- Identify the components of the sentence: "a number cubed" is \(x^3\), "another number" is \(y\).
- Translate phrases into mathematical operations: "ratio" becomes \(\frac{x^3}{y}\), "twice" becomes \(2(\dots)\), "decreased by" becomes subtraction, and "product" becomes \(xy\).
- Combine the components to form the final expression: \(2\left(\frac{x^3}{y}\right) - xy\).
- Select the matching option: The correct choice is \(\boxed{b}\).
Explanation
- Breaking down the sentence
To translate the given sentence into an algebraic expression, we first break it down into smaller, manageable parts. The sentence is: "twice the ratio of a number cubed to another number is decreased by the product of the two numbers." We define the first number as \(x\) and the second number as \(y\).
- Translating into mathematical symbols
Now we translate each part of the sentence:
- "a number cubed": This is represented as \(x^3\).
- "another number": This is represented as \(y\).
- "ratio of a number cubed to another number": This is a fraction, represented as \(\frac{x^3}{y}\).
- "twice the ratio of...": We multiply the ratio by 2, giving \(2\left(\frac{x^3}{y}\right)\).
- "product of the two numbers": This is the multiplication of the two numbers, represented as \(xy\).
- "is decreased by": This indicates subtraction, where we subtract the product from the previous term.
- Finalizing the expression and matching with options
Combining these parts, we get:
\[2\left(\frac{x^3}{y}\right) - xy\]
Comparing this result with the given options:
A\(2\left(\frac{3x}{y}\right) - xy\)
B\(2\left(\frac{x^3}{y}\right) - xy\)
C\(\frac{2x^3}{x} \div xy\)
D\(2(x^3 - y) \div xy\)
The derived expression exactly matches option (b).
Examples
Understanding how to translate word problems into algebraic expressions is a foundational skill in algebra. For example, if you are designing a computer program where you need to calculate the value of an item's volume relative to its density, you might start with a word description like "triple the weight divided by the surface area, minus the constant offset." Learning to map these phrases into accurate mathematical expressions, such as \(3\left(\frac{w}{a}\right) - c\), is how we bridge the gap between descriptive logic and functional calculations. This skill is essential for modeling real-world data and building algorithmic solutions.