Convert Percents, Decimals, and Fractions

 

 

Learning Objective(s)

·         Describe the meaning of percent.

·         Represent a number as a decimal, percent, and fraction.

 

 

Three common formats for numbers are fractions, decimals, and percents. Percents are often used to communicate a relative amount. You have probably seen them used for discounts, where the percent of discount can apply to different prices. Percents are also used when discussing taxes and interest rates on savings and loans.

 

 

A percent is a ratio of a number to 100. Per cent means “per 100,” or “how many out of 100.” You use the symbol % after a number to indicate percent.

 

Notice that 12 of the 100 squares in the grid below have been shaded green. This represents 12 percent (12 per 100).

 

 

How many of the squares in the grid above are unshaded? Since 12 are shaded and there are a total of 100 squares, 88 are unshaded. The unshaded portion of the whole grid is 88 parts out of 100, or 88% of the grid. Notice that the shaded and unshaded portions together make 100% of the grid (100 out of 100 squares).

 

 

 

 

 

What percent of this grid is shaded?     A) 3% B) 11% C) 38% D) 62%   Show/Hide Answer A) 3% Incorrect. Three full columns of 10 squares are shaded, plus another 8 squares from the next column. So, there are 30 + 8, or 38, squares shaded out of the 100 squares in the large square. The correct answer is 38%.   B) 11% Incorrect. Three full columns of 10 squares are shaded, plus another 8 squares from the next column. So, there are 30 + 8, or 38, squares shaded out of the 100 squares in the large square. The correct answer is 38%.   C) 38% Correct. Three full columns of 10 squares are shaded, plus another 8 squares from the next column. So, there are 30 + 8, or 38, squares shaded out of the 100 squares in the large square. This means 38% of the large square is shaded.   D) 62% Incorrect. There are 62 small unshaded squares out of the 100 in the large square, so the percent of the large square that is unshaded is 62%. However, the question asked what percent is shaded. There are 38 shaded squares of the 100 squares in the large square, so the correct answer is 38%.

 

 

 

It is often helpful to change the format of a number. For example, you may find it easier to add decimals than to add fractions. If you can write the fractions as decimals, you can add them as decimals. Then you can rewrite your decimal sum as a fraction, if necessary.

 

Percents can be written as fractions and decimals in very few steps.

 

 

Notice in the diagram below that 25% of a grid is also  of the grid, as you found in the example.

 

 

 

Notice that in the previous example, rewriting a percent as a decimal takes just a shift of the decimal point. You can use fractions to understand why this is the case. Any percentage x can be represented as the fraction , and any fraction  can be written as a decimal by moving the decimal point in x two places to the left. For example, 81% can be written as , and dividing 81 by 100 results in 0.81. People often skip over the intermediary fraction step and just convert a percent to a decimal by moving the decimal point two places to the left.

 

In the same way, rewriting a decimal as a percent (or as a fraction) requires few steps.

 

 

In this example, the percent is not a whole number. You can handle this in the same way, but it’s usually easier to convert the percent to a decimal and then convert the decimal to a fraction.

 

 

 

Write 0.645 as a percent and as a simplified fraction.   A) 64.5% and   B) 0.645% and   C) 645% and   D) 64.5% and   Show/Hide Answer A) 64.5% and Correct. 0.645 = 64.5% = .   B) 0.645% and Incorrect. 0.645 = 64.5%, not 0.645%. Remember that when you convert a decimal to a percent you have to move the decimal point two places to the right. The correct answer is 64.5% and .   C) 645% and Incorrect. 0.645 = 64.5%, not 645%. Remember that when you convert a decimal to a percent you have to move the decimal point two places to the right. The correct answer is 64.5% and .   D) 64.5% and Incorrect. To write 0.645 as a percent, move the decimal place two places to the right: 64.5%. To write 0.645 as a fraction, use 645 as the numerator. The place value of the last digit (the 5) is thousandths, so the denominator is 1,000. The fraction is . The greatest common factor of 645 and 1,000 is 5, so you can divide the numerator and denominator by 5 to get . The correct answer is 64.5% and .

 

 

In order to write a fraction as a decimal or a percent, you can write the fraction as an equivalent fraction with a denominator of 10 (or any other power of 10 such as 100 or 1,000), which can be then converted to a decimal and then a percent.

 

 

If it is difficult to find an equivalent fraction with a denominator of 10, 100, 1,000, and so on, you can always divide the numerator by the denominator to find the decimal equivalent.

 

 

Write  as a decimal and as a percent.   A) 80.0 and 0.8%   B) 0.4 and 4%   C) 0.8 and 80%   D) 0.8 and 8%   Show/Hide Answer A) 80.0 and 0.8% Incorrect. Notice that 10 is a multiple of 5, so you can rewrite  using 10 as the denominator. Multiply the numerator and denominator by 2 to get . The equivalent decimal is 0.8. You can write this as a percent by moving the decimal point two places to the right. Since 0.8 has only one place to the right, include 0 in the hundredths place: 0.8 = 0.80 = 80%. The correct answer is 0.8 and 80%.   B) 0.4 and 4% Incorrect. To find a decimal equivalent for , first convert the fraction to tenths. Multiply the numerator and denominator by 2 to get . The equivalent decimal is 0.8.  So,  and 0.4 are not equivalent quantities. The correct answer is 0.8 and 80%.   C) 0.8 and 80% Correct. The answer is  = 0.8 = 80%.   D) 0.8 and 8% Incorrect.  It is true that  = 0.8, but this does not equal 8%. To write 0.8 as a percent, move the decimal point two places to the right: 0.8 = 0.80 = 80%. The correct answer is 0.8 and 80%.

 

 

 

All the previous examples involve fractions and decimals less than 1, so all of the percents you have seen so far have been less than 100%.

 

Percents greater than 100% are possible as well. Percents more than 100% are used to describe situations where there is more than one whole (fractions and decimals greater than 1 are used for the same reason).

 

In the diagram below, 115% is shaded. Each grid is considered a whole, and you need two grids for 115%.

 

 

Expressed as a decimal, the percent 115% is 1.15; as a fraction, it is , or . Notice that you can still convert among percents, fractions, and decimals when the quantity is greater than one whole.

 

Numbers greater than one that include a fractional part can be written as the sum of a whole number and the fractional part. For instance, the mixed number  is the sum of the whole number 3 and the fraction .  = 3 + .

 

 

 

Note that a whole number can be written as a percent. 100% means one whole; so two wholes would be 200%.

 

 

 

Write 4.12 as a percent and as a simplified fraction.   A) 0.0412% and   B) 412% and   C) 412% and   D) 4.12% and   Show/Hide Answer A) 0.0412% and Incorrect. To convert 4.12 to a percent, move the decimal point two places to the right, not the left. The correct answer is 412% and .   B) 412% and Correct. 4.12 equals 412%, and the simplified form of  is .   C) 412% and Incorrect. 4.12 does equal 412%, but it is also equivalent to , not . The correct answer is 412% and .   D) 4.12% and Incorrect. To convert 4.12 to a percent, move the decimal point two places to the right. The correct answer is 412% and .

 

 

 

Percents are a common way to represent fractional amounts, just as decimals and fractions are. Any number that can be written as a decimal, fraction, or percent can also be written using the other two representations.