If a/b = 3/4, b/c = 8/9 and c/d = 2/3, what is the value of (ad)/b? Express your answer as a common fraction.
Understand the Problem
The question requires finding the value of the expression (ad)/b given the relationships between a, b, c, and d. We will first express a, c, and d in terms of b, then substitute these back into the expression to simplify and solve for (ad)/b.
Answer
$km$
Answer for screen readers
The final answer is $km$.
Steps to Solve
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Express a in terms of b To find $a$, we look at the given relationships. Suppose we know a direct relationship such as $a = kb$ for some constant $k$. We will express $a$ this way.
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Express d in terms of b Next, we express $d$ in terms of $b$. Let's assume $d = mb$ for some constant $m$. This means that $d$ is also directly proportional to $b$.
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Substituting a and d into (ad)/b Now that we have expressed both $a$ and $d$ in terms of $b$, we can substitute these into the expression $\frac{ad}{b}$: $$ \frac{ad}{b} = \frac{(kb)(mb)}{b} $$
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Simplify the expression The expression simplifies as follows: $$ \frac{(kb)(mb)}{b} = k \cdot m \cdot b $$
Thus, we can see that the $(b)$ in the numerator and denominator cancels out.
- Final Result The result we are left with is simply $km$, where $k$ and $m$ are the constants that relate $a$ and $d$ to $b$.
The final answer is $km$.
More Information
The expression $(ad)/b$ can often simplify nicely when both $a$ and $d$ are expressed in terms of a common variable. This is a common technique in algebra used to evaluate expressions based on given relationships.
Tips
- Forgetting to cancel terms in the expression can lead to complicating the final answer.
- Misinterpreting the relationships between the variables can lead to incorrect substitutions.
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