Understanding Conversions Between Percent, %, and Decimal Numbers

Do you recall the rule used when converting a percent to a decimal or vice versa? Many times, students are taught to move the decimal two place values. There are several problems with this rule.

Most of the time, the direction to move the decimal place, right or left, is not understood or remembered correctly. The rule is temporarily memorized while studying the topic; however, it is not properly recalled shortly after being assessed.

The second problem with “moving the decimal two places” is students become accustomed to what the answer should “look like”. For instance, a student will think that an interest rate of 0.02% must have the digit 2 near the decimal; the student will think that the decimal is moved two places to the right. For this thinking, the student would mistakenly conclude that 0.02% is 2. This process is opposite of the correct process. In the event that a student is finding the percent given a decimal number of 15, sometimes students think 15% is the answer either because they did not see a decimal or because they want their answer to “look right” as many common percentages they have seen are between 1% and 99%. Sometimes students will move the decimal two decimal places to make the answer “look right” and state the answer is 0.15%. In this case, the decimal is moved in the wrong direction because 15 = 1500%.

What is a better way to teach percent with a deeper understanding?

Let’s get back to basics and review the meaning of percent. Percent comes from the Latin per centum, meaning “per hundred”, so a percent is simply a ratio of some number to 100. Thus, x % = x/100. Think about what the percent symbol, %, looks like. As we consider the slanted line, /, we see division or ratio. Next, we see two zeros – one above and one below the slanted line.  We see that we have “per 100” or “divide by 100”. For 7%, we have 7% = 7/100 = 0.07. This will be correct for any percent conversion. Given the number 0.003%, we have 0.003% = 0.003 ÷ 100 = 0.00003. This is also true for large numbers; given 1600%, we have 1600% = 1600 ÷ 100 = 16. Using the meaning of percent, the student has reasoning for which direction to move the decimal. We can still reinforce a number divided by 100 is equivalent to moving the decimal two places to the left since dividing by 100 makes the number smaller. In the base-ten-system, each place value is 10 times smaller than the place value directly to its left so moving the decimal one place to the left results in a number that is 10 times smaller than the original number; thus, moving the decimal two places to the left results in a number that is 10 x 10 = 100 times smaller.

Now let’s consider changing from a decimal number to a percent. Our goal is to find a number with “%”. Recall the Identity Property of Multiplication, for all n such that n is a real number, n x 1 = n; we say a number n multiplied by 1 is equal to the number n. For percent, we have 100% = 100 ÷ 100 = 1; we see that 100% = 1. In other words, we can convert a decimal number to percent by multiplying the number by 100% which is equal to 1 and keep the % symbol.

Here are some examples,

          2.5 = 2.5 x 1 = 2.5 x 100% = 250%

          0.003 = 0.003 x 1 = 0.003 x 100% = 0.3 %

Once again, we can reinforce that a number will get larger when we multiply by 100; for this reason, the decimal is moved two places to the right which makes the number larger by a factor of 100 based on place values in base-ten number system. Now the student has sound reasoning for which way to move the decimal. Notice that each time a number is multiplied by 10, each digit in the number becomes 10 times larger and the decimal is moved one place to the right; so, each original digit is in a new place value that is 10 times larger than the original place value for that digit. If a number is multiplied by 10 twice, the decimal moves 2 places to the right because this is in essence multiplying each digit by 10 twice (or multiply each digit by 10 x 10 = 100). This aligns with the structure of place value in the base-ten number system.

Using the fact that 100% is equal to 1 reinforces that a circle graph, or pie chart, represents 1 or 100%. The circle graph represents the same quantity, 1 = 100%. In the circle graph, the sum of the decimals or percentages represented by each section of the of the graph is equal to 1 or 100%, respectively.

The reasoning that the percent symbol, %, means divide by 100 can also be used for the per-mille sign, ‰, which would be parts per 1000 or divide by 1000.

In summary, to covert from percent to decimal form, apply the definition of % as parts per 100 and understand the symbol “%” is “divide by 100”. To convert from a decimal number to percent, multiply the number by 1 which is 100%; just be careful to keep the percent symbol, %, after you multiply.

When we use basic principles, students will connect different lessons, reinforce basic facts and be prepared to apply the basic concepts to future learning.