Question:
Suppose an item costs $\,\$50\,$.
If the price increases by $\,19\%\,$, and then decreases by $\,30\%\,$, the new price is:
Solution:
$(0.7)(1.19)(\$50) = \$41.65$
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:
$\cssId{s46}{(1 + 0.85)(1 - 0.38)(x)}
\cssId{s47}{= (1.85)(0.62)x}
\cssId{s48}{= 1.15x}$
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:
$(1.5)(0.5)(x) = 0.75x$
Question:
Suppose an item costs $\,x\,$.
If the price increases by $\,50\%\,$, and then increases by $\,50\%\,$, the new price is:
Answer:
$(1.5)(1.5)(x) = 2.25x$
Question:
Suppose an item costs $\,\$100\,$.
If the price decreases by $\,50\%\,$, and then decreases by $\,50\%\,$, the new price is:
Answer:
$(0.5)(0.5)(x) = \$25.00$