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Chapter 16: Problem 14

The percentage of integers from 1 through 100 whose squares end with the digit 1 is \(x \%\), and the percentage of integers from 1 through 200 whose squares end with the digit 1 is \(y \%\). Which one of the following is true? (A) \(x=y\) (B) \(\quad x=2 \mathrm{y}\) (C) \(x=4 y\) (D) \(y=2 x\) (E) \(y=4 x\)

Short Answer

Expert verified
x = y = 20%.

Step by step solution

01

Explore the End Digit Pattern

To determine which numbers between 1 and 100 end in 1 when squared, we need to find digits whose squares result in a number ending in 1. Analyze the numbers. Squaring the units digits from 0 to 9 gives the following results: 0^2=0, 1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25, 6^2=36, 7^2=49, 8^2=64, 9^2=81. Only numbers ending in 1 and 9 will have squares ending in 1.
02

Count and Calculate x% for 1 to 100

Identify numbers from 1 to 100 whose units digit is 1 or 9, as only these will end in 1 after squaring. These numbers are: 1, 9, 11, 19, 21, 29, 31, 39,..., 91, 99. Count them: there are 10 numbers ending in 1 and 10 ending in 9, totaling 20. Calculate the percentage: \(x = \frac{20}{100} \times 100\% = 20\%\).
03

Count and Calculate y% for 1 to 200

Apply the same principle to numbers from 1 to 200. Each subset (1-100 and 101-200) contains the same count of numbers with units digits 1 and 9. So in 101-200: 101, 109, ... 191, 199 also count as 20 numbers. Total such numbers from 1 to 200 are \(20 + 20 = 40\). Calculate the percentage: \(y = \frac{40}{200} \times 100\% = 20\%\).
04

Compare x% and y%

From the counts, we find both the percentage for numbers 1-100 and 1-200 whose squares end in 1 is 20%. So \(x = y = 20\). Therefore, the relationship between \(x\) and \(y\) is option (A) \(x=y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integer Properties
Understanding integer properties is a fundamental aspect of many math problems, including GMAT-style math problems. An integer is a whole number that can be either positive, negative, or zero, and they play a crucial role in calculations. Key integer properties include:
  • Even and Odd: Integers are either even or odd. An even integer is divisible by 2, whereas an odd integer is not.
  • Divisibility: This refers to dividing an integer by another without getting a remainder. Knowing divisibility rules helps simplify complex problems.
  • Prime Numbers: Integers greater than 1 that have no divisors other than 1 and themselves.
  • Composite Numbers: Integers greater than 1 that have more than two divisors.
In the exercise, understanding that numbers whose units digit is 1 or 9 will result in those whose square ends in 1 is an application of these integer properties. This specific understanding reflects how square numbers behave concerning integer properties.
Number Patterns
When dealing with number patterns, recognizing these is crucial for solving GMAT math problems efficiently. Here are common types of patterns:
  • Repeating Patterns: Part of the numbers follow a cycle. Recognizing the cycle can simplify counting or prediction tasks.
  • Arithmetic Sequences: A set of numbers with a common difference between consecutive terms.
  • Geometric Sequences: A set where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
In the original problem, identifying that numbers with units digit 1 or 9 repeat every 10 numbers (i.e., 9, 19, 29, etc.) from 1 to 100 helps solve the problem efficiently. Recognizing these cycles enhances calculation speed and accuracy.
Percentage Calculation
Percentage calculations are essential in evaluating numerical relationships or comparisons, especially in standardized tests. Calculating percentages involves:
  • Understanding Ratios: A percentage is a fraction of 100. If you have a part and a total, you can find the percentage by dividing the part by the total and multiplying by 100.
  • Dealing with Proportions: Calculating what portion an individual number is of a total set.
  • Converting Decimals to Percent: Multiply by 100 and place a % sign right next to it.
In the exercise provided, we calculated the percentage of numbers that, when squared, end with 1. For example, for numbers 1 through 100, we found that 20 out of 100 yielded squares ending in 1, which translates to a percentage of 20%.
Square Numbers
Square numbers are obtained by multiplying an integer by itself. Recognizing them can help solve various math problems. Key points about square numbers:
  • Pattern Recognition: Consecutive square numbers increase by an odd number progression (e.g., 1, 4, 9, 16...).
  • Last Digit Impact: The square of a number is heavily influenced by its units digit.
  • Perfect Squares: These are integers derived by squaring whole numbers.
In the context of the exercise, recognizing that only numbers ending in 1 or 9 will have their squares end in 1 can be crucial. This is because the unit's digit in squaring dictates the last digit of the square product. Understanding this helps you quickly identify patterns and solve related problems.

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Most popular questions from this chapter

8 is \(4 \%\) of \(a\), and 4 is \(8 \%\) of \(b\). \(c\) equals \(b / a\). What is the value of \(c\) ? (A) \(1 / 32\) (B) \(1 / 4\) (C) 1 (D) 4 (E) 32

The list price of a commodity is the price after a \(20 \%\) discount on the retail price. The festival discount price on the commodity is the price after a \(30 \%\) discount on the list price. Customers purchase commodities from stores at a festival discount price. What is the effective discount offered by the stores on the commodity on its retail price? (A) \(20 \%\) (B) \(30 \%\) (C) \(44 \%\) (D) \(50 \%\) (E) \(56 \%\)

If \(\frac{x+y}{x-y}=\frac{4}{3}\) and \(x \neq 0\), then what percentage (to the nearest integer) of \(x+3 y\) is \(x-3 y ?\) (A) \(20 \%\) (B) \(25 \%\) (C) \(30 \%\) (D) \(35 \%\) (E) \(40 \%\)

If \(b\) equals \(10 \%\) of \(a\) and \(c\) equals \(20 \%\) of \(b\), then which one of the following equals \(30 \%\) of \(c\) ? (A) \(0.0006 \%\) of \(a\) (B) \(0.006 \%\) of \(a\) (C) \(0.06 \%\) of \(a\) (D) \(0.6 \%\) of \(a\) (E) \(6 \%\) of \(a\)

If \(b\) equals \(10 \%\) of \(a\) and \(c\) equals \(20 \%\) of \(b\), then which one of the following equals \(30 \%\) of \(c\) ? (A) \(0.0006 \%\) of \(a\) (B) \(0.006 \%\) of \(a\) (C) \(0.06 \%\) of \(a\) (D) \(0.6 \%\) of \(a\) (E) \(6 \%\) of \(a\)
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