Write algebraic expressions for the quantities described by the ratio.
Suppose you are given the following problem: "The ratio of boys to girls on a co-ed softball team is 7:4. There are 22 players on the team. How many girls are on the team?"
We would create algebraic expressions by attaching a variable, usually x, to each of the numbers in the ratio. In this example, the number of boys is 7x while the number of girls is 4x.
The first thing to note is that the order of numbers in the ratio will follow the order of the verbal description. Because boys are listed first, the first number in the ratio, seven, represents the number of boys. For similar reasons, the number four represents the number of girls.
Note also that the ratio is not telling us the number of boys or girls on the team. We cannot assume that there are seven boys and four girls on the team. What a ratio tells us is that if we divide the boys into groups of seven and the number of girls into groups of four that the number of groups of boys and the number of groups of girls will be equal even though the size of the groups is different. The variable represents the number of groups.
Set up an equation by adding the two algebraic expressions together and setting them equal to the total number of elements in the entire group. In this case, the total number of elements is the number of people on the softball team, which is 22.
The equation becomes 7x + 4x = 22
Solve the equation for x.
7x + 4x = 22 simplifies to 11x = 22. Dividing both sides by 11 gives x = 2.
Substitute the value of the variable into the algebraic expression for the quantity you are trying to find and evaluate.
In this example, we are trying to find the number of girls on the softball team. The expression for the number of girls is 4x. Substituting the value of x, which is two, gives 4(2) = 8.
There are eight girls on the softball team.
Change the percent into a fraction with a denominator of 100. Suppose you are given the following problem: "Josh's grandmother promises to raise his weekly allowance by 75 percent if he practices his flute every night without being asked. If his usual allowance is $20 per week, what will his new allowance be if he practices every night?"
This problem is a little bit tricky. If Josh's allowance is raised by 75 percent, he will actually receive 175 precent of his usual allowance. The fraction becomes 175/100.
Find the second fraction describing the relationship between his old allowance and his new allowance.
Using this example, you don't know Josh's new allowance. Call it x. Josh's old allowance is $20.
The fraction could be either x/20 or 20/x. Decide which of these fractions to use by looking at the percent. In that expression, the 100 percent represents Josh's old allowance of $20. Since the percent describing his allowance is in the denominator, the number describing his allowance should be in the denominator of the other fraction. Therefore, we are using x/20.
Set up a proportion using the two fractions. Remember that a proportion is two fractions set equal to each other.
175/100 = x/20.
Note the consistency of the equation: The percent representing Josh's original allowance and the number representing Joshua's allowance are both in the denominator while the percent representing the new allowance and the variable representing it are both in the numerator.
Cross-multiply and solve.
175/100 = x/20 becomes 175(20) = 100x.
3500 = 100x
3500/100 =100x/100
35 = x.
Thus, Josh's new allowance is $35.
We could also have taken 75 percent of 20, which is 15, and added it to the original allowance to get 35. Sometimes there is more than one way to solve percentage problems.