Math Problem Statement

Eight times the difference of twice a number and 10 yields the number increased by 40. Find the number.

Solution

The problem provided is:

"Eight times the difference of twice a number and 10 yields the number increased by 40. Find the number."

Let's break this down into an equation step by step:

  1. Let the unknown number be represented as xx.
  2. Twice the number is 2x2x.
  3. The difference of twice the number and 10 is 2x102x - 10.
  4. Eight times this difference is 8(2x10)8(2x - 10).
  5. This is said to equal the number increased by 40, which is x+40x + 40.

Now, we can set up the equation:

8(2x10)=x+408(2x - 10) = x + 40

Solving the equation:

  1. Distribute the 8 on the left side: 16x80=x+4016x - 80 = x + 40

  2. Move xx to the left side by subtracting xx from both sides: 16xx80=4016x - x - 80 = 40 15x80=4015x - 80 = 40

  3. Add 80 to both sides: 15x=12015x = 120

  4. Solve for xx by dividing both sides by 15: x=12015=8x = \frac{120}{15} = 8

Thus, the number is 8\boxed{8}.


Would you like further details on this solution or have any questions? Here are some related questions you might consider:

  1. How do we check if the solution x=8x = 8 is correct?
  2. How does the concept of translating word problems into equations work?
  3. What if the problem had a different multiplier instead of 8?
  4. How would the equation change if it said "decreased by 40" instead of "increased by 40"?
  5. What are other examples of linear equations involving a single variable?

Tip: When solving word problems, always define the variable clearly and build the equation step-by-step based on the given conditions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations

Formulas

8(2x - 10) = x + 40

Theorems

Linear Equation Solving

Suitable Grade Level

Grades 7-9