Percent

GMAT Strategy: How to Deal with Percents in the Answer Choices | Kaplan Test Prep

The GMAT tests your ability to do math-and to NOT do math! By plugging in numbers into percent problems on the GMAT’s quantitative section, you can convert ugly fractions into straightforward arithmetic-and into points!

Percent & Conversions

The word percent means per 100. It is a fraction whose denominator is 100. For example, 26% is equivalent to the fraction 26/100.

A percentage is used to compare values. It can describe the relationship of a part to a whole, or it can describe a magnitude of change.

Decimal to Percent
To change a decimal number to a percentage, multiply by 100; this moves the decimal point 2 places to the right. Then add the percent symbol.

0.32 = 32% 0.07 = 7% 1.75 = 175%

Percent to Decimal
To change a percentage to a decimal, divide by 100; this moves the decimal point 2 places to the left. Then remove the percent symbol.

25% = 0.25 90% = 0.90 = 0.9 32.1% = 0.321 100% = 1

Fraction to Decimal
You can convert a fraction to a decimal by dividing the numerator by the denominator. Sometimes a quicker method is to change the fraction so it has 100 as the denominator.

21/100 = 0.21

43/50 = 86/100 = 0.86

13/16 = 13 ÷ 16 = 0.8125

Decimal to Fraction

To convert decimal to a fraction, move the decimal point 2 places to the right and use 100 as the denominator. Then simplify the fraction. If there is still a decimal in the numerator, move the decimal point to the right as needed, and add a 0 to the denominator for each place value.
0.55 = 55/100 = 11/20

2.18 = 218/100 = 109/50 = 2/9/50 or 2.18 = 2/18/100 = 2/9/50

0.123 = 12.3/100 = 123/1000

Note: There is a chart at the end of this section with common fraction and decimal conversion values.
Fraction and Percent
To change a percentage to a fraction, change the percentage to a decimal, then the decimal to a fraction.

34% = 0.34 = 34/100 = 17/50

Correspondingly, to change a fraction to a percentage, change the fraction to a decimal, then the decimal to percent.

18/90 = 2/10 = 0.2 = 20%

Example

Convert 4% into a decimal and a fraction in lowest terms.

Solution

To convert 4% into a decimal, move the decimal point two places to the left. This requires adding a leading 0.
4% = 0.04

To express 4% as a fraction, put it over a denominator of 100, and then simplify the fraction.
4/100 = 4/4 × 25 = 1/25

Multiplying by Percent

Multiplying by a percentage is a way to find a “part” of the original. Multiplying by a percentage is the same as multiplying by the decimal equivalent.

15% of 40 = 15% × 40 = 0.15 × 40 = 6 8% of 25 = 8% × 25 = 0.08 × 25 = 2

quick trick: Multiplying by 10%

To get 10% of anything, simply slide the decimal point one place to the left.
3456 � 10% = 345.6

For 20%, slide the decimal point one place to the left, then multiply by 2.
3456 � 20% = 345.6 � 2 = 691.2

The same pattern hold true for 30%, 40%, 60%, 70%, 80% and 90%. But for 50%, simply divide by 2.

Example

What is 30% of 210?

Solution

You are trying to find the “part” of 210 that is equal to 30%.

30% of 210 = 210 � 30% Change to multiplication.
= 21 � 3 Slide the decimal point one place to the left, then multiply by 3.
= 63

30% of 210 is 63.

You may also be given two numbers (a part and a whole) and asked to find the percentage of the whole represented by the part. One way to do this is to use a proportion, which sets 2 fractions equal to each other. Change the given numbers to a fraction and set it equal to a fraction with denominator 100. Solve for the numerator and that will be the percentage.

Example

What percent of 24 is 9?

Solution

You are trying to find the percent that matches the given fraction.

part/whole = percent/100 so 9/24 = x/100

Method 1

900 = 24x Use cross multiplication.

x = 900/24 = 3 × 3 × 4 × 25/2 × 3 × 4 = 3 × 25/2 = 75/2 = 37.5

Method 2

Start by simplifying the fraction.
9/24 = 3/8

Since you know 1/8 = 0.125, you can quickly calculate 3/8 = 3(0.125) = 0.375 = 37.5%

Percent Multipliers

https://youtu.be/mLgeFLhyERo

Percent Increase and Decrease

Percentages can be used to describe the magnitude of change. To find the percentage of change, you compare the new amount to the original amount.

Percent increase: If the price of a $30 item increases by 10%, the new price is the original $30 plus 10% of the $30 original, $33. This is 110% or 1.1 times the original price. “Increase” may go by other names such as the “markup” from “wholesale” (cost from the factory) to “retail” (cost to the public).

Percent decrease: If the price of an item decreases by 10%, the new price is the original $30 minus 10% of the $30 original, or $27. This is 100% – 10% = 90% or 0.9 times the original price. “Decrease” may go by other names such as “sale” or “discount.”

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 39.399)
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%.

Example

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 is 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 GMAT 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.
To make calculations easier, use $100 rather than a variable as the retail price of a trade book. Then the wholesale is 70% � retail = 70% � $100 = $70.

new – original/original = percent/100
retail – wholesale/wholesale = markup/100

100 – 70/70 = markup/100

30/70 = 3/7 = 42.8/100 = markup/100

So the markup from wholesale to retail is 43%.

Example

Find the number of residents in a city if 20% of them, or 6200 people, ride bicycles.

Solution

Let R be the number of residents. Translate the words in an equation.

20% of R is 6200 so 0.2R = 6200

R = 6200 = 6200
0.2 2
10
Change the decimal to a fraction. To divide by a fraction, multiply by the reciprocal.

R = 6200 × 10 = 62,000 = 31,000
2 2

The city has 31,000 residents.

Sale Prices

In determining sale prices, be careful not to mix up the amount taken off the original price with the new, sale price.

Example

Kathy buys a bike for $240 after a 40% markdown. What was the original price?

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

The original price was $400.

Example

A sweater, originally $80, is on sale for 25% off. What is the sale price?

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

The sale price is $60.

Example

Kent pays 20% tax on income between $10,000 and $20,000 and 30% on income over $20,000. The first $10,000 is tax-free. If he paid $14,000 in taxes, what was his income?

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,

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 = 60,000
3

Kent’s income was $60,000.

Mixture Problems

Algebra Mixture GMAT Question (Medium)

Dr. Steve Alexandris, MIT PhD and Yale University BA in Mathematics, teaches you how to solve a challenging mixture algebra math question. Feel free to ask any questions below. If 10 gallons of grape juice are added to 40 gallons of a mixture that is 10 percent grape juice then what percentage of the resulting mixture is grape juice?

Example

How many gallons of pure water must be added to 100 gallons of a 4% saline solution to produce a 1% saline solution?

Solution

Let x be the gallons of pure water to be added.

In 100 gallons of a 4% saline solution, there are

0.04(100) = 4 gallons of salt.

In the 1% solution, the total number of gallons will be 100 + x. The amount of salt will remain constant at 4 gallons.

0.01(100 + x) = 4
1 + 0.01x = 4
0.01x = 3
x = 3/0.01 = 300

300 gallons of water need to be added to produce a 1% saline solution.

Taking a Percentage of a Percentage

What happens if you take a percentage of a number and then take a percentage of that new value? You are just multiplying the new total by a second percentage. This is a common GMAT trick.

Example

If the price of a stock starts at $100, increases by 10% during the first year, then increases again by another 20% in the second year, what is the stock’s final price?

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

Fractions / Percentile GMAT Question (Easy)

Fractions and Percentiles in an easy questions. Use fractions as a shortcut to answer these questions quicker. Question taught by Dr. Steve, PhD Massachusetts Institute of Technology.

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.

That $20 gained back 10% of its value, so $20 � 110% = $22.

So 22% is the percentage of the original investment that he still has.

Subtract 100% � 22%. The final percentage loss is 78%.

Notice that this is not 100% � 80% + 10% = -70%, or a 70% loss.

Example

A backpack was marked for sale at 30% off, and an additional 20% was taken off at the register. If the original price was $50, what was the final purchase price?

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%

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