Problem 26 - Algebra 1

Question

Find the quotient. $$\frac{-26}{-\frac{1}{2}}$$

Step-by-Step Solution

Answer

The quotient of \(-26\) and \(-\frac{1}{2}\) is \(52\).

Step 1: Convert Division to Multiplication

An important concept to remember is that dividing by a fraction is the same as multiplying by its reciprocal. In this case, instead of dividing by \(-\frac{1}{2}\), multiply by the reciprocal, which is \(-2\). The expression becomes: \(-26 \times -2\)

Step 2: Multiply the Numbers

The next step is to perform the multiplication. Recall that the product of two negative numbers is a positive number, so, \(-26 * -2 = 52\)

Step 3: Final Result

After performing the above step, the answer to the problem is obtained, so the quotient of \(-26\) and \(-\frac{1}{2}\) is \(52\)

Key Concepts

Division of Fractions

Dividing fractions might seem daunting at first, but once you understand the core idea, it becomes much more approachable. Division of fractions involves finding how many times one fraction fits into another.
When dividing fractions, you actually need to turn the division problem into a multiplication problem. This is a unique property of fractions that makes such calculations more convenient.

• The division of fractions requires converting the divisor into its reciprocal.
• A reciprocal of a number is basically flipping its numerator and denominator.

For example, if you are dividing by \(\frac{1}{2}\), the reciprocal would be \(2\). In essence, division by \(\frac{1}{2}\) becomes multiplication by \(2\).
So, the first step to solving any fractional division is simply changing the operation from division to multiplication, simplifying the process significantly.

Multiplication by Reciprocal

The term "reciprocal" might sound complex, but it's quite simple. The reciprocal of a fraction is found by flipping the fraction.
This means switching the numerator and the denominator. For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). When you multiply by a reciprocal, you are fundamentally changing the operation. Instead of division, you perform multiplication.

• To multiply by a reciprocal, ensure that you convert the fraction correctly.
• This action allows you to follow through usual multiplication rules, thus removing any division from the equation.

Always remember, switching operations from division to multiplication using reciprocals is not just a preference, but a crucial mechanism that simplifies many mathematical problems, just as demonstrated when dividing by \(\frac{1}{2}\).

Negative Numbers

When working with negative numbers, it's important to remember the specific rules that govern multiplication and division involving negatives.

• For multiplication and division, when you have two negative values, the result is always a positive number.
• This is because two negatives cancel each other out.

Remember, for a straightforward multiplication like \(-26 \times -2\), both negatives result in a positive \(52\).
Keeping these rules in mind helps prevent simple mistakes and ensures accuracy in handling expressions that involve negative numbers.

Quotient Calculation

Calculating a quotient involves understanding the relationship between division and multiplication. In mathematics, the quotient refers to the result obtained after division is performed.

• In the context of fractions, the quotient can be derived by multiplying by the reciprocal instead of direct division.
• This technique helps sidestep complexities usually linked with division, especially with fractions.

For example, the expression we are solving, \(\frac{-26}{-\frac{1}{2}}\), uses this method to calculate the quotient by first flipping the divisor fraction and then multiplying. This approach yields the quotient of \(52\), illustrating the advantage of converting division problems into multiplication tasks using reciprocals.