• If the signs are different, the result is negative.

For instance, multiplying \(10\) by \(-5\) yields \(-50\).

In any multiplication involving integers, always pay attention to the signs. This will help ensure the accuracy of your result. Understanding these rules will allow you to tackle multiplication confidently and accurately.

Negatives in Division

Managing negatives in division can look tricky but breaking it down helps. When dividing numbers, similar to multiplication, consider the sign rules:

• Dividing a positive by a negative results in a negative quotient.
• Dividing a negative by a positive also results in a negative quotient.
• Dividing two negatives results in a positive quotient.

In the exercise, the division of \(10\) by \(-\frac{1}{5}\) is resolved by multiplying with \(-5\), leading to \(-50\).

Keeping these rules in mind when you encounter negatives in division ensures you arrive at the correct solution. Understanding negatives and their impact are crucial for solving such problems effectively.