Question

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

600

Step-by-step solution

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 \( r = 3(10u) = 30u \).

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: \( \frac{(30u)(10u)(2u)}{u^3} \)

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: \( \frac{600u^3}{u^3} \)

Cancel Out Common Terms

Canceling out the common \( u^3 \) terms in the numerator and the denominator, the expression simplifies to: \[ \frac{600u^3}{u^3} = 600 \]

Final Answer

The simplified value of the expression is 600.

Key concepts

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:

• \(r = 3s\)
• \(s = 5t\)
• \(t = 2u\)

To solve this, we first express each variable in terms of 'u'. Starting from the innermost relationship, we substitute back to get:

• \(t = 2u\)
• \(s = 10u\) by substituting \(t\)
• \(r = 30u\) by substituting \(s\)

This substitution helps in combining all the variables into a single term, making the expression easier to simplify.

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:

• \(r = 30u\)
• \(s = 10u\)
• \(t = 2u\)

So, our expression becomes: \[ \frac{(30u)(10u)(2u)}{u^3} \]The next step is to simplify this combined expression which involves multiplication and division.

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:

• \(30 \times 10 \times 2 = 600\)
• \(u \times u \times u = u^3\)

So, the numerator is \(600u^3\), and because the denominator is also \(u^3\), we can cancel out the \(u^3\) terms. This leaves us with: \[ \frac{600u^3}{u^3} = 600 \]By canceling common terms, we arrive at our simplified answer.

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.

By breaking the problem into smaller, manageable steps and understanding the relationships between variables, even tough GRE questions can be tackled with confidence. Practice helps in getting faster and more accurate with these techniques.