Question
Part 4 of 4 (d) The ratio \( \frac{a-b}{a-b}=\square \), whereas the ratio \( \frac{a-b}{b-a}=\square \) provided that \( a \neq b \).
Ask by Wang Weaver. in the United States
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(\frac{a-b}{a-b} = 1\) and \(\frac{a-b}{b-a} = -1\).
Solution
Let \( a \) and \( b \) be numbers with \( a \neq b \).
1. First, consider the ratio
\[
\frac{a-b}{a-b}.
\]
Since the numerator and the denominator are the same (and not zero because \( a \neq b \)), we have
\[
\frac{a-b}{a-b}=1.
\]
2. Next, consider the ratio
\[
\frac{a-b}{b-a}.
\]
Notice that
\[
b-a = -(a-b).
\]
Substituting this into the ratio, we obtain
\[
\frac{a-b}{b-a} = \frac{a-b}{-(a-b)} = -\frac{a-b}{a-b} = -1.
\]
Thus, the completed statements are:
- \(\frac{a-b}{a-b}=1\),
- \(\frac{a-b}{b-a}=-1\).
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Simplify this solution Beyond the Answer
The ratio \( \frac{a-b}{a-b} = 1 \) as long as \( a \neq b \) ensures that you’re not dividing by zero. In essence, this reveals the self-duality of any non-zero quantity, where the same value over itself gives you a whole. On the other hand, the ratio \( \frac{a-b}{b-a} = -1 \) reflects the reversal of terms. Since \( b-a \) is simply \( -(a-b) \), this expression illustrates how flipping the order of subtraction switches the sign, reminding us that subtraction is not commutative, and that’s where the fun bits of algebra start to unfold!
