What is the quotient of (a - 3)/7 divided by (3 - a)/21?
Understand the Problem
The question is asking us to find the quotient of the expression (a - 3)/7 divided by (3 - a)/21. This requires simplifying the expression and applying the rules of division for fractions.
Answer
The answer is $-3$.
Answer for screen readers
The final answer is $-3$.
Steps to Solve
- Set Up the Division of Fractions
We need to find the quotient of the fractions (\frac{a - 3}{7}) and (\frac{3 - a}{21}). Dividing by a fraction is the same as multiplying by its reciprocal. So we can rewrite the division as:
$$ \frac{a - 3}{7} \div \frac{3 - a}{21} = \frac{a - 3}{7} \times \frac{21}{3 - a} $$
- Simplify the Expression
Now, we substitute the second fraction's components into the multiplication:
$$ = \frac{(a - 3) \cdot 21}{7 \cdot (3 - a)} $$
Notice that (3 - a) can be rewritten as (-(a - 3)):
$$ = \frac{(a - 3) \cdot 21}{7 \cdot -(a - 3)} $$
- Cancel Common Factors
Now we can cancel out the common factor (a - 3) from the numerator and denominator. Note that (a - 3) cannot be zero, so we assume (a \neq 3):
$$ = \frac{21}{7 \cdot -1} $$
- Simplify Further
Now simplify the fraction:
$$ = \frac{21}{-7} = -3 $$
The final answer is $-3$.
More Information
This problem illustrates the importance of understanding how to work with fractions, especially when dividing them. It also highlights the concept of recognizing when to simplify by canceling common terms.
Tips
- Not recognizing that (3 - a) is equal to (-(a - 3)) can lead to errors in simplification.
- Forgetting to assume that (a \neq 3) might result in an undefined expression during cancellation.
