Algebra Word Problem: Student Population Change

To solve this problem, we will translate the Japanese text into mathematical equations.

Text Translation:
"There is a junior high school where the total number of students (male and female combined) last year was 450. This year, the number of boys decreased by and the number of girls increased by , resulting in a total increase of 18 students in the entire school population.

(1) Let the number of boys last year be and the number of girls last year be . Find and . (Answer in the format without units. Example: )

(2) Find the number of boys this year. (Answer without units. Example: )"

From the problem, we can derive two equations:

Equation 1: Total students last year
The sum of boys and girls last year was 450.

Equation 2: Change in student population
The number of boys decreased by ( ) and the number of girls increased by ( ), leading to a net increase of 18 students.

To make Equation 2 easier to work with, let’s multiply the entire equation by 100 to remove the decimals:

Now we have a system of linear equations:

From Equation 1, we can express in terms of :

Substitute this into Equation 2:

Now, find :

Wait, let me re-check the provided answer in the image. The image says "答え ". Let’s check if those values satisfy the conditions.
If :

Let’s re-read the image carefully.
“男子が 減少し, 女子が 増加したので全校生徒が 人増加した。”

If , then .
There might be a typo in the image’s provided answer or the problem text. Let’s re-calculate based on the text provided.

If the increase was 18:


Subtracting the equations:

Let’s check the image again. Is it possible the increase was 9? Or is it possible the decrease was ?
If and the increase is 18:
.
If the decrease was : .
If the increase was : .

Actually, looking at the image’s answer , let’s see what change would result in 18.
(11%).
Or (4%).

Conclusion on Part 1: Based strictly on the text provided in the image (increase of 18, decrease of 10%, increase of 8%), the mathematical solution is . However, the image provides an answer key of . This suggests a typo in the problem’s numbers in the original source. I will provide the solution based on the text’s numbers.

Using the values derived from the text ( boys last year):
The number of boys decreased by .

If we use the image’s answer key ( ):

(1) 100,350
(2) 90