Question

A rainy day occurs once in an every 10 days. Half of the rainy days produce rainbows. What percent of all days do not produce rainbows?
(a) 95%
(b) 10%
(c) 50%
(d) 5%

Answer 1

Hint: We solve this problem by assuming the number of days as some variable. Then we find the number of days where we do not get rainbows. We have the formula of probability in percentage as \[\Rightarrow P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\times 100\]. By using the above formula we find the required probability.

Complete step by step answer:
Let us assume that the total number of days as \[x\]
Let us assume that the number of rainy days as \[y\]
We are given that a rainy day occurs once in every 10 days
Here we can see that for every 10 days there will be one rainy day then for \[x\] days the number of rainy days is given as
\[\Rightarrow y=\dfrac{x}{10}\]
Now, let us assume that the number of days that are having rainbows as \[r\]
We are given that half of the rainy days produce rainbows
By converting the above statement into mathematical equation we get
\[\begin{align}
  & \Rightarrow r=\dfrac{1}{2}\left( y \right) \\
 & \Rightarrow r=\dfrac{1}{2}\left( \dfrac{x}{10} \right) \\
 & \Rightarrow r=\dfrac{x}{20} \\
\end{align}\]
Now, let us find the number of days where we get no rainbow.
Let us assume that the number of days of getting no rainbow as \[{r}'\] then we get
\[\begin{align}
  & \Rightarrow {r}'=x-r \\
 & \Rightarrow {r}'=x-\dfrac{x}{20} \\
 & \Rightarrow {r}'=\dfrac{19x}{20} \\
\end{align}\]
Now, let us assume that the percent of getting no rainbow as \[P\]
We know that the formula of probability in percentage as
\[\Rightarrow P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\times 100\]
By using the above formula we get the required percentage as
\[\begin{align}
  & \Rightarrow P=\dfrac{{{r}'}}{x}\times 100 \\
 & \Rightarrow P=\dfrac{\left( \dfrac{19x}{20} \right)}{x}\times 100 \\
 & \Rightarrow P=\dfrac{19}{20}\times 100 \\
 & \Rightarrow P=95\% \\
\end{align}\]
Therefore we can conclude that the percentage of days where we get no rainbow is 95%
So, option (a) is the correct answer.

Note:
Students may make mistakes in finding the number of days of not getting a rainbow.
We are given that half of rainy days produce rainbows
By converting the above statement into mathematical equation we get
\[\begin{align}
  & \Rightarrow r=\dfrac{1}{2}\left( y \right) \\
 & \Rightarrow r=\dfrac{1}{2}\left( \dfrac{x}{10} \right) \\
 & \Rightarrow r=\dfrac{x}{20} \\
\end{align}\]
This also means the half of rainy days do not produce rainbows.
By converting the above statement into mathematical equation we get
\[\begin{align}
  & \Rightarrow {r}'=\dfrac{1}{2}\left( y \right) \\
 & \Rightarrow {r}'=\dfrac{1}{2}\left( \dfrac{x}{10} \right) \\
 & \Rightarrow {r}'=\dfrac{x}{20} \\
\end{align}\]
Students take this number as the number of days of getting no rainbow.
This gives the wrong answer because we are missing some possibilities where we get no rainbow when there is no rain.
So, we do not include those days in the above formula.
So \[{r}'=\dfrac{19x}{20}\] is the correct answer.