Problem 39 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ \frac{-\frac{21}{2}}{7} $$

Step-by-Step Solution

Answer

The quotient is \(-\frac{3}{2}\).

Step 1: Rewrite Division as Multiplication

Firstly, remember that dividing by a number is the same as multiplying by its reciprocal. In this case, rewrite the division, \( -\frac{21}{2} \div 7 \), as multiplication: \( -\frac{21}{2} \times \frac{1}{7} \)

Step 2: Multiply the Fractions

Now you can proceed to multiply the fractions. Make sure to handle the negative sign correctly. This yields: \( -\frac{21}{2}\times\frac{1}{7} = -\frac{21}{2}\times\frac{1}{7} = -\frac{21}{14} \)

Step 3: Simplify the Fraction

Finally, simplify the resultant fraction by reducing to lowest terms. This requires finding the greatest common divisor (GCD) of the numerator and denominator. Here, 21 and 14 have GCD of 7, which gives: \( -\frac{21}{14} = -\frac{3}{2} \)

Key Concepts

Division of Fractions

When you encounter a problem involving the division of fractions, it's helpful to remember a crucial rule: dividing by a fraction is the same as multiplying by its reciprocal. This means if you have a division problem like \( \frac{a}{b} \div \frac{c}{d} \), you can rewrite it as \( \frac{a}{b} \times \frac{d}{c} \). The reciprocal of a fraction is simply switching its numerator and denominator.

• For example, the reciprocal of \( \frac{7}{1} \) is \( \frac{1}{7} \).
• This allows division problems to be tackled using multiplication rules.

In the exercise provided, you rewrite \( -\frac{21}{2} \div 7 \) as \( -\frac{21}{2} \times \frac{1}{7} \). This step is essential because working with multiplication is often simpler and more straightforward than division.

Multiplication of Fractions

Once you have set up the problem as a multiplication, the next step is straightforward. To multiply fractions, multiply the numerators together and the denominators together. This multiplies the top parts and the bottom parts of the fractions directly.

• For instance, if you have \( \frac{a}{b} \times \frac{c}{d} \), the result will be \( \frac{a \times c}{b \times d} \).

In the example, \( -\frac{21}{2} \times \frac{1}{7} \) leads to \( -\frac{21}{2 \times 7} \), which is \( -\frac{21}{14} \). Here, the negative sign is carried along with the multiplication, ensuring the solution remains accurate.

Simplifying Fractions

After multiplying the fractions, you often need to simplify the resulting fraction. Simplifying means reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than one.
Finding the greatest common divisor (GCD) of the numerator and the denominator is critical for this process.

• In the solution, for \( -\frac{21}{14} \), both 21 and 14 can be divided by their GCD, which is 7.
• So, divide the numerator and the denominator by 7 to simplify: \( -\frac{21 \div 7}{14 \div 7} \) resulting in \( -\frac{3}{2} \).

This simplification helps express the fraction in a more compact form, making it easier to understand and use in further calculations.