Math Problem Statement

A town has a population of 18,000. If the number of males increases by 20% and the number of females increases by 15%, and the total population becomes 21,000, what is the difference between the current number of males and females?

Solution

Let the current number of males be xx and the current number of females be yy. From the given information, the total population of the town is:

x+y=18,000(equation 1)x + y = 18,000 \quad \text{(equation 1)}

After the increases, the number of males becomes 1.2x1.2x (a 20% increase), and the number of females becomes 1.15y1.15y (a 15% increase). The total population after the increases is 21,000, so we have:

1.2x+1.15y=21,000(equation 2)1.2x + 1.15y = 21,000 \quad \text{(equation 2)}

Now, we have a system of two equations:

  1. x+y=18,000x + y = 18,000
  2. 1.2x+1.15y=21,0001.2x + 1.15y = 21,000

Solving the system:

From equation 1, we can express yy as:

y=18,000xy = 18,000 - x

Substitute this into equation 2:

1.2x+1.15(18,000x)=21,0001.2x + 1.15(18,000 - x) = 21,000

Simplify:

1.2x+1.15(18,000)1.15x=21,0001.2x + 1.15(18,000) - 1.15x = 21,000 1.2x+20,7001.15x=21,0001.2x + 20,700 - 1.15x = 21,000 (1.2x1.15x)=21,00020,700(1.2x - 1.15x) = 21,000 - 20,700 0.05x=3000.05x = 300 x=3000.05=6,000x = \frac{300}{0.05} = 6,000

So, the number of males is x=6,000x = 6,000.

Now, substitute this back into equation 1 to find yy:

6,000+y=18,0006,000 + y = 18,000 y=18,0006,000=12,000y = 18,000 - 6,000 = 12,000

Conclusion:

The current number of males is 6,000, and the current number of females is 12,000. The difference between the number of males and females is:

6,000\boxed{6,000}

Would you like further details or have any questions?

Here are 5 questions related to this problem:

  1. What would the total population be if only the males increased by 20%?
  2. How would the population change if both males and females increased by 10%?
  3. What if the population increase was limited to only males increasing by 25%?
  4. Can you determine the percentage of the population that are females initially?
  5. How would you represent this scenario graphically?

Tip: When solving word problems, always start by translating the conditions into algebraic equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Percentage Increase

Formulas

x + y = 18,000
1.2x + 1.15y = 21,000
y = 18,000 - x
0.05x = 300

Theorems

Linear Equation Solution

Suitable Grade Level

Grades 8-10