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A variable expression is a mathematical phrase that includes variables, numbers, and operations. Variables represent unknown values and are often denoted by letters like \( x \), \( y \), or \( z \). In word problems, identifying which elements correspond to variables and numbers is crucial to formulating an accurate expression. When we say "some number," it implies a number we do not know yet, so we can assign a variable to it, commonly \( x \). This allows us to transform verbal statements into equations or expressions that can be solved or evaluated under different conditions.

To create a variable expression from a word problem, follow these steps:

• Identify the unknowns and represent them with variables.
• Translate operations such as addition, subtraction, multiplication, or division into mathematical symbols.
• Combine these into a coherent expression that represents the initial statement.

In the given exercise, the term "some number" is our variable, \( x \), and we use it to express relationships involving typical operations like forming a quotient.

The word "quotient" refers to the result of dividing one number by another. Division is one of the four basic arithmetic operations and is crucial in forming expressions for problems involving ratios or rates. When you hear "quotient of twenty and some number," this means you are dividing twenty by an unknown number, which our variable represents.

In mathematical terms, the expression for a quotient is written using division:

• Numerator: The number being divided, in this case, 20.
• Denominator: The number by which you divide, represented here by the variable \( x \).

The quotient can then be expressed as \( \frac{20}{x} \). This expression emphasizes the relationship between the specified fixed number (20) and the variable (\( x \)). It is essential to carefully order the terms in a quotient to reflect the correct direction of division.

Subtraction in algebra involves taking a value away from another. It is represented by the minus sign (−). In word problems, phrases like "less than" indicate subtraction. Hence, to express "fifteen less than the quotient," you subtract fifteen from the quotient we calculated earlier.

To break it down:

• Start with the given quantity, which in this case is the quotient \( \frac{20}{x} \).
• Subtract the specified amount (15) from it.

This operation results in the expression \( \frac{20}{x} - 15 \). In algebra, understanding the order of operations is critical, as it dictates how expressions are simplified or calculated. Here, the subtraction is applied directly after forming the quotient.

This complete expression \( \frac{20}{x} - 15 \) now accurately translates the words "fifteen less than the quotient of twenty and some number."