Chapter 17: Problem 1
If \(r=3 s, s=5 t, t=2 u,\) and \(u \neq 0,\) what is the value of \(\frac{r s t}{u^{3}} ?\) $$\begin{array}{l} a. 30 \\ b. 60 \\ c. 150 \\ d. 300 \\ e. 600 \end{array}$$
Short Answer
Expert verified
600
Step by step solution
01
Express All Variables in Terms of One Variable
First, express each variable in terms of the variable 'u'. Given that: - The problem states that \(r = 3s\), \(s = 5t\), and \(t = 2u\).- Start with the innermost relationship and substitute back. - \( t = 2u \) directly.- Substitute \( t = 2u \) into \( s = 5t \) to get \( s = 5(2u) = 10u \).- Substitute \( s = 10u \) into \( r = 3s \) to get \( r = 3(10u) = 30u \).
02
Substitute into the Expression
You need to find the value of \( \frac{rst}{u^3} \). Substitute \( r, s, t \) in terms of 'u' into the expression: \[ r = 30u \ s = 10u \ t = 2u \]So, the expression becomes: \( \frac{(30u)(10u)(2u)}{u^3} \)
03
Simplify the Expression
Simplify the numerator and denominator separately: - The numerator: \((30u)(10u)(2u) = 30 \times 10 \times 2 \times u^3 = 600u^3 \)- The denominator: \( u^3 \)Now, the expression becomes: \( \frac{600u^3}{u^3} \)
04
Cancel Out Common Terms
Canceling out the common \( u^3 \) terms in the numerator and the denominator, the expression simplifies to: \[ \frac{600u^3}{u^3} = 600 \]
05
Final Answer
The simplified value of the expression is 600.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Variable Substitution in Algebra
Variable substitution in algebra involves expressing variables in terms of other variables. This technique is extremely useful in simplifying complex problems. In our exercise, we start with the relationships:
- \(r = 3s\)
- \(s = 5t\)
- \(t = 2u\)
- \(t = 2u\)
- \(s = 10u\) by substituting \(t\)
- \(r = 30u\) by substituting \(s\)
Algebraic Expressions
Algebraic expressions are mathematical phrases that use variables, numbers, and operations. In the given problem, we are dealing with the algebraic expression \( \frac{rst}{u^3}\). To solve this, substitute the values we found:
- \(r = 30u\)
- \(s = 10u\)
- \(t = 2u\)
Rational Expressions Simplification
Rational expressions are fractions that have polynomials in the numerator and the denominator. Our task is to simplify the rational expression:\[ \frac{(30u)(10u)(2u)}{u^3} \]First, simplify the products in the numerator:
- \(30 \times 10 \times 2 = 600\)
- \(u \times u \times u = u^3\)
GRE Quantitative Reasoning
The GRE Quantitative Reasoning section tests your ability to interpret and analyze quantitative information. A key skill is simplifying complex algebraic expressions just like in this exercise. Always follow these steps:
- Substitute variables whenever possible to simplify the expressions.
- Simplify algebraic and rational expressions carefully by performing all arithmetic operations.
- Look for opportunities to cancel common terms.