Word based problem with percentages

Question

If your salary is increased by 10% and then after 6 months by a further 50%, what is the overall % increase?

Answer 1

Assume that in the question the further increase is based on the new salary and not the original one. Let the original salary be $x$.

After the first increase, it becomes $\frac{110}{100}x$. After the second increase it becomes $\frac{150}{100}(\frac{110}{100}x)$. Hence the overall percentage increase from the original is $100(\frac{150}{100}(\frac{110}{100}x)-x)=65$.

Answer 2

NOTE: $\text{orig-salary}$ represents the salary you had before it was changed. $\text{next-salary}$ represents the salary you had after it was increased by 10%. $\text{final-salary}$ represents the salary you had after it was increased by 50%.

Whenever we say "x was increased by y%" we mean this: $$x + (x * y\%) = (x * 100\%) + (x * y\%) = x * (100\% + y\%)$$

So, using the information in your problem, we have the following two equations: \begin{align*} \text{next-salary} &= \text{orig-salary} * (100\% + 10\%)\\ \text{final-salary} &= \text{next-salary} * (100\% + 50\%) \end{align*} Replace $\text{next-salary}$ with $\text{orig-salary} * (100\% + 10\%)$ \begin{align*} \text{final-salary} &= \text{next-salary} * (100\% + 50\%)\\ \text{final-salary} &= (\text{orig-salary} * (100\% + 10\%)) * (100\% + 50\%)\\ \text{final-salary} &= \text{orig-salary} * (110\%) * (150\%)\\ \text{final-salary} &= \text{orig-salary} * 1.10 * 1.50\\ \text{final-salary} &= \text{orig-salary} * 1.650 \end{align*}

The overall increase would be $1.650 - 100\%$ OR $165.0\% - 100\% = 65\%$

The overall increase is $65\%$.

Answer 3

If x is your original salary, 10 % increase in salary is obtained by multiplying 1.1 (since 0.1 is 10% of 1) Similarly 50 % increase in salary is obtained by multiplying 1.5 ( since 0.5 is 50% of 1)

so with two consecutive increments of 10% and 50%. your new salary becomes 1.5 * (1.1*(x)) = 1.65*x.
Work from here on...

Answer 4

I would like to suggest three quick formulas for tackling similar kind of problems:

• If there is successive increase of $x\%$ and $y\%$, then the net change will be $$x + y + \frac{x \cdot y}{100}\%$$
• If there is successive decrease of $x\%$ and $y\%$, then the net change will be $$x + y - \frac{x \cdot y}{100}\%$$
• If there is $x\%$ increase and then $y\%$ decrease, then the net change is $$ x - y - \frac{x \cdot y}{100}\%$$

Exercise: How does the last result change if there is $x\%$ decrease followed by $y\%$ increase?

These results can be easily verified, but it's always better to memorize them before any quantitative aptitude test.