Question

What number do you start with if you multiply it by 5 , then divide by 4 and then halve that to get 5 ?

Short answer

The starting number is 8.

Step-by-step solution

Set Up the Equation

Let the unknown number be denoted as \( x \). According to the problem, the sequence of operations on \( x \) is: multiply by 5, divide by 4, and then halve the result to get 5.

Translate Operations into an Equation

Translate the given operations into the equation: \[ \frac{1}{2} \left( \frac{5x}{4} \right) = 5 \] .

Simplify the Equation

Multiply both sides of the equation \[ \frac{1}{2} \left( \frac{5x}{4} \right) = 5 \] by 2 to eliminate the fraction: \[ \frac{5x}{4} = 10 \] .

Isolate the Variable

Multiply both sides by 4 to get rid of the denominator: \[ 5x = 40 \] .

Solve for \( x \)

Divide both sides by 5 to solve for \( x \): \[ x = 8 \] .

Key concepts

algebraic equations

To solve this type of problem, we need to set up an equation. An algebraic equation uses a variable (in this case, let's call it \(x\)) to represent the number we're trying to find. The problem gives us a series of steps which we can write as an equation.
Here’s how you translate words into an equation:
First, identify the operations: multiply by 5, divide by 4, and then halve the result. According to the problem, all these steps result in 5.

• Start by letting \(x\) be the unknown number: \(x\)
• Multiply by 5: \(5x\)
• Then divide by 4: \(\frac{5x}{4}\)
• Halve the result: \(\frac{1}{2} \left( \frac{5x}{4} \right)\)

So, we set it equal to 5:
\[\begin{equation}\frac{1}{2} \left( \frac{5x}{4} \right) = 5\end{equation}\]Understanding this helps in translating everyday math problems into algebraic equations, making problem-solving straightforward.

mathematical operations

Mathematical operations are the building blocks of algebra. In this problem, we deal with multiplication, division, and dealing with fractions. Let’s break down these operations:
1. **Multiplication**: When you multiply a variable by a number (e.g., \( 5x \)), you scale the variable by that number.
2. **Division**: Dividing a variable term (e.g., \( \frac{5x}{4} \)) means splitting the term into parts. Here, we're dividing by 4, distributing the multiplication result.
3. **Fractions**: Halving the term (\( \frac{1}{2} \left( \frac{5x}{4} \right) \)) means we take what's inside and divide it by 2.
Understand how each operation transitions into the other. Always apply operations step-by-step and look at how each transformation affects the variable. These basic building blocks help ensure we get to the correct solution.

step-by-step problem solving

Breaking down the problem into steps is key to solving it correctly. Here are our main steps:

• **Step 1: Set Up the Equation**
  Identify your starting point and plan your equation by understanding the problem.
• **Step 2: Translate Operations into an Equation**
  Write down each step as part of the equation. Here, we transform words (multiply, divide, halve) into mathematical symbols.
• **Step 3: Simplify the Equation**
  We multiplied both sides by 2 to eliminate the fraction.
  \[\begin{equation}2 \times \frac{1}{2} \left( \frac{5x}{4} \right) = 2 \times 5\end{equation}\]leading to
  \[\begin{equation}\frac{5x}{4} = 10\end{equation}\]
• **Step 4: Isolate the Variable**
  To isolate \(x\), multiply both sides by 4:
  \[\begin{equation}5x = 40\end{equation}\]
• **Step 5: Solve for \( x \)**
  Finally, divide by 5:
  \[\begin{equation}x = \frac{40}{5} = 8\end{equation}\]

Each of these steps ensures we simplify and solve the problem systematically. It helps to write down each operation and transformation clearly, making the path to the solution easy to follow.