#### How much is a 100% increase?

An increase of 100% is the original plus 100% of the original, which is the same as doubling or multiplying by 2. So 30 increased by 100% is 60 because you are adding 100% of 30 to 30.

##### Example

If the price of a stock drops from $60 to$45, what is the percentage decrease?

### Solution

You are trying to find the percent decrease based on the original value and the new value.

change/original = percent/100
so
new – original/original = percent/100

45 – 60/60 = x/100
-15/60 = x/100

The negative tells you this is a decrease. You can leave it out of the rest of the calculations as long as you remember to use the correct terminology (e.g., “decreased” or “dropped”).

15/60 = 1/4 = 25/100 = 25%

For this fraction, it is easier to simplify than cross-multiply.

The price dropped by 25%.

##### Example

The monthly cost of cable internet service went from the introductory price of $39.99 a month to$50.99 a month. What was the percentage increase from the introductory price?

### Solution

The original price here is the introductory price.

change/original = percent/100
which is
new – original/original = percent/100

50.99 – 39.99/39.99 = x/100
11/39.99 = x/100
39.99x = 1100 (Round and use 40 instead of 39.99)
x = 27.5

The price increased by 27.5%.

#### Discounts and Markups

A percentage can be used to apply the same proportion of change to multiple values.

Discount is the decrease in price of an item when the price is decreased by a certain percentage.

Markup is the increase in price when the cost of an item is increased by a certain percentage. The following examples illustrate this concept.

For markups and discounts, calculate:
new – original/original = percent/100

If the value is negative, it is a discount. If the value is positive, it is a markup.

##### Example

A pair of aerobic shoes was priced $115 and is now discounted to$69. What is the percentage discount?

### Solution

new – original/original = percent/100

69 – 115/115 = x/100
-46/115 = x/100

The negative tells you this is a decrease. You can leave it out of the rest of the calculations.

46/115 = 2 × 23/5 × 23 = 2/5 = 40/100 = x/100
x = 40

The discount is 40%.

##### Example

A pair of aerobic shoes is purchased at wholesale for $69 and sold by the store for$115. What is the percentage markup?

### Solution

new – original/original= percent/100

115 – 69/69 = x/100
46/69 = x/100

Notice the original price here is $69, not$115.

46/69 = 2 × 23/3 × 23 = 2/3 = 67/100 = x/100
x = 67

The markup is 67%.

An employee is to mark up the price of a piece of jewelry by 120%. If its wholesale cost was $110, what will be its selling price? ### Solution Notice that the price is being marked up by 120%, not to 120%. The amount of the markup is 120% of$110 so it’s 1.2 × 110 = $132. The selling price is then the original price plus the markup, so$110 + $132 =$242.

Another way to calculate the selling price is 120% + 100% = 220%, so 2.2 × $110 =$242

##### Example

A college bookstore purchases trade books on a 30% margin, i.e., it purchases a trade book for 30% less than its retail price. What is the percentage markup from the wholesale price?

### Solution

The wholesale price is the retail price minus 30% of the retail price.

wholesale = retail – (30% of retail) = (100% – 30%) × retail = 70% × retail

So the wholesale price is 70% of the retail price.

But don’t stop there. It is a common GRE trick to require you to use the result of one part of a question to get the final answer.

This question is asking for the markup from wholesale to retail.

### Solution

Since the markdown is 40%, the purchase price is 60% of the original price.
100% – 40% = 60%
purchase price = 60% of original price

Let x be the original price.

0.6x = 240
x = 240/0.6 = 240/6/10 = 240 × 10/6 = 400

### Solution

Since the markdown is 25%, the purchase price is 75% of the original price.
100% – 25% = 75%
sale price = 75% of original price

Let s be the sale price.

0.75(80) = s
s = 0.75 × 80 = 3/4 × 80 = 60

### Solution

Let Kent’s income be k. Divide his income into the tax brackets:

• For $0 to$10,000, there is no tax.
• The $10,000 between$10,000 to $20,000 is taxed at 20%. • Income beyond$20,000 is taxed at 30%.

So, the equation is:

10,000 (0%) + 10,000 (20%) + (k – 20,000)(30%) = 14,000
0 + 2000 + 0.3(k – 20,000) = 14,000
2000 + 0.3k – 6000 = 14,000
0.3k = 18,000
k = 18,000 × (10/3) = 60,000

### Solution

In the first year, $100 × 110% =$110.

For the second year, $110 × 120% =$132.

The stock price would be $132. Notice that this is not the same price change as 10% + 20% = 30%. A 30% increase would have been resulted in a$130 price.

#### Fractions and Percentile

##### Example

Joe’s portfolio lost 80% of its value, then gained back 10% of its value. What was Joe’s final percentage loss?

### Solution

Use $100 for the original value to help solve the question. The portfolio lost 80%. 100% − 80% = 20%, so$20 was left.

### Solution

The discount is not 30% + 20% = 50%.

30% off means you pay 70%.
Another 20% off means paying 80% of the 70%.
$50 × 0.8 × 0.7 =$50 × 0.56 = $28 A way to check this is to calculate the dollar amount coming off the price. 30% off of$50 = 0.3 × $50 =$15. So the purchase price before getting to the register is $50 ×$15 = $35. 20% off of$35 = 0.2 × $35 =$7
So the final purchase price is $35 ×$7 = \$28.

#### Common Conversions

1 = 100%
3⁄4 = 75%
1⁄2 = 50%
1⁄4 = 25%

1⁄10 = 10%
3⁄10 = 30%
7⁄10 = 70%
9⁄10 = 90%

1⁄5 = 2⁄10 = 20%
2⁄5 = 4⁄10 = 40%
3⁄5 = 6⁄10 = 60%
4⁄5 = 8⁄10 = 80%

1⁄3 = 33.33
2⁄3 = 66.66

For many other denominators, you can remember the first value then multiply to get other values.
For example, if you remember that 1/8 = 0.125, then 3/8 = 3 × 0.125 = .375

1⁄6 = 16.66%
5⁄6 = 83.66%

1⁄7 = 14.28%
2⁄7 = 28.56%
3⁄7 = 42.85%
4⁄7 = 57.14%
5⁄7 = 71.42%
6⁄7 = 85.71%

1⁄8 = 0.125 = 12.5%
3⁄8 = 0.375 = 37.5%
5⁄8 = 0.625 = 62.5%
7⁄8 = 0.875 = 87.5%

1⁄9 = 11.1%
2⁄9 = 22.2%
4⁄9 = 44.4%
5⁄9 = 55.6%
7⁄9 = 77.7%
8⁄9 = 88.8%

1⁄11 = 9.09%
2⁄11 = 18.18%
3⁄11 = 27.27%
4⁄11 = 36.36%
5⁄11 = 45.45%
6⁄11 = 54.54%
7⁄11 = 36.36%
8⁄11 = 72.72%
9⁄12 = 81.81%
10⁄11 = 90.90%

1⁄12 = 8.3%
5⁄12 = 41.7%
7⁄12 = 58.3%
11⁄12 = 91.7%