Three Less Than Six Times A Number

The phrase "three less than six times a number" is a common example of translating verbal expressions into algebraic equations. Understanding how to interpret and solve such problems is essential for students learning algebra, as it forms the foundation for more complex mathematical reasoning. In this article, we will break down the meaning of this phrase, explain how to translate it into an equation, solve it step-by-step, and explore its real-world applications.

Understanding the Phrase

The phrase "three less than six times a number" can be broken down into two main parts: "six times a number" and "three less than." The key to translating this phrase into an algebraic expression is to identify the unknown quantity, which is typically represented by a variable such as x. The phrase "six times a number" translates to 6x, while "three less than" indicates that we subtract 3 from the result. Therefore, the algebraic expression for "three less than six times a number" is 6x - 3.

Translating to an Algebraic Equation

To translate the phrase into an equation, we need to set it equal to a value or another expression. For example, if the problem states "three less than six times a number is equal to 15," we can write the equation as 6x - 3 = 15. This equation can then be solved to find the value of x.

Solving the Equation

Solving the equation 6x - 3 = 15 involves isolating the variable x. Here are the steps:

1. Add 3 to both sides of the equation to eliminate the constant term on the left side: $6x - 3 + 3 = 15 + 3$ Simplifying, we get: $6x = 18$

2. Divide both sides by 6 to solve for x: $x = \frac{18}{6}$ Simplifying, we find: $x = 3$

Therefore, the solution to the equation is x = 3. This means that when the number is 3, six times the number minus three equals 15.

Real-World Applications

Understanding how to translate and solve expressions like "three less than six times a number" is not just an academic exercise; it has practical applications in various fields. For instance, in economics, such expressions can be used to model cost functions, where the cost of producing x items might be represented as 6x - 3. In physics, similar expressions can describe relationships between variables, such as distance, time, and speed.

Common Mistakes to Avoid

When working with algebraic expressions, students often make common mistakes. One frequent error is misinterpreting the phrase "less than." For example, some might incorrectly write the expression as 3 - 6x instead of 6x - 3. Another mistake is forgetting to perform the same operation on both sides of the equation when solving, which can lead to incorrect solutions.

Practice Problems

To reinforce your understanding, try solving the following problems:

1. Three less than six times a number is 21. Find the number.

   • Equation: 6x - 3 = 21
   • Solution: x = 4

2. Six times a number minus five equals 25. Find the number.

   • Equation: 6x - 5 = 25
   • Solution: x = 5

3. Four less than three times a number is 11. Find the number.

   • Equation: 3x - 4 = 11
   • Solution: x = 5

Conclusion

Translating verbal phrases into algebraic expressions and solving the resulting equations is a fundamental skill in algebra. The phrase "three less than six times a number" translates to the expression 6x - 3, which can be solved using basic algebraic techniques. By practicing these types of problems, students can develop a strong foundation in algebra, which is essential for success in higher-level mathematics and real-world applications. Remember to pay close attention to the wording of the problem and to perform operations correctly to avoid common mistakes. With practice and patience, mastering these concepts will become second nature.