3.2 Solve Percent Applications
- Translate and solve basic percent equations
- Solve percent applications
- Find percent increase and percent decrease
- Solve simple interest applications
- Solve applications with discount or mark-up
Assignment
Translate and Solve Basic Percent Equations
Remember that to find part of something, you use multiplication. Half of a number is $\frac{1}{2}x$, twice something is $2x$, and 35% of something is $0.35x$. Percentage problems are generally multiplication problems, so you will be keying into the word âofâ a lot.
The only catch is remembering to convert percentages to decimals when doing the math. The word percent literally means âper 100â, so if you see the symbol, remember to divide by $100$ or move the decimal two places.
Example 1
What number is $35\%$ of $90$?
âWhat numberâ is our variable, then we have âisâ for equals, then our multiplication.
\[\begin{align} x = 0.35 \cdot 90 \end{align}\]After the set-up, this is calculator work to find the exact value.
\[\begin{align} x &= 0.35 \cdot 90\\ x &= 31.5 \end{align}\]$\blacksquare$
Example 2
$6.5\%$ of what number is $\$1.17$?
This one is the reverse order, and now âofâ is attached to our unknown number.
\[\begin{align} 0.065x &= \$1.17 \\ x &= \$18 \end{align}\]$\blacksquare$
Example 3
$144$ is what percent of $96$?
Now we are looking for the percentage. Donât be distracted that the part is larger than the whole. Percents over $100\%$ are perfectly acceptable.
\[\begin{align} 144 &= x\cdot96 \\ 1.5 &= x &\text{Or $150\%$} \end{align}\]$\blacksquare$
Solve Applications with Discount or Markup
Finding percent is one thing, but often that amount has to be applied to some other value. Sale discounts are a common example of decreases, and increases can range from markups to taxes and tips.
Example 4
Dezohn and his girlfriend enjoyed a nice dinner at a restaurant and his bill was $\$68.50$. He wants to leave an $18\%$ tip. If the tip will be $18\%$ of the total bill, what is the new total?
Letâs pare this down to a more manageable problem by cutting out some of the fluff.
The bill was $\$68.50$. He wants to leave an $18\%$ tip. How much is does he have to pay?
Both versions of the problem donât say it outright, but it wants to know how much is $18\%$ of $\$68.50$?
\[\begin{align} x &= 0.18\cdot \$68.50 \\ x &= \$12.33 \end{align}\]Then we need to add that to the original bill.
\[\begin{align} \$68.50+\$12.33 = \$80.83 \end{align}\]Some unsolicited advice on tips: if youâre at a place that requires you to come up with amount yourself, itâs easier to just move the decimal over one, round to a dollar, and double it. That will give you a tip of about $20\%$.
In the problem above that would mean $\$68.50$ becomes $\$6.85$, which then rounds to $\$7.00$, and then doubles to $\$14.00$.
$\blacksquare$
Example 5
The label on Masaoâs breakfast cereal said that one serving of cereal provides $85$ milligrams (mg) of potassium, which is $2\%$ of the recommended daily amount. What is the total recommended daily amount of potassium?
Again, try to cut out the uncessecary parts.
\[\begin{align} 85 &= 0.02x \\ 4250 &= x \end{align}\]$85$ milligrams is $2\%$ of what amount?
$\blacksquare$
Find Percent Increase and Percent Decrease
Percent increases and decreases require finding the difference between the new and original amounts of something, then finding out what percent that is of the original.
Example 6
In 2011, the California governor proposed raising community college fees from $\$26$ a unit to $\$36$ a unit. Find the percent change.
First, the change, which is $\$36-\$26 = \$10$. Now we see what percentage that is of the original.
\[\begin{align} \$10 &= x\cdot \$26 \\ 0.385 &= x \end{align}\]Since the price went up, we can say it was a $38.5\%$ increase.
$\blacksquare$
Example 7
The average price of a gallon of gas in one city in June 2014 was $\$3.71$. The average price in that city in July was $\$3.64$. Find the percent change.
The difference between the prices is $\$0.07$.
\[\begin{align} \$0.07 &= x \cdot \$3.71 \\ 0.019 &= x \end{align}\]The price dropped, so it was a $1.9\%$ decrease.
$\blacksquare$
Solve Simple Interest Applications
Interest is money generated by other money. Itâs more complicated than that, but our goal here is to quickly figure out how much money you would earn on an investment, or how much extra money you would have to pay back on a loan. And that we can do with this formula.
Simple Interest
Let $P$ be the principal (or initial amount of money), $r$ the annual interest rate, and $t$ be years. The amount of interest $I$ earned is defined by
\[\begin{align} I= Prt \end{align}\]
Example 8
Nathaly deposited $\$12\,500$ in her bank account where it will earn $4\%$ interest. How much interest will Nathaly earn in $5$ years?
Since we are looking for interest, this just requires substitution and multiplication.
\[\begin{align} I &= Prt \\ I &= (\$12\,500)(0.04)(5) \\ I &= \$2\, 500 \end{align}\]After four years, the account will have earned $\$2\, 500$.
$\blacksquare$
Example 9
Loren loaned his brother $\$3\,000$ to help him buy a car. In $4$ years his brother paid him back the $\$3,000$ plus $\$660$ in interest. What was the rate of interest?
The interest rate is missing this time, so weâll have to solve for it.
\[\begin{align} I &= Prt \\ \$660 &= (\$3\, 000)\cdot r\cdot (4) \\ \$660 &= \$12\, 000r \\ \frac{\$660}{\$12\,000} &= r \\ 0.055 &= r \end{align}\]The interest rate his brother gave him on the loan was $5.5\%$.
$\blacksquare$