Problems Involving Percent Increase and Decrease

Want more practice with percents and related concepts?

Here, you will practice solving problems involving percent increase and decrease. You may use a calculator for these exercises.

Examples

Question: Suppose an item costs $\,\$50\,.$ If the price increases by $\,19\%\,,$ and then decreases by $\,30\%\,,$ the new price is:

Solution:
$$\cssId{s8}{(0.7)(1.19)(\$50) = \$41.65}$$

Why? To increase any amount by $\,19\%\,,$ just multiply by $\,1.19\,$:

$$ \cssId{s11}{x + 0.19x} \cssId{s12}{= 1x + 0.19x} \cssId{s13}{= 1.19x} $$

Notice that when you increase, you multiply by a number greater than $\,1\,.$

If you decrease any amount by $\,30\%\,,$ then $\,70\%\,$ remains:

$$ \cssId{s16}{x - 0.3x} \cssId{s17}{= 1x - 0.3x} \cssId{s18}{= 0.7x} $$

Thus, to decrease any amount by $\,30\%\,,$ just multiply by $\,0.7\,.$ Notice that when you decrease, you multiply by a number less than $\,1\,.$

Combining these ideas:

$\$50$	original amount
$(1.19)(\$50)$	new amount, after the $\,19\%\,$ increase
$(0.7)\cdot (1.19)(\$50)$	new amount, after the $\,30\%\,$ decrease
$(0.7)(1.19)(\$50) = \$41.65$	round dollar amounts (as needed) to two decimal places

What if we switch the order of applying the increase/decrease?

$\$50$	original amount
$(0.7)(\$50)$	new amount, after the $\,30\%\,$ decrease
$(1.19)\cdot (0.7)(\$50)$	new amount, after the $\,19\%\,$ increase
$(1.19)(0.7)(\$50) = \$41.65$	round dollar amounts (as needed) to two decimal places

Same result! Since $\,(1.19)(0.7) = (0.7)(1.19)\,,$ you can do the multiplication in whatever order you prefer.

Question: Suppose an item costs $\,x\,.$ If the price decreases by $\,38\%\,,$ and then increases by $\,85\%\,,$ the new price is:

Answer:

$$ \begin{align} &\cssId{s46}{(1 + 0.85)(1 - 0.38)(x)}\cr\cr &\qquad \cssId{s47}{= (1.85)(0.62)x}\cr\cr &\qquad \cssId{s48}{= 1.15x} \end{align} $$

In this exercise, all answers are rounded to two decimal places.

Question: Suppose an item costs $\,x\,.$ If the price decreases by $\,50\%\,$ and then increases by $\,50\%\,,$ the new price is:

Answer:

$$\cssId{s54}{(1.5)(0.5)(x) = 0.75x}$$

Question: Suppose an item costs $\,x\,.$ If the price increases by $\,50\%\,,$ and then increases by $\,50\%\,,$ the new price is:

Answer:

$$\cssId{s59}{(1.5)(1.5)(x) = 2.25x}$$

Question: Suppose an item costs $\,\$100\,.$ If the price decreases by $\,50\%\,,$ and then decreases by $\,50\%\,,$ the new price is:

Answer:

$$\cssId{s64}{(0.5)(0.5)(x) = \$25.00}$$

(Here, $\,x = \$100\,.$)

Concept Practice

All answers are rounded to two decimal places.