Question

The average (arithmetic mean) of all scores on a certain algebra test was $90.$ If the average of the 8 male students' grades was $87,$ and the average of the female students' grades was $92,$ how many female students took the test? A. $$8$$ B. $$9$$ C. $$10$$ D. $$11$$ E. $$12$$

Short answer

12

Step-by-step solution

Define the variables

Let the number of male students be represented by $m$ and the number of female students be represented by $f$.

Express total scores

The sum of the scores of male students is $87m$ and the sum of the scores of female students is $92f$. Since the average score of all students is $90$, we know the total sum of all scores is $90(m+f)$.

Set up the equation

We have the equation from the previous step: $$87m+92f=90(m+f).$$

Plug in the known value for male students

We know there are 8 male students, so substitute $m=8$ into the equation: $$87(8)+92f=90(8+f).$$

Simplify the equation

Calculate $87(8)=696$ and $90(8)=720$, then substitute these values into the equation: $$696+92f=720+90f.$$

Solve for the number of female students

Firstly, move all terms involving $f$ to one side: $$696+92f-90f=720.$$This simplifies to: $$696+2f=720.$$Then, subtract 696 from both sides: $$2f=24.$$Finally, divide both sides by 2: $$f=12.$$

Key concepts

Arithmetic Mean

The arithmetic mean, often known as the average, is a fundamental concept in mathematics. To find the arithmetic mean, you add up all the numbers in a set and then divide by the number of elements in that set.
For example, if your test scores are 85, 90, 78, and 92:

• Add the scores: 85 + 90 + 78 + 92 = 345.
• Count the number of scores: 4.
• Divide the sum by the count: 345 ÷ 4 = 86.25.

Therefore, the arithmetic mean is 86.25. This measure helps summarize a set of numbers using a single value, which is very useful in fields like statistics, business, and education.

Algebra Problems

Solving algebra problems involves finding the values of unknown variables that make an equation true. These problems often require you to set up one or more equations based on the information given.
In the exercise, you are given the average scores of male and female students along with the total average. You set the scene by defining your variables, which represent the numbers of male and female students. Here, we used m for male students and f for female students.
In algebra problems, it's important to translate written words into mathematical expressions. The relationships provided in the problem are translated into equations. By breaking down the text into small parts, you can slowly transform all the relations into an equation.

Equation Solving

Solving equations is a key skill in algebra. It involves finding the value for a variable that makes the equation true. Let's revisit the equation from the exercise:
Step 1:
The equation provided is 87m + 92f = 90(m + f).
Step 2:
We know m (males) is 8. So, substitute m with 8:
87(8) + 92f = 90(8 + f).
Step 3:
Calculate 87(8) = 696, and 90(8) = 720. The equation transforms to
Step 4:
696 + 92f = 720 + 90f.Move all terms involving f to one side:
696 + 92f - 90f = 720,which simplifies to: 696 + 2f = 720,subtract 696 from both sides: 2f = 24. Step 5: Divide both sides by 2:
f = 12. So, there are 12 female students.
Remember, solving equations often involves several steps: simplifying expressions, moving variables to one side, and isolating the variable to find its value.

Test Score Analysis

Test score analysis helps us understand performance through statistical measures. The concept of averages is crucial here.
For instance, knowing the average score can provide insights into the overall performance of a group. If the average score of a class is high, it suggests that most students performed well. Conversely, a low average might indicate that the test was challenging for many students.

• Individual scores give specific insights.
• Average scores give a general performance overview.
• Comparing averages can show differences between groups.

In this exercise, analyzing different averages (overall, male, and female students) helps us find the number of female students by setting up relationships between these averages. This is practical for educational assessments and understanding student performance trends.