Counting Three-Digit Even Multiples of Five: A Comprehensive Guide
Understanding how to count numbers that fit specific criteria, such as being three-digit numbers that are even multiples of five, is an important skill in both mathematics and everyday problem-solving. This article explores the methodology behind counting such numbers, providing a detailed explanation and clarifying common misunderstandings.
Introduction
When asked to identify the number of three-digit even numbers that are multiples of five, it’s crucial to break down the problem carefully. The question provides constraints that we need to adhere to: the numbers must be three-digit, even, and multiples of five. Understanding these constraints is key to solving the problem correctly.
The Constraints and the Solution
The term "three-digit" is defined here as a number ranging from 100 to 999. Additionally, the problem specifies "even numbers," which means the numbers in question must be divisible by 2. However, since multiples of five must end in either 0 or 5, and we are dealing with even numbers, the only option left is numbers that end in 0.
Step-by-Step Calculation
1. Start with the range: The numbers in question must be between 100 and 999. Let’s first list a few examples to understand the pattern:
100, 110, 120, ..., 9902. Identify the pattern: Notice that each hundred has a sequence of ten numbers that are multiples of both 5 and 10 (i.e., even multiples of five): 100, 110, 120, ..., 190; 200, 210, 220, ..., 290; and so on.
3. Generalize the pattern: For every hundred numbers, there are 10 such multiples. Since the range from 100 to 999 includes 9 full sets of 100 numbers (100-199, 200-299, ..., 900-999), we have:
Total numbers 10 numbers/hundred × 9 hundreds 90 numbers
Permutations and Combinations Approach
We can also solve this using permutations and combinations. The number must be a three-digit number (N P Q) where:
N (hundreds place) can be any of the 9 digits from 1 to 9 (9 choices) P (tens place) can be any of the 10 digits from 0 to 9 (10 choices) Q (units place) must be 0 (1 choice)Combining these choices:
Total numbers 9 (choices for N) × 10 (choices for P) × 2 (choices for Q) 180
The Importance of Precision in Problem Statement
It is crucial to pay close attention to the wording of the problem. The original question mentions "even number" instead of "even numbers" or "strictly positive integers," which could lead to misunderstandings. The use of "numbers" suggests that negative and positive integers could be included, but the typical context in such questions usually restricts to positive integers.
Conclusion
Counting three-digit even multiples of five is a straightforward process when understood correctly. By carefully analyzing the constraints and applying logical steps, we can determine that there are 90 such numbers between 100 and 999. Understanding this concept can help in solving similar problems more efficiently and accurately.