Question

So far in your volleyball practice, you have put into play 37 of the 44 serves you have attempted. Solve the equation \(\frac{90}{100}=\frac{37+x}{44+x}\) to find the number of consecutive serves you need to put into play in order to raise your serve percentage to \(90 \%\).

Short answer

You need to successfully serve 26 more times in a row to raise your serving percentage to 90%.

Step-by-step solution

Understand the Given Equation

The given equation represents the fraction of successful serves. Here, \(\frac{37+x}{44+x}\) is the fraction of successful serves where 37 is the current number of successful serves, 44 is the total number of serves and 'x' is the number of additional successful serves. This fraction is equated to \(\frac{90}{100}\), which is the goal success rate.

Simplify the Equation

First, simplify the right-hand side of the equation. \(\frac{90}{100}\) simplifies to \(0.9\). Hence, the equation now is \(0.9=\frac{37+x}{44+x}\).

Cross Multiply

Cross multiply the equations so you can solve for x. This gives: \(0.9*(44+x)=37+x\).

Expand and Rearrange

Expand the left side which leads to \(39.6+0.9x=37+x\). Then, rearrange to get x terms on one side and constant on the other: \(0.1x=2.6\).

Solve for x

Divide by 0.1 on both sides to solve for x. You obtain \(x=2.6/0.1=26\).

Key concepts

Cross Multiplication

Cross multiplication is a method used for solving equations that involve two ratios or fractions set equal to each other. It's particularly handy when dealing with problems where the goal is to find an unknown variable that's part of one of the fractions.

In the volleyball practice example, you had to determine the number of additional successful serves needed to achieve a certain serve percentage. By setting the equation \(\frac{37+x}{44+x}=0.9\), you employed cross multiplication to get rid of the fractions, making it easier to solve for the unknown 'x'. Here is how it works: You multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the numerator of the second fraction and the denominator of the first fraction. So, \(0.9*(44+x)=37+x\). This one step transforms a complex fractional equation into a more straightforward linear equation which can be solved with basic algebra.

Linear Equations

Linear equations are algebraic expressions that represent a straight line when graphed on a coordinate plane. They are composed of variables and constants and obey the general form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.

In our example, after using cross multiplication, you arrived at a linear equation involving the variable 'x'. The equation \(39.6 + 0.9x = 37 + x\) is linear because it can be simplified to a similar form, where x represents the number of successful serves you need to add. The process involved expanding the terms and then rearranging the equation to get all the 'x' terms on one side and the constant terms on the other. This makes the structure of the equation clear and highlights that we're looking for the 'x' that will make the equation true, representing the solution to our problem.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations. Unlike equations, algebraic expressions do not contain an equality sign. They are the building blocks for creating equations that describe relationships and solve problems.

Throughout the process of solving for 'x', you manipulated algebraic expressions. You expanded them, combined like terms, and simplified. When you started with the equation \(0.9 = \frac{37+x}{44+x}\) and turned it into \(0.9*(44+x) = 37 + x\), you engaged in algebraic manipulations to make the variables and constants clear. Adding, subtracting, and dividing terms to isolate 'x' and solve the problem requires an understanding of how algebraic expressions work and how they can be transformed while maintaining balance on both sides of the equation.