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Chapter 5: Problem 32

Ian said, "I thought of a number, doubled it, and then added \(10 .\) I multiplied the answer by \(-0.5\) and divided that result by \(2 .\) The answer was \(-3.5 .\) What was my number?"

Short Answer

Expert verified
Ian's number was 2.

Step by step solution

01

Define the variable

Let the unknown number be represented by the variable, say, \( x \).
02

Translate the problem into an equation

According to the problem, Ian doubled the number, added 10, multiplied the result by \(-0.5\), and then divided by 2. Set up an equation representing these steps leading to the result \(-3.5\).
03

Form the equation

Starting with the initial number \( x \), double it to get \(2x\). Add 10 to get \(2x + 10\). Multiply the result by \(-0.5\), giving \(-0.5(2x + 10)\). Then, divide by 2: \[ \frac{-0.5(2x + 10)}{2} = -3.5 \]
04

Simplify the equation

Simplify the left side of the equation step-by-step. First, calculate \(-0.5(2x + 10)\): \(-0.5 \cdot 2x - 0.5 \cdot 10 = -x - 5\). The equation becomes: \[ \frac{-x - 5}{2} = -3.5 \]
05

Solve for \( x \)

Multiply both sides of the equation by 2 to eliminate the denominator: \[ -x - 5 = -7 \]. Then, solve for \( x \) by isolating \( x \): \[ -x = -7 + 5 \] which simplifies to \( -x = -2 \). Therefore, \( x = 2 \).
06

Verify the solution

Substitute \( x = 2 \) back into the original sequence of operations to check if it satisfies the conditions. Doubling 2 gives 4. Adding 10 results in 14. Multiplying by \(-0.5\) yields \(-7\). Dividing by 2 gives \(-3.5\), which matches the given result. Hence, the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are combinations of variables, constants, and operators (like addition, subtraction, etc.). In the given problem, the algebraic expression is initially set up by identifying the unknown value as a variable, commonly denoted as \( x \). Ian's number manipulation is turned into algebraic expressions step-by-step: doubled to become \( 2x \), then added 10 to make \( 2x + 10 \), and then manipulated further by mathematical operations to set up the equation.
Variable Manipulation
Variable manipulation involves performing arithmetic operations to solve for the unknown variable. The problem is translated into an equation which is then simplified step-by-step:
1. Start with the initial number \( x \).
2. Double it: \( 2x \).
3. Add 10: \( 2x + 10 \).
4. Multiply by \( -0.5 \): \( -0.5(2x + 10) \) which becomes \( -x - 5 \).
5. Divide by 2: \( \frac{-x - 5}{2} = -3.5 \).
Solving for \( x \) involves removing fractions through multiplying both sides by 2 and isolating \( x \):
\[ -x - 5 = -7 \]
Simplifies to:
\[ -x = -2 \]
Thus, \( x = 2 \).
Equation Verification
Verifying your solution ensures the steps and results satisfy the original problem. Substitute \( x = 2 \) back into the sequence of operations:
鈥� Double 2: \( 2 \times 2 = 4 \).
鈥� Add 10: \[ 4 + 10 = 14 \].
鈥� Multiply by \( -0.5 \): \( 14 \times -0.5 = -7 \).
鈥� Divide by 2: \( \frac{-7}{2} = -3.5 \).
The verification shows the solution is correct and consistent with the given result.

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