Question

# A number increases from $30$ to $40$ and then decreases from $40$ to $30$. Compare the percent of increase from $30$ to $40$ and that of the decrease from $40$ to $30$.

Hint:
Let us consider a number $a$ and then $a$ increases to $b.$
So, the difference is $b - a$.
Then the percentage of increase $\dfrac{{b - a}}{a} \times 100\%$.
Again, consider, the number $b$ decreases to $a$.
So, the difference is $a - b$
Then the percentage of decrease $\dfrac{{a - b}}{b} \times 100\%$.
With the help of the above formula we will find the percentage of increase and decrease.

It is given that a number increases from $30$ to $40$ and then decreases from $40$ to $30$.
When it increases, $30$ to $40$
The initial number is taken as $30$.
The final number is taken as $40$.
So, it increases by the difference in initial number from final number $40 - 30 = 10$
Now, the percentage of increasing is found using the formula given in the hint, we get $\dfrac{{10}}{{30}} \times 100$
Let us solve the above equation to find the increase in percentage,
$\dfrac{{10}}{{30}} \times 100 = \dfrac{{100}}{3}\% = 33.33\%$
Hence the percentage of increase is $33.33\%$
Next let us consider when the number decreases from $40$ to $30$
The initial number is taken as $40$.
The final number is taken as $30$.
So, it decreases by the difference in final number from initial number $40 - 30 = 10$
Now, the percentage of decreasing is found using the formula given in the hint, we get $\dfrac{{10}}{{40}} \times 100 = 25\%$
Let us solve the above term to find the decrease in percentage,
$\dfrac{{10}}{{40}} \times 100 = \dfrac{{100}}{4}\% = 25\%$
Hence the percentage of decrease is $25\%$.
Now let us compare both the percentage as follows,
So, the percentage of increase from $30$ to $40$ and that of the decrease from $40$ to $30$ is,
$33.33\% - 25\%$
Let us solve the above term we get,
$8.33\%$
Hence,
The percent of increase from $30$ to $40$ is more than that of the decrease from $40$ to $30$.
The percent of increase from $30$ to $40$ and that of the decrease from $40$ to $30$ is $8.33\%$.

Note:
The percent of increase from $30$ to $40$ is more than that of the decrease from $40$ to $30$ because the initial value plays a major role in finding the value that is when the number decreases the initial value is high so the percentage value decreases whereas in the other hand the initial value is low so percentage increases in high compared to that.