Question

If \(x+y=10\), what percentage of \(x\) is \(x+2 y\) ? A. 10 B. 200 C. 150 D. 100 E. Cannot be determined

Short answer

D. 100.

Step-by-step solution

- Understand the Given Equation

The problem states that we have the equation: \(x + y = 10\).

- Express y in terms of x

Using the equation \(x + y = 10\), solve for \(y\) in terms of \(x\): \(y = 10 - x\).

- Substitute y into the New Expression

Substitute \(y = 10 - x\) into the expression \(x + 2y\): \[x + 2y = x + 2(10 - x)\].

- Simplify the Expression

Simplify the expression: \[x + 2(10 - x) = x + 20 - 2x = 20 - x\].

- Calculate the Percentage

\[\frac{20 - x}{x} \times 100\] Substitute \(x = 10\): \[\frac{20 - x}{x} \times 100 = \frac{20 - 10}{10} \times 100 = \frac{10}{10} \times 100 = 1 \times 100 = 100\]

- Identify the Correct Answer

The expression \(x + 2y\) is found to be 100% of \(x\). Therefore, the correct answer is D. 100.

Key concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this problem, we're given an equation: \(x + y = 10\). The goal is to understand the relationship between \(x\) and \(y\). Algebra helps in expressing variables in terms of one another and solving equations involving these variables. Here, we first express \(y\) in terms of \(x\): if \(x + y = 10\), then \(y = 10 - x\). This substitution allows us to work with only one variable at a time in further calculations. Remember that algebra is fundamental for breaking down complex problems into manageable steps. By rephrasing one variable in terms of another, we make it possible to solve more parts of the problem independently.

Percentage Calculations

Percentage calculations are a way of expressing a number as a fraction of 100. This concept is very useful for comparing ratios or showing the relative size of different quantities. In our example, we need to find what percentage \(x + 2y\) is of \(x\). This translates into the mathematical expression: \[ \frac{x + 2y}{x} \times 100 \]. It's essential to simplify the expression inside the fraction first. In our case, \(x + 2y\) simplifies with the substitution we obtained from the algebra work: \[ x + 2(10 - x) = 20 - x \]. Once we have the simplified form, we plug it into our percentage formula: \[ \frac{20 - x}{x} \times 100 \]. Substituting values of \(x\), especially extreme ones like \(x = 10\) and \(x = 0\), provides insights into the relationship. Always calculate percentages with care, ensuring each step of simplification is correct.

Equations

Equations are mathematical statements indicating that two expressions are equal. In this problem, understanding and manipulating equations are key. We began with the equation \(x + y = 10\). This is a linear equation showing a direct relationship between \(x\) and \(y\). Solving for one variable in terms of another, as we did by expressing \(y = 10 - x\), is a common technique. We then used this information to rewrite the expression \(x + 2y\) in a different form: \[ x + 2(10 - x) \. \] Through this manipulation, we translated the original problem into a format where we can easily apply percentage calculations: \( \frac{20 - x}{x} \times 100 \). It's pivotal to understand each step in solving equations to avoid mistakes and to effectively simplify expressions. This manipulation skill is vital for various mathematical problems beyond just this example.