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Lesson Explainer: Converting between Fractions, Decimals, and Percentages Mathematics

In this explainer, we will learn how to convert between fractions, terminating decimals, and percentages.

Let’s begin by making sure we are familiar with the vocabulary we will be using.

Definition: Fraction

A fraction, in everyday language, means a part of something. In math, a fraction describes a proportion that compares a part, π‘Ž, to a whole, 𝑏. It tells if one divides the whole in equal portions, how many of these portions make the part we are considering. We call π‘Ž the numerator and 𝑏 the denominator, then write the fraction as the ratio π‘Žπ‘. A fraction can also be understood as a quotient, π‘ŽΓ·π‘.

For example, let’s consider the fraction 25.

The denominator of 5 gives us the number of equal portions in the whole. The numerator of 2 tells us to consider the part of the whole made of two portions. This can be visually demonstrated with a rectangle, where 2 out of 5 equal portions are shaded. The entire rectangle represents the whole and the two shaded portions represent the part.

Definition: Terminating Decimal

A decimal is another way to write a fraction. A terminating decimal is one where the numbers end (or terminate) at some point. Terminating decimals are equivalent to fractions with denominators of 10 or 100 or 1β€Žβ€‰β€Ž000 or 10β€Žβ€‰β€Ž000 or 100β€Žβ€‰β€Ž000 and so on. The number of digits to the right of the decimal point tells us which denominator the equivalent fraction has.

A decimal with only one digit after the decimal point is the number of portions out of 10. We call this the number of tenths. For example, 0.7=710.

A decimal with two digits after the decimal point is the number of portions out of 100. We call this the number of hundredths. For example, 0.09=9100 and 0.79=79100.

A decimal with three digits after the decimal point is the number of portions out of 1β€Žβ€‰β€Ž000. We call this the number of thousandths. For example, 0.001=11000,0.091=911000, and 0.791=7911000.

Any digits to the left of the point represent the whole number in a mixed number. For example, 8.7=8710=8710.

Definition: Percentage

We write a percentage as 𝑝%, meaning 𝑝 out of 100. A percentage is a way to express a given proportion π‘Žπ‘ such that π‘Žπ‘=𝑝100.

We can use a double line diagram to show how π‘Ž and 𝑏 compare in the same way as 𝑝 and 100.

For example, we can express the fraction 34 as a percentage. To find 𝑝%, we start with the equal ratios, 34=𝑝100. Let’s use a double line diagram to help us reason. We may notice that 100 can be split in four portions of 25 each. So, if one-quarter of 100 is 25, then three-quarters is 75. Therefore, 𝑝=75. So, 34 can be expressed as 75%.

By now, we should see that both decimals and percentages are closely related to fractions. This allows us to convert between fractions, decimals, and percentages based on the relationship between the different representations.

To convert a fraction π‘Žπ‘ to a terminating decimal, it can be helpful to first convert the denominator to exactly 10 or any number of tens multiplied together, such as 10Γ—10=100,10Γ—10Γ—10=1000,10Γ—10Γ—10Γ—10=10000,10Γ—10Γ—10Γ—10Γ—10=100000.

It will help us here to think of a fraction as a value in the numerator divided by the denominator. A numerator divided by ten becomes a value terminating in the tenths place. A numerator divided by 100 becomes a value terminating in the hundredths place. To consider larger denominators, we can use a decimal place value table to remind us where different place values terminate after the decimal point. We should note that whole number place values are to the left of the decimal point.

If we have a calculator, this process can be simplified by just doing a division on a calculator, specifically the numerator divided by the denominator. The result of this division on the calculator will be a decimal. Because we may not always have access to a calculator, we should still be familiar with the general method that can be done without technology.

How To: Expressing a Fraction as a Terminating Decimal

Step 1: Find a number that you can multiply (or divide) the denominator by, such that it becomes 10, 100, 1β€Žβ€‰β€Ž000, or any number with a one followed by zeros.

Step 2: Multiply (or divide) both the numerator and denominator by this number.

Step 3: Use the denominator to decide the decimal place (tenths for 10, hundredths for 100, and so on). A decimal system place value table can be used to help determine this.

Step 4: Write the numerator value so that it terminates in the corresponding decimal place, writing extra zeros if necessary.

For example, we can express 25 as 410 by multiplying the numerator and denominator by two. Four portions out of ten means we put the value 4 in the tenths place, resulting in the terminating decimal 0.4.

Let’s see how converting between fractions and decimals can be useful to answer a question about weighing apples.

Example 1: Answering a Word Problem Using Conversion between Fractions, Terminating Decimals, and Percentages

A woman bought 420 kg of apples. She places these apples on a digital scale that displays their weight (in kilograms) as a decimal. What numbers should appear on the display of the digital scale?

Answer

We want to convert the given fraction to a decimal. The easiest way to find the decimal value is by doing a division on a calculator, specifically 4Γ·20. The result of this division on the calculator is 0.2. When we cannot use technology, we should also know how to convert by hand.

Recall that the first three decimal places to the right of the decimal point are tenths, hundredths, and thousandths. Thus, we would like to rewrite our fraction to have a denominator of 10, 100, or 1β€Žβ€‰β€Ž000. There are two ways to proceed.

One option is to simplify the fraction as 4Γ·220Γ·2=210, which gives us a denominator of 10. We then place the value of 2 in the tenths place to the right of the decimal point, giving us 0.2 kilograms.

Another option is to rewrite the fraction as 4Γ—520Γ—5=20100, which gives us a denominator of 100. We then place the value of 20 in the hundredths place to the right of the decimal point, giving us 0.20 kilograms. It is worth noting that there are 10 hundredths in one tenth, so a 20 in the hundredths place is equal to a 2 in the tenths place. This means we could write the answer more simply as 0.2 kilograms.

Now that we have been reminded of how to convert from a fraction to a decimal, let’s try converting in the other direction. To convert a terminating decimal to a fraction, we will first determine what place the decimal terminates. A value terminating in the tenths place becomes the numerator of a fraction with denominator ten. A value terminating in the hundredths place becomes the numerator of a fraction with denominator 100. Then, we can use a decimal place value table to remind us where other place values terminate after the decimal point. We should note that whole number place values are to the left of the decimal point.

How To: Expressing a Terminating Decimal as a Fraction

Step 1: Identify the place value where the decimal terminates. A decimal system place value table can be used to help determine this.

Step 2: Take the numerical digits of the decimal and make that the numerator of a fraction with the denominator identified in step 1 (for tenths, use 10, for hundredths, use 100, and so on).

Step 3: Once the decimal is converted to fraction form, it should be written in its simplest form, where the numerator and denominator have no whole number common factors other than 1. If the resulting improper fraction has a value greater than 1, it can be written as a mixed number.

For example, because the decimal 9.415 terminates in the thousandths place, we will write the value 9β€Žβ€‰β€Ž415 over a denominator of 1β€Žβ€‰β€Ž000. We should check to see if 9415 and 1β€Žβ€‰β€Ž000 have a highest common factor (HCF). Since the HCF is five, we will divide the numerator and denominator by 5, resulting in 9415Γ·51000Γ·5=1883200. This improper fraction is greater than 1, so we need to find the largest multiple of 200 that is also less than 1β€Žβ€‰β€Ž883. That is 9, because 200Γ—9=1800, leaving us with a remainder of 83. The 9 becomes our whole number part and the remainder over 200 becomes the fraction part. Taking the improper fraction apart can be written like this: 1883200=1800200+83200=983200.

Example 2: Converting a Terminating Decimal to a Fraction

Write βˆ’0.15 as a fraction in its simplest form.

Answer

We want to convert the given decimal to a fraction. We should note that converting between decimals and fractions does not change the positive or negative value of the number. Since we are given a negative decimal, the equivalent fraction will also be negative. That means we may reason through our answer without regard for the negative sign, as long as we remember to reinstate the negative sign in our final answer.

Recall that the second decimal place to the right of the decimal point is the hundredth. Thus, we will create a fraction with a denominator of 100. The numerator will be the numerical digits from the original decimal, 15. Putting that together gives us the fraction 15100.

Finally, we must determine if the fraction can be simplified or written as a mixed number. Since, 15100 is less than 1, it cannot be written as a mixed number. However, 15 and 100 have a highest common factor of 5. So, we proceed to divide the numerator and denominator by 5, as shown here: 15Γ·5100Γ·5=320.

Since the terminating decimal 0.15 can be written as 320, βˆ’0.15 can be written as a βˆ’320.

Recall that the word percent literally means β€œout of one hundred.” That is why we model percentages as amounts out of 100. A fraction π‘Žπ‘ can be expressed as a percentage 𝑝%, read β€œπ‘ percent” and meaning 𝑝 out of 100. For this to work, π‘Ž and 𝑏 must compare in the same way as 𝑝 and 100. This is mathematically expressed with equal ratios 𝑝100=π‘Žπ‘. We can use a double line diagram to visualize this proportional relationship.

Multiplying both sides of the equation by 100 looks like 100×𝑝100=π‘Žπ‘Γ—100.

On the left side of the equation, we should notice that 𝑝 is both divided by 100 and multiplied by 100. Those are opposite operations, so we are left with just 𝑝: 𝑝=π‘Žπ‘Γ—100.

As we have shown, multiplying both sides of the equation by 100 gives us the value of 𝑝. Therefore, to express a fraction as a percentage, we can simply multiply the fraction by 100.

Formula: Express a Fraction as a Percentage

A fraction of the form π‘Žπ‘ can be expressed as a percentage 𝑝%, where 𝑝=π‘Žπ‘Γ—100.

For example, 920 can be expressed as a percentage by multiplying the numerator and denominator by 5. This gives us 45100 , where the numerator indicates our percentage is 45%.

In general, we could have multiplied it by 100, resulting in 𝑝=320Γ—100; then, 𝑝=15.

Therefore, 320 can be expressed as 15%.

Let’s practice converting between decimals, fractions, and percentages.

Example 3: Converting a Terminating Decimal to a Fraction, Then to a Percentage

Write 5.07 as a fraction in its simplest form. Then, write 5.07 in percentage form.

Answer

We begin by considering the decimal 5.07. We can use a decimal place value table to identify where 5.07 terminates. The last digit, 7, is in the hundredths place.

So, we can write this as 507 out of 100, which is the fraction 507100. This fraction is in its simplest form, because 507 and 100 have no common factor other than 1. However, 507100 is greater than 1, so we can still write it as a mixed number. The whole number part is 5 because 100Γ—5=500, leaving us with a remainder of 7. The 7 becomes the fraction part of our mixed number. We can write this out as 507100=500100+7100=57100.

Next, we want to write this value as a percentage. Recall that, in general, a fraction π‘Žπ‘ can be expressed as a percentage 𝑝% using the formula 𝑝=π‘Žπ‘Γ—100. And when we calculate 507100Γ—100, we get 𝑝=507. So, 5.07=507100=507%. It is also relevant to recall that if a fraction is of the form 𝑝100, it can be expressed as 𝑝%.

In conclusion, 5.07 can be written as the fraction 57100 or the percentage 507%.

We have seen how to express a fraction as a percentage and now we are ready to explore how to express a percentage as a fraction.

By definition, a percentage 𝑝% means 𝑝 out of 100 and represents the proportion𝑝100. To convert from a percentage 𝑝% to a fraction π‘Žπ‘, we must find values π‘Ž and 𝑏 that compare in the same way as 𝑝 and 100. This is mathematically expressed with equal ratios𝑝100=π‘Žπ‘. We can use a double line diagram to visualize this proportional relationship.

According to this proportional relationship, 𝑝 divided by 100 equalsπ‘Žπ‘. Therefore, to express a percentage as a fraction, we simply write the fraction𝑝100 in its simplest form.

Formula: Expressing a Percentage as a Fraction

A percentage 𝑝% can be expressed as the fraction𝑝100. Then, we can write the fraction in its simplest form, if necessary.

For example, the percentage 32% can be expressed as a fraction by taking 32100 and writing it in its simplest form. Therefore, 32% can be expressed as the simplified fraction 825 because 32Γ·4100Γ·4=825.

Let’s practice writing a percentage as a fraction in its simplest form.

Example 4: Converting a Percentage to a Fraction

Write 120% as a fraction in its simplest form.

Answer

Recall that a percentage 𝑝% can be expressed as the fraction 𝑝100. Therefore, 120% can be expressed as 120100. To ensure our answer is in its simplest form, we will consider whether 120 and 100 have a highest common factor (HCF). Twenty is the HCF of 120 and 100, so we divide the numerator and denominator by 20, resulting in 120Γ·20100Γ·20=65. Because 65 is greater than 1, we can write it as a mixed number. We can think of 65 as 55+15, which gives us the mixed number 115.

Therefore, 120% can be expressed as the simplified fraction 115.

Let’s return to decimals once again to discover how to express a terminating decimal as a percentage.

Recall how a percentage 𝑝% can be expressed as the proportion 𝑝100. To convert from a decimal 𝑑 to a percent, 𝑑 must equal the ratio of 𝑝 to 100. This is mathematically expressed with the equation 𝑝100=𝑑. However, since we are converting from a decimal to a percentage, we want to know what 𝑝 equals.

Multiplying both sides of the equation by 100 looks like 100×𝑝100=𝑑×100.

On the left side of the equation, we should notice that 𝑝 is both divided by 100 and multiplied by 100. Those are opposite operations, so we are left with just 𝑝: 𝑝=𝑑×100.

As we have shown, multiplying both sides of the equation by 100 gives us the value of 𝑝. Therefore, to express a terminating decimal as a percentage, we can simply multiply the decimal by 100.

It is worth noting that each place value is multiplied by 10 moving from right to left. So, moving two places to the left means we are multiplying by 10 twice. Therefore, to quickly find the product of a number and 100, we move all digits to the left two place values.

Formula: Expressing a Terminating Decimal as a Percentage

A terminating decimal 𝑑 can be expressed as a percentage 𝑝%, where 𝑝=𝑑×100.

Multiplying a decimal by 100 results in all digits being moved two place values to the left. Using a decimal place value table may help you move the correct number of places.

For example, we can express 0.185 as a percentage by multiplying by 100, which is to move all digits two decimal place values to the left.

Therefore, the terminating decimal 0.185 can be expressed as the percentage 18.5%.

Now that we have seen how multiplying a decimal by 100 gives us the percentage, it should not be too surprising that we will divide by 100 to convert from a percentage to a decimal.

A percentage can be expressed as a terminating decimal 𝑑 as long as 𝑑 equals the ratio of 𝑝 to 100. This is mathematically expressed with the equation 𝑑=𝑝100. Therefore, to express a percentage as a terminating decimal, we can simply divide the decimal by 100.

It is worth noting that each place value is divided by 10 moving from left to right.

Therefore, to quickly find the quotient of a number and 100, we move all digits to the right two place values.

Formula: Expressing a Percentage as a Terminating Decimal

A percentage 𝑝% can be expressed as a terminating decimal 𝑑, where 𝑑=𝑝100.

Dividing a decimal by 100 results in all digits being moved two place values to the right.

Using a decimal place value table may help you move the correct number of places. Fill in zeros to the left, as needed.

For example, we can express 48% as a decimal by dividing by 100, which is to move all digits two decimal place values to the right. We can then place a zero in the ones place to show there is no whole number part of the decimal.

Therefore, the percentage 48% can be expressed as the terminating decimal 0.48.

In the next example, we will practice converting from a percentage to a decimal.

Example 5: Converting a Percentage to a Terminating Decimal

What is the decimal form of 12%?

Answer

We begin by considering the percentage 12%. By definition, a percentage can be expressed as 𝑝 out of 100. This means the percentage 12% can be written as 12100. Recall that, in general, a percentage 𝑝% can be expressed as a terminating decimal 𝑑 using the formula 𝑑=𝑝100. So, we can express 12% by dividing by 100, which is to move all digits two decimal place values to the right. We can then place a zero in the ones place to show there is no whole number part of the decimal.

Therefore, the percentage 12% can be expressed as the terminating decimal 0.12.

We may face problems where we are given an equivalence of a decimal, percent, and fraction, but some parts are missing. Now that we have reviewed the relationship between all three representations, let’s put that knowledge to work in our final example.

Example 6: Finding the Percentage and the Part Knowing the Decimal and the Whole

If 0.05=π‘Ž%=𝑏20, find π‘Ž and 𝑏.

Answer

We begin by expressing the terminating decimal, 0.05, as a percentage. To express a terminating decimal as a percentage, we can simply multiply the decimal by 100. To quickly find the product of a number and 100, we move all digits to the left two decimal place values. Moving the digits 0 and 5 two places to the left gives us 5%.

Therefore, the terminating decimal 0.05 can be expressed as the percentage 5%. So, π‘Ž=5.

Next, we look for the value of 𝑏, for which 0.05=5%=𝑏20. In other words, we are looking for what part of 20 is 5%. We can use a double line diagram to visualize this proportional relationship.

Since 5% can be represented as 5 out of 100, we have 5%=5100. We want to find the value 𝑏 that compares with 20 in the same way 5 compares with 100. This is mathematically expressed with equal ratios 5100=𝑏20. We can simplify 5Γ·5100Γ·5=120. Since, 120=𝑏20, we have just shown that 𝑏=1. The terminating decimal 0.05 and percentage of 5% can both be expressed as 120.

In conclusion, π‘Ž=5 and 𝑏=1.

Let’s finish by recapping some important points from the explainer.

Key Points

  • A decimal place value table is a helpful visual to convert between a decimal and a fraction or percentage.
  • To convert a fraction to a terminating decimal, find an equivalent fraction over 10 or any number of tens multiplied together. The numerator of the new fraction contains the digits that should terminate in
    • the tenths place if the denominator is 10,
    • the hundredths place if the denominator is 100,
    • the thousandths place if the denominator is 1 000, and so on.
  • To convert a fraction to a terminating decimal with a calculator, divide the numerator by the denominator.
  • To convert a terminating decimal to a fraction without a calculator, use the digits of the decimal as the numerator of a fraction with
    • a denominator of 10 if the decimal terminates in the tenths place,
    • a denominator of 100 if the decimal terminates in the hundredths place,
    • a denominator of 1β€Žβ€‰β€Ž000 if the decimal terminates in the thousandths place, and so on.
      Then, simplify the fraction.
  • To express a fraction as a percentage, multiply the fraction by 100.
  • A percentage 𝑝% can be expressed as the fraction 𝑝100. We can then write the fraction in its simplest form, if necessary.
  • To express a decimal as a percentage, multiply the decimal by 100 or move the digits two places to the left.
  • To express a percentage as a decimal, divide the percentage by 100 or move the digits two places to the right.

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