Question

The probability of choosing at random a number that is divisible by 6 or 8 from among 1 to 90 is equal to :

• A. $\frac{1}{6}$
• B. $\frac{1}{30}$
• C. $\frac{11}{80}$
• D. $\frac{23}{90}$

Answer 1

The Correct Option is D

Nos. divisible by 6 are 6, 12, 18, ......, 90. Nos. divisible by 8 are 8, 16, 24, ......, 88. Now, total no. divisible by 6 = 15 and total no. divisible by 8 = 11 Now, the no. divisible by both 6 and 8 are 24, 48, 72. $\therefore$ Probability (number divisible by 6 or 8) $ = \frac{15 + 11 - 3}{90} = \frac{23}{90}$

Answer 2

90 is the 15th number divisible by 6
There are 15 numbers divisible by 6 from 1 to 90
Total number divisible by 8:
In the same manner, as 90 is divisible by \(\frac{88}{8}\) is the 11th number divisible by 8.
To count integers divisible by 6 or 8, add the numbers we computed previously and remove the number of numbers divisible by
LCM (6,8)=24 Because we have counted twice.
Total numbers divisible by 24: first find out how many numbers are divisible by 6 or 8 from 1 to 90
Total numbers divisible by 6:
We know that 90 is divisible by 6 and the next number that is divisible by 6 is 96 which is greater than 90.
So, the \(n^{th}\) number divisible by 6 is 90.
\(6n=90\)
\(n=\frac{90}{6}\)
\(n=15\)
We have that 72 is divisible by 24 and the next number divisible by 24 is 96 which is greater than 90. So, the amount of numbers divisible by 6 or 8 from 1 to 90 is: 15+11−3=23
Now, the probability of an event P(E) is given by
\(P(E)=\frac{number\,of\,favorable\,outcome}{total\,number\,of\,outcome}\)
So, the probability of choosing a number is divisible by 6 or 8 from 1 to 90.
\(P(6\,or\,8)=\frac{total\,numbers\,divisible\,by\,6\,or\,8}{total\,numbers\,chosen}\)
\(P(6\,or\,8)=\frac{23}{90}\)

Answer 3

Nos. divisible by 6 are 6,12,18...90. Nos. divisible by 8 are 8,16,24...88. Now, total no. divisible by 6=15 And total no. divisible by 8=11 Now, the no. divisible by both 6 and 8 are 24, 48, 72. So, that no. divisibly by both 6 and 8=3 ∴ Probability (number divisible by 6 or 8) =15+11−390=2390