In an examination, a student was asked to divide a certain number by 8. By mistake he multiplied it by 8 and got the answer 2016 more than the correct answer. What was the number?
256
Solving the Student's Calculation Error Problem
This problem involves setting up an algebraic equation based on the information given about a student's calculation mistake. We need to find the original number that the student was supposed to divide by 8.
Understanding the Problem Setup
Let the unknown number be represented by the variable \(x\).
- The correct operation was to divide the number by 8. The correct answer should have been \(\frac{x}{8}\).
- The incorrect operation performed was multiplying the number by 8. The incorrect answer obtained was \(8x\).
According to the problem statement, the incorrect answer was 2016 more than the correct answer. This can be written as an equation:
Incorrect Answer = Correct Answer + 2016
\(8x = \frac{x}{8} + 2016\)
Step-by-Step Solution to Find the Number
Now, we need to solve the equation \(8x = \frac{x}{8} + 2016\) for \(x\).
- To eliminate the fraction, multiply every term in the equation by 8: \[ 8 \times (8x) = 8 \times \left(\frac{x}{8}\right) + 8 \times 2016 \] \[ 64x = x + 16128 \]
- Subtract \(x\) from both sides of the equation to gather the terms with \(x\) on one side: \[ 64x - x = 16128 \] \[ 63x = 16128 \]
- Divide both sides by 63 to find the value of \(x\): \[ x = \frac{16128}{63} \]
- Perform the division: \[ x = 256 \]
So, the original number is 256.
Verifying the Answer
Let's check if our answer, \(x=256\), satisfies the condition given in the problem.
- Correct Answer: Divide 256 by 8. \(\frac{256}{8} = 32\).
- Incorrect Answer: Multiply 256 by 8. \(256 \times 8 = 2048\).
- Difference: Subtract the correct answer from the incorrect answer. \(2048 - 32 = 2016\).
The difference is indeed 2016, which matches the problem statement. Thus, our calculated number 256 is correct.
Summary of the Number Problem
The number the student was asked to divide by 8 was 256.
| Original Number | \(x\) |
|---|---|
| Correct Calculation | \(x \div 8\) |
| Incorrect Calculation | \(x \times 8\) |
| Equation | \(8x = \frac{x}{8} + 2016\) |
| Solution for \(x\) | 256 |
Revision Table: Key Concepts in Solving Word Problems
| Step | Description |
|---|---|
| Read Carefully | Understand the problem, identify knowns and unknowns. |
| Assign Variables | Use variables (like \(x\)) for unknown quantities. |
| Formulate Equation | Translate the problem statement into a mathematical equation. |
| Solve Equation | Use algebraic techniques to find the value of the variable. |
| Verify Answer | Check if the solution makes sense in the context of the original problem. |
Additional Information: Algebraic Equations
An algebraic equation is a statement that two mathematical expressions are equal. In this problem, we used a linear equation with one variable. Solving such equations involves isolating the variable on one side of the equation using inverse operations (addition/subtraction, multiplication/division) while maintaining equality.
For example, to solve \(63x = 16128\), we perform the inverse operation of multiplication, which is division. Dividing both sides by 63 gives \(x = \frac{16128}{63}\).
Setting up the correct equation is often the most crucial step in solving word problems like this one involving calculation errors or comparisons between different results.
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