Percentages

A percentage is a way of expressing a number as a fraction of 100. To convert any number into a percentage you would multiply the number by 100 and then add the ‘%’ sign to the end. (Remember, when multiplying by 100, simply move the decimal place two spaces to the right).

Since you multiply by 100 to convert a decimal number to a percentage, you divide by 100 to convert a percentage to a decimal number. For example, 74%=0.74 (dividing by 100 means you move the decimal place two spots to the left).

Can you calculate what 44% of 80 is? To do so you would first convert 44% to its decimal form of 0.44 and then multiply this by 80 to get the answer.

Here’s a slightly trickier question: if 80% of a class passes a test and 6 students fail the test, how many students are in the class?

To answer this question observe that since 80% passed then 20% failed, so 20% of the total class size is 6. We can let x denote the total class size and then set up a simple equation to solve.

Solving the equations shows that the class has 30 students. The number of students that passed would be:

Percentages are usually used for numbers between 0 and 1 (or between 0% and 100%), but it is possible to have more then 100% of something. For example if we earned $15,000 one year, then earned $40,000 the next year, our income would have increased by 167%.

As in the previous example, percentages are often used to represent a change in something. Take the Super Bowl for example: 111 million people tuned in to watch Super Bowl 45 and 106.5 million watched Super Bowl 44. From these numbers we can calculate that viewership of the game grew by 4.23% between these years.

To calculate the percent change between two quantities divide the absolute change in quantities by the original quantity then multiply by 100 to get a percentage. The absolute change is equal to the second (newer) quantity minus the first (older) quantity.

Notice that we do not need the ‘million’ to calculate the percentage change; it factors out of the equation:

Percentage changes can also be negative. Say you bought a new car for $15,000. If you appraised the car two years later, it would likely have a much lower market value of say $11,500. The market value of the car would have changed by -23.33%.

Percentages don’t always need to involve a change of something. Taxes are usually denoted in percentage terms. If sales tax in your state is 6%, how much tax would you pay on a $150 purchase?

You would pay $9 in sales tax on the purchase.

Here is one last example: say you purchased an item on sale for 70% off. You paid $15 for the item (ignore tax). How much did the item originally cost?

Since you purchased the item for 70% off, you paid 30% of the original price. Letting x be the original price we have:

So, the item originally cost $50.

Practice questions

Solve

1. 15% of 100
2. 15% of 90
3. 85% of 90
4. What percent of 200 is 320?
5. If 25% of a number is 13, then what is the number?
6. If 30% of a number is 60, then what is the number?
7. If a store sells 120 lamps one year, then 165 lamps the next year, by what percentage has their lamp sales increased by?
8. If a movie ticket costs $8.00 one year, then $9.50 two years later, by what percentage has the ticket price increased by?
9. If you spend $35 on a jacket during a 60% off clearance sale, what is the original price of the jacket? (ignore tax)
10. Only 12% of the people who applied to a certain university were accepted. If 1,020 people were accepted to the university, how many applied?

Solutions

1. \( 0.15\cdot100=15 \)
2. \( 0.15\cdot90=13.5 \)
3. Solution: $$ \begin{align*} & 0.85\cdot90=76.5 \\ & \text{ also observe: } 90-13.5=76.5 \end{align*} $$
4. \( 320/200=1.6=160\% \)
5. \( 0.25\cdot x=13\rightarrow x=13/0.25=52 \)
6. \( 0.30\cdot x=60\rightarrow x=60/0.3=200 \)
7. \( \frac{165-120}{120}=0.375=37.5\% \)
8. \( \frac{9.5-8}{8}=0.1875=18.75\% \)
9. \( 0.4\cdot x=\$35\rightarrow x=\$35/0.4=\$87.50 \)
10. Solution: $$ \begin{align*} & 0.12\cdot x=1,020 \\ & \rightarrow x=1,020/0.12=8,500 \text{ applicants} \end{align*} $$