1. If from each of two numbers you subtract half of the smaller one, the remainder of the larger one is three times the remainder of the smaller one. How many times larger is the larger number than the smaller one?
1. The first way to solve the problem is to represent the condition of the problem in the form of a drawing.

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-079.jpg?height=226&width=556&top_left_y=732&top_left_x=790)

From the drawing, it is clear that the first number is twice as large as the second.

The second way to solve the problem is to solve it without relying on a drawing. Since the remainder of the smaller number is half of the number itself, the larger number contains 4 halves of the smaller number and is therefore twice as large as the smaller one.

The third way to solve (Shabanov Konstantin, student of Gymnasium No. 24).

Let there be two numbers \(a\) and \(b\), where \(a > b\). Then

\[
\begin{aligned}
& a - b / 2 = x \\
& b - b / 2 = y, \quad y = b / 2
\end{aligned}
\]

By the condition

\[
x : y = 3, \quad x = 3 y
\]

that is

\[
x = 3 b / 2 \text{.}
\]

Since \(a - b / 2 = x\) and \(x = 3 b / 2\), we can find

\[
a = b / 2 + x = x = b / 2 + 3 b / 2 = 2 b \text{.}
\]

The larger number \(a\) is twice as large as the smaller \(b\).

The fourth way to solve (Lakhov Evgeny, student of School No. 12).

It became: the larger one \(-3x\), the smaller one \(-x\).

It was: the larger one \(3x + x = 4x\)

The smaller one \(x + x = 2x, \quad 4x : 2x = 2\) (times).

Answer: the larger number is 2 times larger.