English:Fractions, Decimals, and Percentages
Fractions, Decimals, and Percentages
Introduction
Fractions, decimals, and percentages are three ways to describe parts of a whole. You may see them when you share food, measure ingredients, read sports results, compare prices, or look at survey results. In this aiMOOC, you will learn how these three forms are connected and how to move confidently from one form to another.
A fraction such as 3/4, a decimal such as 0.75, and a percentage such as 75% can all name the same amount. The symbols look different, but the value can be equal. Learning to recognize these connections makes many everyday calculations easier.
Learning goals: By the end of the course, you should be able to explain fractions, decimals, and percentages; find equivalent forms; compare and order values; use visual models; and solve simple real-life problems involving parts of a whole.
The shaded fraction circles above help you see unit fractions such as one half, one third, and one fourth as equal parts of a whole.
This video uses a visual model to connect a fraction, a decimal, and a percentage.
Understanding Fractions
A fraction shows how many equal parts of a whole you are considering. In the fraction 3/5, the numerator is 3 and tells how many parts are selected. The denominator is 5 and tells how many equal parts make one whole.
Equal parts matter. If a shape is cut into pieces of different sizes, simply counting pieces does not make a fair fraction model.
Equivalent Fractions
Equivalent fractions have the same value even though they use different numerators and denominators. For example, 1/2, 2/4, and 5/10 all name the same amount.
You can make an equivalent fraction by multiplying the numerator and denominator by the same non-zero whole number. You can also simplify a fraction by dividing the numerator and denominator by the same common factor.
A useful set of benchmark fractions is 1/2, 1/4, 3/4, 1/5, and 1/10. Benchmarks help you estimate and compare unfamiliar values.
Fractions on a Number Line
A fraction can be placed on a number line. To place 3/4 between 0 and 1, divide the distance from 0 to 1 into four equal intervals and count three intervals from 0. This shows that fractions are numbers, not just pieces of shapes.
This video shows how division connects fractions and decimals.
Understanding Decimals
A decimal uses place value based on powers of ten. The first place to the right of the decimal point is the tenths place. The second is the hundredths place.
In 0.47, the 4 means four tenths and the 7 means seven hundredths. That is the same as 47/100. In 2.35, the 2 means two wholes, the 3 means three tenths, and the 5 means five hundredths.
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A place-value chart helps you line up digits correctly. When comparing decimals, compare the whole-number part first, then tenths, then hundredths. For example, 0.62 is greater than 0.58 because six tenths is greater than five tenths.
Decimal Fractions
Fractions with denominators 10, 100, or 1000 are especially easy to write as decimals. For example, 7/10 = 0.7 and 34/100 = 0.34.
You can also write many other fractions as decimals by dividing the numerator by the denominator. For example, 3/4 means 3 divided by 4, which equals 0.75.
Remember that adding a zero to the end of a decimal does not change its value: 0.5 = 0.50. The extra zero can be useful when comparing numbers with different numbers of decimal places.
Understanding Percentages
The word percent means per hundred. A percentage tells how many parts out of 100 are being described. For example, 25% means 25 out of 100, so 25% = 25/100 = 1/4 = 0.25.
A 10 by 10 hundred grid contains 100 equal squares. If 36 squares are shaded, the shaded part is 36/100, 0.36, or 36%.
Pie charts can also show how percentages divide a whole into parts. The percentages for all parts of one complete pie chart should total 100%.
This video explains how to convert decimals to percentages.
Connecting the Three Forms
Fractions, decimals, and percentages can represent the same value. The goal is not to memorize every possible conversion. Instead, understand the meaning of each form and use a method that fits the number.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
Fraction to Decimal
To change a fraction into a decimal, divide the numerator by the denominator. For example, 3/5 = 3 ÷ 5 = 0.6.
For common fractions, you can also make an equivalent fraction with denominator 10 or 100. Since 3/5 = 6/10, you can immediately see that the decimal is 0.6.
Decimal to Percentage
To change a decimal into a percentage, multiply by 100 and write the percent sign. For example, 0.42 × 100 = 42, so 0.42 = 42%.
You can think of this as asking, "How many hundredths are there?" The decimal 0.42 is forty-two hundredths, so it is 42 out of 100.
Percentage to Decimal
To change a percentage into a decimal, divide by 100. For example, 65% = 0.65. A percentage smaller than 1%, such as 0.5%, becomes a decimal smaller than one hundredth.
Percentage to Fraction
Write the percentage as a fraction with denominator 100, then simplify when possible. For example, 40% = 40/100 = 2/5.
Fraction to Percentage
One method is to turn the fraction into a decimal and then multiply by 100. Another method is to make an equivalent fraction with denominator 100. For example, 3/4 = 75/100, so 3/4 = 75%.
A double number line can help you reason about percentages without relying only on rules.
Comparing and Ordering Values
When fractions, decimals, and percentages appear together, convert them to one common form before comparing. For example, compare 1/2, 0.45, and 60%. Writing all three as percentages gives 50%, 45%, and 60%, so 0.45 is the smallest and 60% is the largest.
You can also use benchmarks. If a value is greater than one half, it is greater than 0.5 and greater than 50%. Benchmarks are especially helpful when you only need an estimate.
Real-Life Applications
Fractions are useful when sharing, measuring, and describing ratios of parts. Decimals are common in measurements and money. Percentages are useful for survey results, test scores, discounts, battery levels, and many other comparisons out of 100.
Suppose a jacket costs 40 units of money and is discounted by 25%. Since 25% is one quarter, the discount is one quarter of 40, which is 10. The sale price is therefore 30.
If 18 out of 30 students choose a particular activity, the fraction is 18/30 = 3/5. The decimal is 0.6 and the percentage is 60%.
Helpful Strategies and Common Mistakes
Always ask what the whole is. A fraction or percentage only makes sense when you know the whole being described.
Keep place value in mind. The decimal 0.7 means seven tenths, while 0.07 means seven hundredths. These values are not equal.
Do not assume that a larger denominator makes a larger fraction. With unit fractions, 1/8 is smaller than 1/4 because the whole is divided into more, smaller equal parts.
When converting a percentage to a decimal, divide by 100. When converting a decimal to a percentage, multiply by 100. Check whether your answer is reasonable by using familiar benchmarks such as 25%, 50%, and 75%.
Interactive Tasks
Quiz: Test Your Knowledge
Which decimal is equal to one half? (0.5) (!0.2) (!0.25) (!0.05)
Which percentage is equal to three quarters? (75 percent) (!25 percent) (!30 percent) (!40 percent)
What does the denominator tell you? (Number of equal parts in the whole) (!Number of selected parts) (!Number of decimal places) (!Number of percentages)
Which fraction is equal to 0.2? (One fifth) (!One half) (!One fourth) (!Three fourths)
What is 0.36 written as a percentage? (36 percent) (!3.6 percent) (!360 percent) (!0.36 percent)
What is 40 percent written as a simplified fraction? (Two fifths) (!Four fifths) (!One fourth) (!Two thirds)
Which value is greatest? (80 percent) (!0.75) (!Three fifths) (!One half)
Which value is equal to 25 percent? (0.25) (!0.025) (!2.5) (!0.75)
What should you do to convert a decimal to a percentage? (Multiply by one hundred) (!Divide by ten) (!Add one hundred) (!Subtract one hundred)
Which statement about equivalent forms is correct? (They can look different but have the same value) (!They always use the same symbols) (!They always have the same denominator) (!They can never be compared)
Memory Game
| Numerator | The number that tells how many fraction parts are selected |
| Denominator | The number that tells how many equal parts make the whole |
| Tenths | The first place to the right of a decimal point |
| Hundredths | The second place to the right of a decimal point |
| Percent | A way to describe an amount per hundred |
| Equivalent | Having the same value in a different form |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Fraction form | Shows selected equal parts of a whole |
| Decimal form | Uses place value to the right of a point |
| Percentage form | Describes an amount per hundred |
| Equivalent values | Look different but represent the same amount |
| Hundred grid | Helps connect parts of a whole with hundredths |
Crossword Puzzle
| Numerator | What is the top number in a fraction called? |
| Denominator | What is the bottom number in a fraction called? |
| Decimal | What form uses a point and place value? |
| Percent | What word means per hundred? |
| Equivalent | What word describes two forms with the same value? |
| Hundredths | What place comes second to the right of the decimal point? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Fraction Hunt: Find five examples of fractions in your home or classroom, draw or photograph them, and explain what the whole and the equal parts are in each example.
- Hundred Grid Art: Shade a hundred grid to create a simple picture, then write the shaded part as a fraction, decimal, and percentage.
- Recipe Fractions: Choose a simple recipe and identify at least three fractional measurements; explain what would happen to the amounts if the recipe were doubled.
- Benchmark Cards: Make learning cards for one half, one fourth, three fourths, one fifth, and one tenth, showing each as a fraction, decimal, percentage, and quick sketch.
Standard
- Class Survey: Ask classmates one yes-or-no question, record the results, and express one response group as a fraction, decimal, and percentage.
- Discount Detective: Collect three examples of percentage discounts from advertisements or shop signs and calculate the discount amount for a simple sample price.
- Conversion Explainer: Create a short poster or video that teaches how to convert a fraction to a decimal and then to a percentage using one example of your choice.
- Number Line Challenge: Draw a number line from zero to one and place at least eight mixed values written as fractions, decimals, or percentages in the correct order.
Advanced
- Price Comparison Project: Compare two imagined shop offers by calculating discounts, final prices, and percentage savings, then explain which offer gives better value.
- School Data Interview: Interview a teacher or school staff member about a simple set of non-sensitive school data, convert one part-to-whole result into three equivalent forms, and explain your method.
- Fair Game Design: Design a simple chance game with colored cards or counters, predict outcomes using fractions and percentages, test the game repeatedly, and compare experimental results with your prediction.
- Teach the Connection: Produce a two- to three-minute teaching video for younger learners that uses a visual model and a real-life example to connect fractions, decimals, and percentages.
Learning Assessment
- Multiple Representations: Show two different visual models for 3/5 and explain how both models support the decimal and percentage forms.
- Conversion Reasoning: A learner says that 0.4 equals 4 percent; explain the mistake and show a correct conversion using place value.
- Comparison Strategy: Order 2/3, 0.7, and 65% from least to greatest, and justify your order by changing them to a common form.
- Discount Application: A game costs 60 units of money and is reduced by 20%; calculate the discount and final price, and explain why your answer is reasonable.
- Survey Transfer: In a survey, 21 of 35 people choose option A; simplify the fraction, find a decimal and percentage, and interpret what the result means.
- Create and Defend: Invent three equivalent values in fraction, decimal, and percentage form, then prove their equivalence with a calculation and a visual model.
Evidence of Learning
Strong evidence of learning includes knowledge of numerator, denominator, place value, percent, and equivalent values. It also includes skills such as converting among forms, comparing mixed representations, estimating with benchmarks, and checking whether answers are reasonable.
Useful products can include a hundred-grid model, a conversion poster, a short explanation video, a number line, a survey report, or a worked real-life problem. Good products show clear mathematical thinking, not only final answers.
Important transfer achievements include choosing a useful form for a new problem, explaining why two different representations are equal, using percentages in simple price or data situations, and applying fraction-decimal-percentage connections to unfamiliar examples.
OERs on the Topic
The following English Wikipedia pages provide further reference material about the three main representations.
Linked Learning Areas
Fractions, decimals, and percentages connect to place value, ratio, measurement, data, probability, and everyday financial mathematics. The links below can help you continue learning.