Problem 62 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. One less than some number is multiplied by three less than five times the number and the entire result is divided by six less than the number. The result is twenty-seven less than eleven times the number.

Step-by-Step Solution

Answer

Question: Translate the statement "One less than a number, multiplied by three less than five times the number, divided by six less than the number, equals twenty-seven less than eleven times the number" into a mathematical equation. Answer: \(\frac{(x-1)(5x-3)}{x-6} = 11x-27\)

Step 1: Declare The Unknown Number

Let x be the unknown number.

Step 2: One less than the number

Subtract 1 from x: x-1 .

Step 3: Three less than five times the number

Multiply x by 5 and then subtract 3: 5x-3 .

Step 4: Multiply the two expressions

(x-1)(5x-3) .

Step 5: Six less than the number

Subtract 6 from x: x-6 .

Step 6: Divide the result

We need to divide the product by the expression six less than the number: \frac{(x-1)(5x-3)}{x-6} .

Step 7: Twenty-seven less than eleven times the number

Multiply x by 11 and then subtract 27: 11x-27 . Now let's write the complete equation.

Step 8: Write The Final Equation

\frac{(x-1)(5x-3)}{x-6} = 11x-27 . This is the required equation that represents the given statement. The student can now proceed to solve this equation or use it based on the further requirements of the exercise.

Key Concepts

Mathematical Expressions

When translating word problems into mathematical expressions, we are essentially converting words into a format that can be manipulated mathematically. In this exercise, we start by identifying an unknown number, commonly represented by a variable like \( x \). This sets the foundation for constructing our expression.

For example, the phrase "one less than some number" implies a subtraction operation from our variable \( x \), yielding \( x - 1 \). Similarly, to express "five times a number, less three" as a mathematical expression, we show it as \( 5x - 3 \). Mathematical expressions capture specific operations such as addition, subtraction, multiplication, and division, which correspond directly to instructions given in the problem.

An essential tip for building these expressions is to carefully take note of keywords and phrases. Words like "less than" suggest subtraction, while "times" indicates multiplication. Understanding these hints helps in accurately forming the initial components of our equation.

Algebraic Equations

Once we have our expressions set up, the next task is to build them into a complete algebraic equation. An algebraic equation uses two expressions set equal to one another, separated by an equals sign, to demonstrate the relationship between the elements.

In the given problem, after breaking down individual phrases into expressions—like \( x - 1 \) and \( 5x - 3 \)—we combine these into more complex expressions as shown in the steps. For instance, multiplying "one less than the number" by "three less than five times the number" results in the expression \((x-1)(5x-3)\).

To express the relationship described in the problem fully, this product is then divided by "six less than the number", resulting in the complete left-side expression \( \frac{(x-1)(5x-3)}{x-6} \). This whole expression equals "twenty-seven less than eleven times the number", or \( 11x - 27 \), creating the full equation \( \frac{(x-1)(5x-3)}{x-6} = 11x-27 \). This equation expresses the entire word problem as a single mathematical statement that can be solved.

Problem Solving Steps

Breaking down word problems into manageable steps is key to finding a viable solution. Each step brings us closer to the final equation by methodically interpreting and expressing each mathematical component.

• **Step 1:** Identify the unknown, often marked as \( x \), to set the base for all future calculations.
• **Step 2 to 4:** Translate descriptive phrases into individual expressions by dissecting the word problem. This includes subtraction for "one less than" or multiplication for "five times."
• **Step 5 to 6:** Combine these expressions to reflect operations described in the scenario. Here, it's critical to respect the mathematical order of operations to avoid mistakes.

Finally, writing the complete equation in Step 8 encapsulates the entire problem into a format ready for algebraic solution. These systematic problem-solving steps demystify complex problems. They break them into components that are more straightforward to work with, ensuring clarity and accuracy throughout the process.