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Chapter 1: Problem 92

Evaluate \(\frac{a-b}{3}\) when \(a\) is three times \(b\) and \(a=18\)

Short Answer

Expert verified
4

Step by step solution

01

Substitute the value of a

Given that \(a = 18\), substitute this value into the expression \(\frac{a-b}{3}\) to get \(\frac{18-b}{3}\).
02

Express b in terms of a

It is given that \(a\) is three times \(b\). So, we can write \(a = 3b\).
03

Substitute a = 18 to find b

From \(a = 3b\) and \(a = 18\), we substitute 18 for \(a\) to get \(18 = 3b\). Solving for \(b\) gives \(b = \frac{18}{3} = 6\).
04

Substitute b back into the expression

Now that we know \(b = 6\), replace \(b\) in the expression \(\frac{18-b}{3}\) to get \(\frac{18-6}{3}\).
05

Simplify the expression

Subtract 6 from 18 to get 12. This changes the expression to \(\frac{12}{3}\). Now, divide 12 by 3 to get 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
When solving algebraic expressions, substitution is a crucial first step. It involves replacing a variable with its known value. In our exercise, we start by substituting the value of `a = 18` into the given expression \(\frac{a-b}{3}\). This step simplifies the expression to \(\frac{18-b}{3}\).
Variable Manipulation
Variable manipulation is the process of rewriting expressions in various forms to facilitate easier computation. In this exercise, we use the information that `a` is three times `b`. This allows us to write the equation `a = 3b`. By substituting `a = 18` into this relationship, we can solve for `b`.
So, we get:
  • \textbf{Step 1}: Start with the equation \(\boldsymbol{18 = 3b}\).
  • \textbf{Step 2}: Solve for `b` by dividing both sides by 3 to get \(\boldsymbol{b = \frac{18}{3} = 6}\).
With this `b`, we make the expression manageable.
Simplification
Simplification is the final stage where we make the expression as simple as possible to get the final answer. Using our previous results: \(\frac{18-b}{3}\) and knowing \(\boldsymbol{b=6}\), we substitute `b` back into the expression:

\(\frac{18-6}{3}\), which simplifies to \(\frac{12}{3}\).
This gives our final answer of 4.

Simplifying complex expressions in stages makes the problem more approachable and ensures accuracy.

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