Problem 60 - Elementary Algebra

Question

Answer the question by setting up and solving an appropriate equation. \(220 \%\) of what number is 66 ?

Step-by-Step Solution

Answer

The number is 30.

Step 1: Understanding the Problem

We are asked to find a number such that 220% of it equals 66. This means we need to translate the problem into a mathematical equation.

Step 2: Setting Up the Equation

To find the number, let's denote it by \(x\). According to the problem, 220% of \(x\) is 66. We translate this to an equation as follows: \( 2.2x = 66 \). Here, 220% is converted to a decimal, which is 2.2.

Step 3: Solving the Equation

Now, solve the equation \( 2.2x = 66 \) for \(x\). To do this, divide both sides of the equation by 2.2. \[ x = \frac{66}{2.2} \]

Step 4: Performing the Division

Calculate \( \frac{66}{2.2} \). This gives us \( x = 30 \).

Step 5: Conclusion

The number we're looking for is 30, because 220% of 30 equals 66.

Key Concepts

Percentages

Percentages are a way to express a number as a fraction of 100. They are widely used in various fields, such as finance, mathematics, and statistics. Understanding how to convert percentages into decimals is a crucial skill. Here is how you can make conversions simple:

• Move the decimal point two places to the left. For example, 220% becomes 2.2.
• This represents the part of the whole you are considering. In this exercise, 2.2 indicates 220 out of 100.

Percentages are handy for comparing proportions or understanding changes relative to a baseline of 100. This approach allows you to solve the given problem by setting it into an equation format. It's important to understand the correspondence between percentages and their decimal equivalents to accurately solve related mathematical problems.

Equation Solving

Solving equations is about finding the unknown value that makes the equation true. Equations are mathematical statements with an equals sign, highlighting the balance between expressions on either side. Here, we identify our unknown as 'x'. We are told that 220% of 'x' equals 66. We translate this verbally understood statement into a mathematical equation:

• The percentage becomes a decimal multiplier, i.e., 2.2.
• Our expression becomes: \(2.2x = 66\).

With the equation in place, the goal is to isolate 'x'. Here's a straightforward move:

• Divide both sides of the equation by the coefficient of 'x', in this case, 2.2.
• The calculation becomes \(x = \frac{66}{2.2}\).

Solving the division yields the answer swiftly. Recognizing how to rearrange and solve equations isn't just a math skill but a real-world problem-solving technique.

Understanding Word Problems

Word problems often challenge students because they require translating everyday language into mathematical expressions. The key to mastering them is a step-by-step approach:

• Identify what you need to find out—in this case, 'what number' relates to 220% becoming 66.
• Translate the language into a mathematical statement. The expression "220% of what number is 66" becomes the equation \(2.2x = 66\).

It's essential to carefully parse each part of the statement:

• "Of what number" hints that a multiplication is involved.
• "Is 66" suggests an equality, where the left-hand side equals 66.

Approach word problems with calm logic. This mindset helps break them down into understandable parts, making the math involved much more approachable.