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Chapter 6: Problem 5

a. How many five-digit integers (integers from 10,000 through 99,999 ) are divisible by 5 ? b. What is the probability that a five-digit integer chosen at random is divisible by \(5 ?\)

Short Answer

Expert verified
There are 18,000 five-digit integers divisible by 5, and the probability that a randomly chosen five-digit integer is divisible by 5 is 0.2.

Step by step solution

01

1. Identify the range of five-digit integers divisible by 5

The given range is from 10,000 to 99,999. We need to find the first and the last five-digit integers divisible by 5.
02

2. Find the first and last five-digit integers divisible by 5

The first five-digit integer divisible by 5 is 10,000, while the last one is 99,995.
03

3. Calculate the number of integers divisible by 5

To find the total number of numbers divisible by 5 within the range, we can use the following formula: $$Total \ Numbers = \frac{Last \ Number \ - \ First \ Number}{Divisor} + 1$$ Where, Last Number = 99,995 First Number = 10,000 Divisor = 5 Plugging the values, $$Total \ Numbers = \frac{99,995 - 10,000}{5} + 1 = \frac{89,995}{5} + 1 = 17,999 + 1 = 18,000$$ There are 18,000 five-digit integers divisible by 5.
04

4. Calculate the probability that a randomly chosen five-digit integer is divisible by 5

The probability is given by the ratio of the number of integers divisible by 5 to the total number of five-digit integers. Total five-digit integers = 99,999 - 10,000 + 1 = 90,000 So, the probability can be calculated as: $$Probability = \frac{Number \ of \ integers \ divisible \ by \ 5}{Total \ number \ of \ five-digit \ integers}$$ $$Probability = \frac{18,000}{90,000} = 0.2$$ The probability that a randomly chosen five-digit integer is divisible by 5 is 0.2.

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Key Concepts

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

Five-Digit Integers
Five-digit integers are numbers that range from 10,000 to 99,999. These numbers are important when studying divisibility since they represent a substantial range. Understanding these numbers means knowing how many you have: exactly 90,000 of them.
The term "five-digit" stems from each number having digits ranging from 1 to 9 for the largest digit, and every subsequent digit can range inclusively from 0 to 9. This means that the smallest is 10,000 and the largest is 99,999. Knowing these boundaries helps when considering problems that involve them, such as calculating probabilities or finding divisible numbers.
Probability Calculation
Probability is a measure of how likely an event is to occur. It鈥檚 a value between 0 and 1, where 0 means the event will not occur, and 1 implies certainty. When computing probabilities for number-related events, we often use ratios.
For five-digit integers divisible by 5, probability involves determining how many such integers exist and comparing that to the total set of five-digit integers. The calculation is straightforward:
  • Calculate the total number of favorable outcomes: integers divisible by 5.
  • Divide this by the total possible outcomes: all five-digit integers.
This results in a probability that tells us the likelihood that a randomly picked five-digit integer will be divisible by 5. In this case, the probability is 0.2, meaning there鈥檚 a 20% chance.
Integer Division by 5
Dividing an integer by 5 involves determining whether an integer results in a quotient without a remainder. In simpler terms, you want to find numbers that fit evenly into 5. For a number to be divisible by 5, it must end in either 0 or 5.
To find the amount of five-digit numbers divisible by 5, you need to locate the smallest and largest five-digit numbers that can be divided by 5. The smallest is 10,000, and the largest is 99,995. Using these, you can employ a formula:\[\text{Total Numbers} = \frac{\text{Last Number} - \text{First Number}}{\text{Divisor}} + 1\]Here, you substitute in:
  • Last Number: 99,995
  • First Number: 10,000
  • Divisor: 5
The calculation gives you 18,000, showing there are 18,000 numbers fitting the criteria vs. the total numbers available, which is 90,000.

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Most popular questions from this chapter

A study was dont to determine the efficacy of three different drugs-A, \(B\), and \(C\)-in relieving headache pain. Over the period covered by the study, 50 subjects were given the chance to use all three drugs. The following results were obtained: 21 reported relief from drug \(A\). 21 reported relief from drug \(B\). 31 reported relief from drug \(C\). 9 reported relief from both drugs \(A\) and \(B\). 14 reported relief from both drugs \(A\) and \(C\). 15 reported relief from both drugs \(B\) and \(C\). 41 reported relicf from at least one of the drugs. Note that some of the 21 subjects who reported relief from drug \(A\) may also have reported relief from drugs \(B\) or \(C . A\) similar occurrence may be true for the other data. a. How many people got relief from none of the drugs? b. How many people got relief from all three drugs? c. Let \(A\) be the set of all sabjects who got relief from drug \(A, B\) the set of all subjects who got relief from drug \(B\), and \(C\) the set of all subjects who got relief from drug \(C\). Fill in the numbers for all eight regions of the diagram below. d. How many subjects got relief from \(A\) only?

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