To solve the problem, we need to find the percentages of integers from 1 to 100 and from 1 to 200 whose squares end with the digit 1. Let's break it down step by step.
### Step 1: Identify the integers from 1 to 100 whose squares end with the digit 1.
The squares of integers can end with the digit 1 if the integer itself ends with either 1 or 9.
- **Integers ending with 1:** 1, 11, 21, 31, 41, 51, 61, 71, 81, 91
- **Integers ending with 9:** 9, 19, 29, 39, 49, 59, 69, 79, 89, 99
Counting these:
- There are 10 integers ending with 1.
- There are 10 integers ending with 9.
Thus, the total number of integers from 1 to 100 whose squares end with the digit 1 is:
\[ 10 + 10 = 20 \]
### Step 2: Calculate the percentage \( x \) for integers from 1 to 100.
The total number of integers from 1 to 100 is 100. Therefore, the percentage \( x \) is calculated as:
\[
x = \left( \frac{20}{100} \right) \times 100 = 20\%
\]
### Step 3: Identify the integers from 1 to 200 whose squares end with the digit 1.
We will use the same logic as before. The integers from 1 to 200 that end with 1 or 9 are:
- **Integers ending with 1:** 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191
- **Integers ending with 9:** 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199
Counting these:
- There are 20 integers ending with 1.
- There are 20 integers ending with 9.
Thus, the total number of integers from 1 to 200 whose squares end with the digit 1 is:
\[ 20 + 20 = 40 \]
### Step 4: Calculate the percentage \( y \) for integers from 1 to 200.
The total number of integers from 1 to 200 is 200. Therefore, the percentage \( y \) is calculated as:
\[
y = \left( \frac{40}{200} \right) \times 100 = 20\%
\]
### Step 5: Compare \( x \) and \( y \).
From our calculations:
- \( x = 20\% \)
- \( y = 20\% \)
Thus, we can conclude that:
\[
x = y
\]
### Final Answer:
The correct option is that \( x \) equals \( y \).
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