Demographics Of Students The Strength Of The School

Hey guys! Let's break down this interesting math problem that explores the demographics of a school, specifically focusing on the distribution of students by gender and age. This isn't just about crunching numbers; it's about understanding how data can paint a picture of a school's population. So, grab your thinking caps, and let's dive in!

Unraveling the Student Population: Gender and Age Dynamics

In this section, we're going to dissect the core of the problem. The question presents a scenario within a school setting, challenging us to understand the relationships between different student groups. Specifically, we're dealing with the fraction of girls in the school, the fraction of boys under a certain age, and the number of girls at or above a certain age. Let's break down each component to make sure we're all on the same page.

First off, we know that 5/12 of the total students are girls. This crucial piece of information immediately tells us the remaining fraction, which represents the boys, is 7/12. Now, why is this important? Because it sets the stage for understanding the proportions we'll be working with later. Think of it like laying the foundation for a building; we need this basic understanding to build upon.

Next, the problem states that 4/7 of the boys are under 14 years old. This is another key fraction that we need to keep in our mental toolkit. It doesn't tell us the exact number of boys under 14, but it gives us a ratio. We know that for every 7 boys, 4 of them are younger than 14. This is where the problem starts to get interesting, as we're now dealing with a fraction within a fraction – a concept that can sometimes trip people up if not approached carefully. To avoid confusion, remember to think of this as two separate pieces of information that we'll eventually need to connect.

Then, we're given a seemingly straightforward piece of information: 3 girls are aged 14 or older. This might seem like a small detail, but it's actually a crucial anchor point in the problem. It gives us a concrete number to work with, which is often essential when dealing with fractions and proportions. Without this specific number, we'd be stuck working with relative amounts, but with it, we can start to deduce the actual size of the student population.

The question then presents a condition: if the number of students under 14 is something, we must find the total strength of the school. This sets the goal for our mathematical journey. It's like having a destination in mind before setting out on a trip. Knowing what we need to find helps us strategize our approach and keep track of our progress.

To successfully solve this problem, we need to carefully combine these pieces of information. We'll need to use the fractions to represent proportions of the student population, and then use the concrete number of girls aged 14 or older to bridge the gap between these proportions and actual numbers. It's a bit like piecing together a puzzle, where each piece of information is a crucial part of the overall picture. So, let's roll up our sleeves and get ready to put our mathematical skills to the test!

Setting Up the Equation: A Mathematical Blueprint

Alright, let's get our hands dirty with some math! To solve this problem effectively, we need to translate the word problem into a mathematical equation. This is like creating a blueprint before starting construction – it gives us a clear plan to follow. So, let's break down how we can set up our equation.

First things first, we need to define our variables. This is a crucial step in algebra because it allows us to represent unknown quantities with symbols. In this case, the most important unknown is the total number of students in the school. Let's call this 'x'. It's a common choice, but you could use any letter you like – just make sure you define what it represents!

Now that we have our variable, we can start expressing the other quantities in terms of 'x'. We know that 5/12 of the students are girls, so the number of girls can be represented as (5/12)x. Similarly, the number of boys is the remaining fraction of the total students, which is 7/12, so we can represent the number of boys as (7/12)x. See how we're turning the word problem into mathematical expressions? That's the magic of algebra!

Next, we need to consider the information about the ages of the students. We know that 4/7 of the boys are under 14 years old. To find the number of boys under 14, we need to multiply the fraction of boys (7/12)x by 4/7. This gives us (4/7) * (7/12)x, which simplifies to (1/3)x. So, (1/3)x represents the number of boys under 14.

We also know that there are 3 girls aged 14 or older. This is a concrete number, which is super helpful. However, we need to relate this to the total number of girls to eventually link it to the total student population. We'll come back to this piece of information shortly.

The problem states that the number of students under 14 is a certain value. Let's call this value 'y' for now. This 'y' is the sum of the boys under 14 (which we know is (1/3)x) and the girls under 14. Now, this is where things get a little tricky. We don't have a direct expression for the number of girls under 14 yet, but we do know the total number of girls and the number of girls aged 14 or older. So, we can find the number of girls under 14 by subtracting the number of girls aged 14 or older from the total number of girls. This gives us (5/12)x - 3 girls under 14.

Now, we can set up our main equation! The total number of students under 14 ('y') is the sum of the boys under 14 ((1/3)x) and the girls under 14 ((5/12)x - 3). So, our equation looks like this: y = (1/3)x + (5/12)x - 3. This equation is the heart of our solution. It relates the total number of students ('x') to the number of students under 14 ('y'), and it incorporates all the key information given in the problem. The next step is to simplify and solve this equation to find the value of 'x', which will give us the total number of students in the school.

Solving the Equation: Finding the Total Number of Students

Okay, guys, we've set up our equation, and now it's time for the fun part – solving it! This is where we put on our algebra hats and manipulate the equation to isolate our unknown variable, 'x', which represents the total number of students in the school. Let's recap our equation first to make sure we're all on the same page:

y = (1/3)x + (5/12)x - 3

Remember, 'y' represents the number of students under 14, which is a value that the problem will give us. Our goal is to find 'x', the total number of students. To do this, we need to simplify the equation and rearrange it so that 'x' is on one side and all the known values are on the other.

The first step in simplifying this equation is to combine the terms that contain 'x'. We have (1/3)x and (5/12)x. To add these fractions, we need a common denominator. The least common multiple of 3 and 12 is 12, so we'll rewrite (1/3)x as (4/12)x. Now we can add the terms:

(4/12)x + (5/12)x = (9/12)x

We can simplify the fraction 9/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us:

(9/12)x = (3/4)x

Now, let's substitute this simplified term back into our equation:

y = (3/4)x - 3

Our next goal is to isolate the term with 'x'. To do this, we need to get rid of the -3 on the right side of the equation. We can do this by adding 3 to both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other to keep it balanced:

y + 3 = (3/4)x - 3 + 3

This simplifies to:

y + 3 = (3/4)x

We're almost there! Now, we need to get 'x' by itself. It's currently being multiplied by 3/4. To undo this multiplication, we can multiply both sides of the equation by the reciprocal of 3/4, which is 4/3:

(4/3)(y + 3) = (4/3)(3/4)x

On the right side, the (4/3) and (3/4) cancel each other out, leaving us with just 'x'. On the left side, we have to distribute the 4/3 to both terms inside the parentheses:

(4/3)y + (4/3)(3) = x

This simplifies to:

(4/3)y + 4 = x

And there you have it! We've successfully solved for 'x'. Our equation now tells us that the total number of students in the school is equal to (4/3) times the number of students under 14, plus 4. So, if we're given the number of students under 14 ('y'), we can plug it into this equation and easily find the total number of students ('x').

For instance, let’s assume the number of students under 14, represented by 'y', is 279. We can substitute this value into our equation to find 'x', the total number of students.

So, the equation becomes:

x = (4/3) * 279 + 4

First, let's multiply (4/3) by 279:

(4/3) * 279 = (4 * 279) / 3

To simplify, we can divide 279 by 3:

279 / 3 = 93

Now, multiply the result by 4:

4 * 93 = 372

So, (4/3) * 279 equals 372.

Next, we add 4 to this result:

372 + 4 = 376

Therefore, if the number of students under 14 is 279, the total number of students, x, is 376.

This solution illustrates how the formula x = (4/3)y + 4 can be used to quickly calculate the total number of students in the school given the number of students under 14.

Practical Applications: Why This Matters

Now, you might be thinking,