Fractions | CBSE Class 6 Maths Notes

This note covers equal shares, fractional units, parts of a whole, measuring with paper strips, fractions on the number line, mixed fractions, equivalent fractions, lowest terms, comparing fractions, addition, subtraction, the history of fraction notation, and puzzles with fractional units.

What do fractions tell us about equal shares?

A fraction describes a share obtained when a whole number of things is shared equally. The whole is the complete quantity being considered. A unit is the quantity taken as one, such as one complete roti.

When one roti is shared equally between two children, each receives one half, written 1/2. The slash separates the top and bottom numbers of a fraction. In 1/2, it represents the fraction bar. One roti shared equally among four children gives each one 1/4, or one quarter.

The numerator is the top number of a fraction; the denominator is its bottom number. The denominator identifies how many equal parts make one whole. The numerator counts how many of those parts are being considered.

Why does sharing among more people reduce each share?

Keep the total quantity fixed. If more children share that quantity equally, each receives less. Thus one half of a roti is greater than one quarter of the same roti. Comparing shares requires keeping track of both the total amount and the number sharing it.

Definition: A fractional unit, also called a unit fraction, is one equal part obtained by dividing a whole unit into equal parts. Examples include 1/2, 1/3, 1/4 and 1/5.

Property: More equal parts give smaller fractional units

For the same whole, 1/9 is less than 1/5. The symbol < means “is less than”; > means “is greater than”; and = means “is equal to”. Nine being greater than five does not make one ninth greater than one fifth.

Worked example 1. Three guavas together weigh 1 kilogram, written kg. They are roughly of the same size. What will each guava roughly weigh?

Answer: Each will roughly weigh 1/3 kg. The qualification “roughly” matters: the information describes approximately equal guavas, rather than giving their individual measured weights.

How do equal parts represent fractions of a whole?

A piece represents a particular fractional unit when it is one of the equal parts of the chosen whole. Counting pieces is not enough: their size relative to that whole matters. Two pieces cut from a whole need not each be one half.

Consider a whole chikki, a sweet used here as an object that can be divided. One piece can be one quarter of it, while the remaining piece contains three quarters. There are two pieces, but they represent unequal amounts of the original whole.

How can pieces with different shapes represent the same fraction?

A whole chikki can be divided into six equal parts in different ways. A piece from either division represents one sixth of the original chikki. Different shapes can therefore represent the same fraction when they account for equal amounts of the same whole.

A rectangular whole chikki appears above a larger remaining piece and a separate quarter. Below, two divisions into six equal pieces produce differently shaped sixths, including a rectangular piece and a triangular piece.

Reference: NCERT Class 6, page 154, unnumbered illustration

The larger piece in the quarter illustration contains three pieces of size 1/4. Its fraction is therefore 3/4. The numerator counts three quarter-sized pieces, while the denominator indicates that four such equal pieces make the original whole.

Worked example 2. A larger piece of chikki contains three pieces, each equal to 1/4 of the original whole. What fraction of the whole is the larger piece?

Answer: It is 3/4. Each counted piece is a quarter, so three such pieces are three quarters of the same whole.

This interpretation connects a fraction to measurement. First identify the whole, then identify the equal-sized part used to measure it, and finally count how many of those parts make the quantity being described.

How can paper strips help us measure fractional quantities?

Take one complete paper strip as a unit of length. Folding it into two equal parts creates halves. Folding the folded strip in half again creates four equal parts of the original strip. Each of these smaller parts has length 1/4 unit.

Two quarter-length parts give 2/4 of the original strip; three give 3/4; four give 4/4, which is one complete strip. The number of pieces changes, but the fractional unit used for measuring remains one quarter throughout.

A full strip is followed by a strip marked in halves and another marked in quarters. Brackets below the quartered strip identify one quarter, two quarters, three quarters and four quarters.

Reference: NCERT Class 6, page 156, unnumbered illustration

How do we read a fraction as a count of fractional units?

We usually read 3/4 as “three quarters” or “three upon four”. Reading it as “three times one quarter” makes both pieces of information clear: the fractional unit is 1/4, and there are three such units.

The symbol × means multiplication, or “times”. The symbol + means addition, or combining quantities. Thus 3 × 1/4 and 1/4 + 1/4 + 1/4 describe the same collection of three quarter-sized parts.

Folding once more creates eighths. Eight equal eighths make the original whole, so 8/8 = 1. These folds change the size of the measuring part; they do not change the length chosen as one unit.

Worked example 3. In the fraction 5/6, identify the numerator, denominator and fractional unit.

Answer: The numerator is 5 and the denominator is 6. The fractional unit is 1/6, and 5/6 means five such sixths.

How are fractions marked on a number line?

A number line represents numbers by positions along a line. The distance from 0 to 1 is one unit. To mark a fraction, divide that unit length into the number of equal intervals indicated by its denominator, then count the required intervals from zero.

An interval here is a length between neighbouring division marks. When the unit is divided into two equal intervals, each interval measures 1/2 unit. A line starting at zero and ending at the halfway mark has length one half.

Number lines extend from 0 to 2. Blue segments start at zero. The first shows a half-unit length; later lines use thirds and fifths, with some endpoint labels left blank.

Reference: NCERT Class 6, page 159, unnumbered illustration

What changes when the fraction is greater than one?

Equal fractional steps can continue beyond 1. A fraction is not restricted to a position between 0 and 1. The same unit length and the same subdivision must be used on both sides of 1, so each step still represents the same fractional unit.

In fractions less than one, the numerator is smaller than the denominator. In fractions greater than one, the numerator is larger. If the numerator and denominator are equal, the collected fractional units complete one whole, as in 4/4 and 8/8.

Worked example 4. The distance from 0 to 1 on a number line is one unit. Divide it into two equal parts. What length is obtained by taking three half-unit steps from zero?

Answer: The length is 3/2 units. Two half-unit steps reach 1, and the third adds another half. Therefore, 3/2 = 1 + 1/2.

Always distinguish the position reached from the count of marks drawn. The numerator counts fractional intervals from zero. The starting mark at zero does not itself contribute an interval of length.

How do we change between fractions and mixed numbers?

Definition: A mixed number, or mixed fraction, contains a whole number part and a fractional part less than one. The whole part counts complete units; the fractional part describes the remaining amount.

For example, 8/3 consists of eight thirds. Two groups of three thirds make two whole units, leaving two thirds. Thus 8/3 = 2 + 2/3, also written 2 2/3 and read “two and two thirds”.

How are complete wholes collected?

The denominator tells us how many fractional units make one whole. In 3/2, every two halves form a whole, giving 1 + 1/2. In 5/2, two pairs of halves form two wholes, giving 2 + 1/2.

Worked example 5. Express 8/3 as a mixed number and explain its parts.

Answer: Collect the eight thirds into two complete groups of three thirds, leaving two thirds. The mixed number is 2 2/3. Its whole part is 2, and its fractional part is 2/3.

How can a mixed number be written as one fraction?

Reverse the grouping process. Express each whole using the fractional unit of the mixed number, then include the remaining fractional units. The denominator stays connected to the size of each part throughout this calculation.

Worked example 6. Write 3 3/4 as a single fraction.

Answer: Each whole contains four quarters. The three wholes contain twelve quarters, and the fractional part adds three more quarters. Altogether there are fifteen quarters, so 3 3/4 = 15/4.

Mixed notation puts the whole and fractional parts beside each other, but their values are added. In this example, 3 3/4 means 3 + 3/4. Thinking in quarters explains the conversion without losing the meaning of either part.

Why can different fractions represent the same quantity?

Equivalent fractions represent the same share, length or number. Their numerators and denominators may differ because they measure that quantity using different fractional units. One half, two quarters and four eighths have the same value.

Equal-length strips have matching blue portions labelled 1/2, 2/4 and 4/8. A fraction wall below places rows of halves, thirds, quarters, fifths and sixths beneath a one-unit strip.

Reference: NCERT Class 6, page 164, unnumbered illustration

A fraction wall arranges equal whole lengths in rows, with different rows divided into different numbers of equal parts. Aligning their starting points makes equal fractional lengths visible. Three sixths match one half, while two sixths match one third.

Property: Equivalent fractions preserve the share

Multiplying the numerator and denominator by the same positive whole number gives an equivalent fraction. This matches equal sharing: increasing both the quantity shared and the number sharing it by the same factor preserves each person's share.

Worked example 7. Anil belongs to a group sharing 2 cakes equally among 5 children. How many cakes would 10 children need to receive the same share each?

Answer: Each original share is 2/5 of a cake. Doubling both the cakes and the children gives 4 cakes for 10 children. Each new share is 4/10, which equals 2/5.

The same relationship appears when one roti is shared between two children, two rotis among four children, and three rotis among six children. The respective shares are 1/2, 2/4 and 3/6, all equal.

The symbol ÷ means division. Sharing one roti among four children can be written 1 ÷ 4 = 1/4. Collecting the shares gives 1/4 + 1/4 + 1/4 + 1/4 = 1, or 4 × 1/4 = 1.

How do we find a common denominator and lowest terms?

A common denominator is the same denominator used when expressing different fractions as equivalent fractions. It gives those fractions the same fractional unit. This makes their numerators counts of equally sized pieces.

A multiple of a whole number is obtained by multiplying it by a whole number. A common multiple is a multiple shared by the numbers being considered. The least common multiple is their smallest positive common multiple.

For 3/4 and 7/10, the product of the denominators is 40. Here, product means the result of multiplication. Equivalent forms are 30/40 and 28/40. The denominator 20 also works: the equivalent forms are 15/20 and 14/20.

Property: Dividing both parts by a common factor preserves value

A factor divides a whole number exactly, with nothing left over. A common factor divides both numbers exactly. Dividing a fraction's numerator and denominator by the same common factor preserves its value. Using a common factor greater than 1 reduces the numerator and denominator.

A fraction is in lowest terms, or its simplest form, when numerator and denominator have no common factor other than 1. The highest common factor is the greatest factor shared by both. Dividing both numbers by it gives lowest terms directly.

Worked example 8. Express 16/20 in lowest terms.

Answer: Both 16 and 20 are divisible by 4. Dividing each by 4 gives 4/5. Since 4 and 5 have no common factor other than 1, 4/5 is the simplest form.

Can simplification be done in stages?

For 36/60, divide both numbers by 2 to get 18/30. Divide both by 2 again to get 9/15, then by 3 to get 3/5. Alternatively, dividing 36 and 60 by 12 gives 3/5 directly.

Both methods give the same answer. Sometimes it can be easier to go in steps. After each division, check whether the new numerator and denominator still share a factor greater than 1 before deciding that simplification is complete.

How can we compare and order fractions?

When fractions have the same denominator, their fractional units have the same size. Compare the numerators to find which contains more of those units. For example, 5/7 is greater than 4/7 because five sevenths exceed four sevenths.

When the numerators are equal, think about equal sharing. Four chikkis shared among seven children give each child more than four chikkis shared among eight children. Therefore, 4/7 > 4/8. Since 4/8 = 1/2, this also shows 4/7 > 1/2.

How does a common denominator help?

For fractions with different numerators and denominators, first convert them into equivalent forms with the same denominator. Then compare the new numerators. Using the same fractional unit makes the comparison one between counts of equally sized parts.

Worked example 9. Compare 4/5 and 7/9.

Answer: Use 45 as a common denominator. Multiplying both parts of 4/5 by 9 gives 36/45; multiplying both parts of 7/9 by 5 gives 35/45. Since 36/45 > 35/45, we have 4/5 > 7/9.

Another pair is 7/9 and 17/21. A common denominator is 63. The equivalent fractions are 49/63 and 51/63, so 7/9 < 17/21. The chosen denominator need not be the product of the original denominators.

What do ascending and descending order mean?

Ascending order runs from smallest to largest. Descending order runs from largest to smallest. Convert all the fractions in a list to a common denominator, compare the resulting numerators, and then write the original fractions in the required order.

For instance, the ascending order of 7/10, 11/15 and 2/5 is 2/5 < 7/10 < 11/15. Keep each original numerator paired with its own denominator when writing the ordered answer.

How do we add fractions?

The sum is the result of addition. When adding fractions that have the same denominator, count their fractional units together. The numerator changes to give the total count; the denominator stays the same because the size of each unit stays the same.

Worked example 10. Find 2/5 + 1/5.

Answer: Both fractions use fifths. Two fifths and one fifth make three fifths, so 2/5 + 1/5 = 3/5. The denominator remains 5.

Rectangular strips are divided into five equal parts. One strip has two red parts, another has one, and the resulting strip has three red parts, representing 2/5 + 1/5 = 3/5.

Reference: NCERT Class 6, page 176, unnumbered illustration

Addition can produce more than one whole. Four sevenths and six sevenths make ten sevenths: 4/7 + 6/7 = 10/7 = 1 3/7. Seven of the ten pieces form one whole, leaving three sevenths as the fractional part.

How does Brahmagupta's method handle different denominators?

Brahmagupta's method expresses the fractions with a common fractional unit before adding their counts. This removes the difficulty of trying to count differently sized pieces together as though they were identical.

1. Find equivalent fractions with the same denominator, using a common multiple of the original denominators.
2. Add their numerators while retaining that common denominator.
3. Express the result in lowest terms if needed.

Worked example 11. Find 1/4 + 1/3.

Answer: Use twelfths. We have 1/4 = 3/12 and 1/3 = 4/12. Therefore, 1/4 + 1/3 = 3/12 + 4/12 = 7/12.

For 1/6 + 1/3, use sixths: 1/3 = 2/6. The sum is 3/6, which simplifies to 1/2. Changing to equivalent fractions before adding preserves both original quantities and makes the addition meaningful.

How do we subtract fractions?

The symbol − means subtraction, or taking away a quantity. The difference is the result of subtraction. Fractions with the same denominator already use the same-sized parts, so subtract their numerators and keep their denominator.

Worked example 12. Subtract 4/7 from 6/7.

Answer: Begin with six sevenths and remove four sevenths. Two sevenths remain, so 6/7 − 4/7 = 2/7. The pieces are still sevenths after the removal.

A seven-part strip has six red parts. In the next drawing, four red parts are shown shifted down for removal, leaving two red parts in place.

Reference: NCERT Class 6, page 180, unnumbered illustration

What if the denominators differ?

First change both fractions to equivalent fractions with the same fractional unit. Then subtract the new numerators. Keeping the common denominator records the size of the parts that remain after subtraction.

Worked example 13. Find 3/4 − 2/3.

Answer: Use twelfths. Multiply both parts of 3/4 by 3 to get 9/12. Multiply both parts of 2/3 by 4 to get 8/12. Therefore, 3/4 − 2/3 = 9/12 − 8/12 = 1/12.

The steps in Brahmagupta's subtraction method are to find equivalent fractions with the same denominator, subtract their numerators while retaining that denominator, and simplify the result into lowest terms if needed.

Why does the wording of subtraction matter?

“Subtract 4/7 from 6/7” means start with 6/7. The quantity following “from” is the starting amount. Read the instruction carefully before writing the subtraction, particularly when the fractions are presented in words rather than as a calculation.

The reason for retaining the denominator is the same in addition and subtraction. Combining or removing equal-sized parts changes their number. It does not change the fractional unit used to measure those parts.

How do fractions solve measurement and sharing problems?

A word problem connects a calculation to a quantity. Identify the whole, the given fractional amounts and what must be found. Combining amounts calls for addition; finding a remaining amount or a difference calls for subtraction.

Keep the measurement unit, the named standard used to express a measurement, with the result. A metre, written m, measures length; a kilometre, written km, measures distance; a litre measures volume, the amount of space occupied. Minutes measure time.

How are two quantities combined?

Worked example 14. Rahim mixes 2/3 litres of yellow paint with 3/4 litres of blue paint to make green paint. Find the volume of green paint.

Answer: The total is 2/3 + 3/4 litres. Expressing both in twelfths gives 8/12 + 9/12 = 17/12 litres, or 1 5/12 litres.

In another problem, Geeta buys 2/5 metre of lace and Shamim buys 3/4 metre. Their tablecloth has a perimeter, or total boundary length, of 1 metre. Together they have 1 3/20 metres, enough to cover the whole border.

How is a remaining distance found?

Worked example 15. Jaya's school is 7/10 km from her home. She travels 1/2 km by auto and walks the rest. How far does she walk to school?

Answer: Subtract the auto journey from the total distance. Since 1/2 = 5/10, the walking distance is 7/10 − 5/10 = 2/10 = 1/5 km.

For the park problem, Jeevika takes 10/3 minutes for a complete round and Namit takes 13/4 minutes. Namit takes less time. Converting to twelfths gives a difference of 1/12 minute. Compare the times before choosing which one to subtract.

A complete answer therefore names both the fractional value and what it measures. A distance answer needs its distance unit, while a time comparison should identify the person taking less time and the size of the difference.

What history and puzzles are connected with fractions?

In Sanskrit, bhinna means “broken”. Fractions were also called bhaga or ansha, meaning “part” or “piece”. These terms connect naturally to dividing a whole and describing the resulting pieces.

The Bakshali manuscript, from around the year 300 CE, used a numerator above a denominator in a form similar to present notation. CE means Common Era, the era used here for numbering historical years.

The line between numerator and denominator was introduced later by the Moroccan mathematician Al-Hassar in the 12th century. Brahmagupta described general procedures for adding and subtracting fractions in 628 CE. These procedures work by expressing fractions with a common denominator.

What are Egyptian fractions?

Ancient Egyptian and Babylonian civilisations primarily used fractional units. More general fractions were expressed as sums of fractional units, now called Egyptian fractions. An example is 19/24 = 1/2 + 1/6 + 1/8.

Fraction methods travelled from India to Europe via the Arabs. They came into general use in Europe in around the 17th century. These historical developments connect the notation for fractions with the methods used to calculate with them.

Can different fractional units add up to one?

Among the fractional units smaller than one, 1/2 is the largest. Two different such units cannot sum to one: replacing either half in 1/2 + 1/2 by a smaller unit makes the total less than one.

Three different fractional units can make one: 1/2 + 1/3 + 1/6 = 1. This is the only solution apart from changing their order. With four different fractional units, the puzzle has six solutions.

A circle is divided into six equal sectors, or wedge-shaped parts. Three are green, two are orange and one is purple. The equation beneath is 1/2 + 1/3 + 1/6 = 1.

Reference: NCERT Class 6, page 185, unnumbered illustration

Glossary

• Fraction — A number describing equal shares or a quantity measured using fractional units.
• Fractional unit — One equal part formed by dividing a whole unit into equal parts.
• Numerator — The top number in a fraction, counting its fractional units.
• Denominator — The bottom number indicating how many equal parts form one whole.
• Number line — A line on which numbers, including fractions, have associated positions.
• Mixed fraction — A number containing a whole part and a fractional part less than one.
• Equivalent fractions — Fractions with different forms that represent the same share, length or number.
• Fraction wall — Equal whole lengths arranged in rows with different equal subdivisions.
• Common denominator — A shared denominator that expresses fractions using the same fractional unit.
• Common factor — A number that divides each of the given whole numbers exactly.
• Lowest terms — A fraction's form when numerator and denominator share no factor except one.
• Ascending order — An arrangement of quantities from the smallest value to the largest.
• Descending order — An arrangement of quantities from the largest value to the smallest.
• Egyptian fractions — Sums of fractional units used to express more general fractional quantities.

Common errors and misconceptions

• Misconception: 1/9 is greater than 1/5 because 9 is greater than 5. Correct: Dividing the same whole into more equal parts makes each part smaller, so 1/9 < 1/5.
• Misconception: Any two pieces of a whole must be halves. Correct: Each piece must be an equal share to represent a half; the chikki example has a quarter and three quarters.
• Misconception: Fractions are all less than one. Correct: Fractions such as 3/2 and 5/2 are greater than one and can represent whole units plus a fractional part.
• Misconception: Changing just the denominator gives an equivalent fraction. Correct: Multiply both numerator and denominator by the same positive whole number, or divide both by a common factor.
• Misconception: Every common denominator must be the product of the original denominators. Correct: Any suitable common multiple works; 20 works for 3/4 and 7/10, as does 40.
• Misconception: Adding fractions means adding their denominators too. Correct: First obtain the same denominator, then add the numerators while retaining the common denominator.
• Misconception: One division guarantees that a fraction is in lowest terms. Correct: Continue until the numerator and denominator have no common factor other than 1.
• Misconception: Subtracting 4/7 from 6/7 starts with 4/7. Correct: Start with the quantity after “from”, giving 6/7 − 4/7 = 2/7.

Exam-style questions with model answers

Q1. One roti is shared equally among 5 children, and another equally among 9 children. Which group receives the larger share per child? Explain. [2 marks]

1. The shares are 1/5 roti and 1/9 roti respectively.
2. The group of 5 receives the larger share per child: dividing the same whole among fewer children gives each more, so 1/5 > 1/9.

Q2. Express 8/3 as a mixed number, showing the whole and fractional parts. [3 marks]

1. The fraction 8/3 contains eight thirds. Three thirds make one whole, so collect the thirds into groups of three.
2. Two complete groups use six of the eight thirds. Therefore, the whole number part is 2.
3. Two thirds remain after forming the two wholes. The fractional part is 2/3, giving the mixed number 2 2/3.

Q3. Express 36/60 in lowest terms by dividing in stages, and explain when to stop. [4 marks]

1. Both numbers are divisible by 2. Divide the numerator and denominator by 2 to obtain 36/60 = 18/30.
2. Both new numbers are again divisible by 2, giving the equivalent fraction 18/30 = 9/15.
3. Now divide both 9 and 15 by 3, which gives 9/15 = 3/5.
4. Stop because 3 and 5 have no common factor other than 1. Thus 3/5 is the required simplest form.

Q4. Compare 4/5 and 7/9 using equivalent fractions, explaining the role of the common denominator. [5 marks]

1. The denominators are 5 and 9, so the original fractions count different-sized parts. Express them with the same fractional unit before comparing their numerators.
2. Choose 45 as a common denominator because it is a multiple of both 5 and 9.
3. Multiply both the numerator and denominator of 4/5 by 9. This preserves its value and gives 36/45.
4. Multiply both the numerator and denominator of 7/9 by 5. This gives the equivalent fraction 35/45.
5. Thirty-six forty-fifths exceed thirty-five forty-fifths. Therefore, 36/45 > 35/45, and the original comparison is 4/5 > 7/9.

Q5. Rahim mixes 2/3 litres of yellow paint with 3/4 litres of blue paint to make green paint. Find the total volume as a mixed number, showing the common-denominator method. [5 marks]

1. The required volume is the combined amount of the two paints, so calculate 2/3 + 3/4 litres.
2. Choose 12 as a common denominator, since both 3 and 4 divide 12 exactly. Each new fractional unit is one twelfth of a litre.
3. Convert both fractions without changing their values: 2/3 = 8/12 and 3/4 = 9/12.
4. Add the counts of twelfths and retain their denominator: 8/12 + 9/12 = 17/12 litres.
5. Twelve twelfths make one litre, leaving five twelfths. Rahim therefore makes a total of 1 5/12 litres of green paint.

Q6. Jaya's school is 7/10 km from home. She takes an auto for 1/2 km and walks the remaining distance. Find her walking distance to school in lowest terms. [3 marks]

1. The walking distance is the total home-to-school distance less the auto journey, so the required calculation is 7/10 − 1/2 km.
2. Express the half-kilometre journey in tenths: 1/2 = 5/10. Subtract to obtain 7/10 − 5/10 = 2/10 km.
3. Divide the numerator and denominator by 2 to simplify 2/10 to 1/5. Jaya walks 1/5 km to reach school.

Q7. Jeevika takes 10/3 minutes for a complete round of a park, while Namit takes 13/4 minutes for the same round. Who takes less time, and by how much? Show your comparison and subtraction. [5 marks]

1. Both times describe the same complete round, so compare the fractions 10/3 and 13/4 to identify the smaller time.
2. Use 12 as a common denominator. Jeevika's time becomes 40/12 minutes after multiplying both parts of 10/3 by 4.
3. Namit's time becomes 39/12 minutes after multiplying both parts of 13/4 by 3. Since 39/12 < 40/12, Namit takes less time.
4. Find the difference by subtracting the smaller time from the larger time: 40/12 − 39/12 = 1/12 minute.
5. Therefore, Namit takes 1/12 minute less than Jeevika to complete the round. The difference is already in lowest terms.

Q8. Three guavas together weigh 1 kg and are roughly of the same size. What does each guava roughly weigh? Explain why the answer is approximate. [2 marks]

1. Each guava roughly weighs 1/3 kg, treating the total as three approximately equal shares.
2. Their sizes are described as roughly equal, so the data do not establish identical measured weights for the individual guavas.

Key takeaways

• A fractional unit is one equal part of a whole; more equal divisions of the same whole produce smaller parts.
• The numerator counts fractional units, while the denominator identifies how many such units make one whole.
• Fractions have positions on the number line, including positions beyond one when the numerator exceeds the denominator.
• Mixed fractions combine complete whole units with a fractional part; counting the fractional units converts between the two forms.
• Equivalent fractions describe the same value using different fractional units, as shown by matching lengths in a fraction wall.
• Lowest terms means the numerator and denominator share no common factor except one; simplification preserves the fraction's value.
• Use a common denominator to compare fractions, then compare their numerators as counts of equal-sized parts.
• For addition and subtraction, obtain common fractional units, operate on the numerators, retain the denominator, and simplify if needed.

Test yourself

What is the fractional unit in 5/6, and how many such units does it contain?

The fractional unit is 1/6, and the fraction contains five such units.

Which is greater, 1/100 or 1/200, for the same whole?

One hundredth is greater: dividing the same whole into fewer equal parts gives larger parts.

What mixed number represents 3/2?

It is 1 1/2, because two halves make a whole and one half remains.

Why are 1/2, 2/4 and 4/8 equivalent?

They represent the same length or share, measured in halves, quarters and eighths respectively.

Is 16/20 in lowest terms? What is its simplest form?

No. Divide both numbers by 4 to obtain 4/5, whose only common factor is 1.

What is 1/6 + 1/3 in lowest terms?

Convert 1/3 to 2/6. The sum is 3/6, which simplifies to 1/2.

What is 3/4 − 2/3?

Use twelfths: 9/12 − 8/12 = 1/12, retaining the common denominator when subtracting.

Which three different fractional units add up to one?

They are 1/2, 1/3 and 1/6; together they make one complete whole.