YOUR STUDY SPACE · CBSE

Fractions — Class 6 Mathematics

Ganita Prakash · Chapter 7

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Chapter summary

This chapter develops fractions as equal shares. When one whole is divided into equal parts, each part is a fractional unit, such as ; the more parts the whole is split into, the smaller each share, so . A fraction has a numerator and denominator , and fractions can be shown on the number line. The chapter explains equivalent fractions \u2014 different fractions that represent the same amount, such as \u2014 obtained by multiplying or dividing numerator and denominator by the same number, and shows how to reduce a fraction to its simplest form. Fractions are compared by making denominators equal. Mixed numbers like are converted to and from improper fractions. Finally, the chapter teaches adding and subtracting fractions: like fractions are added directly, while unlike fractions are first rewritten with a common denominator. Real-life sharing situations with rotis and lengths run throughout to keep the meaning of fractions concrete.

Fractional Units and Equal SharesFractions on the Number LineEquivalent Fractions and Simplest FormComparing FractionsMixed and Improper FractionsAdding and Subtracting Fractions

Key ideas & terms

Fraction: A number representing equal parts of a whole, with numerator and denominator .
Fractional unit: One equal part of a whole, such as or .
Numerator and denominator: In , is the numerator (parts taken) and is the denominator (equal parts of the whole).
Equivalent fractions: Different fractions that represent the same value, e.g. .
Mixed number: A number with a whole part and a fractional part, such as .
Like fractions: Fractions having the same denominator, which can be added or subtracted directly.

Extra questions & answers

What is a fractional unit?

When one whole is divided into several equal parts, each part is a fractional unit, such as or .

Which is greater, or ?

is greater, because sharing one whole among fewer people gives a larger share.

What are the numerator and denominator of ?

The numerator is 3 (the parts taken) and the denominator is 4 (the equal parts the whole is divided into).

What are equivalent fractions?

Different fractions that represent the same amount, such as .

How do you get an equivalent fraction?

Multiply or divide both the numerator and the denominator by the same non-zero number.

What is the simplest form of ?

, obtained by dividing numerator and denominator by 2.

What is a mixed number?

A number made of a whole number part and a proper fraction, such as .

How do you convert to an improper fraction?

Multiply the whole part by the denominator and add the numerator: .

How do you add two fractions with the same denominator?

Add the numerators and keep the common denominator, e.g. .

How do you add fractions with different denominators?

Rewrite them as equivalent fractions with a common denominator, then add the numerators.

Figure it Out

Fill in the blanks with fractions. 1. Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh ____ kg.

If 1 kg is divided equally among 3 guavas, each guava weighs kg because .

Fill in the blanks with fractions. 2. A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is ___ kg.

kg divided into 4 equal packets gives each packet kg.

Fill in the blanks with fractions. 3. Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank ____ glass of sugarcane juice.

glasses shared among 4 friends is . Each friend drank glass.

Fill in the blanks with fractions. 4. The big fish weighs kg. The small one weighs kg. Together they weigh ____ kg.

Together they weigh kg.

5. Arrange these fraction words in order of size from the smallest to the biggest in the empty box below: One and a half, three quarters, one and a quarter, half, quarter, two and a half.

Smallest to biggest: quarter half three quarters one and a quarter one and a half two and a half .

Figure it Out

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? a.

Piece (a) is a small rectangular part that is one of 12 equal parts of the whole, so it is of the whole chikki.

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? b.

Piece (b) is a large triangular part that is one of 4 equal parts, so it is of the whole.

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? c.

Piece (c) is a triangular part that is one of 8 equal parts, so it is of the whole.

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? d.

Piece (d) is a long vertical strip that is one of 6 equal strips, so it is of the whole.

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? e.

Piece (e) is an L-shaped part that covers two parts, but as a single fractional unit shown it is ? From the wall it corresponds to of the whole.

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? f.

Piece (f) is a diamond-shaped part that is one of 6 equal parts, so it is of the whole.

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? g.

Piece (g) is a tiny square that is one of 24 equal small squares, so it is of the whole.

The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? h.

Piece (h) is a small triangle that is one of 24 equal parts, so it is of the whole.

Do it once more! Fill in the blank boxes.

2 times = ____

4 times = ____

6 times = ____

8 times = = 1

Eight copies of make one whole, so .

Figure it Out

1. Continue this table of for 2 more steps.

and .

2. Can you create a similar table for ?

, , , , and so on, each step adding one more .

3. Make using a paper strip. Can you use this to also make ?

Yes. Fold the strip into 3 equal parts to get . Then fold each part in half; each new part is of the whole because .

4. Draw a picture and write an addition statement as above to show: a. 5 times of a roti

. The picture shows five quarter-circles making more than one whole roti.

4. Draw a picture and write an addition statement as above to show: b. 9 times of a roti

added nine times . The picture shows two whole rotis and one quarter.

5. Match each fractional unit with the correct picture:

matches the circle divided into 3 equal sectors with one shaded.

5. Match each fractional unit with the correct picture:

matches the circle divided into 5 equal parts with one shaded.

5. Match each fractional unit with the correct picture:

matches the circle divided into 8 equal parts with one shaded.

5. Match each fractional unit with the correct picture:

matches the circle divided into 6 equal parts with one shaded.

Now, can you find the lengths of the various blue lines shown below? Fill in the boxes as well.

1. Here, the fractional unit is dividing a length of 1 unit into three equal parts. Write the fraction that gives the length of the blue line in the box or in your notebook.

The blue line covers 2 of the 3 equal parts, so its length is unit.

2. Here, a unit is divided into 5 equal parts. Write the fraction that gives the length of the blue lines in the respective boxes or in your notebook.

The two blue lines are and units long respectively.

3. Now, a unit is divided into 8 equal parts. Write the appropriate fractions in your notebook.

Dividing into 8 parts gives fractional unit . The marks correspond to etc.

Figure it Out

1. On a number line, draw lines of lengths , , and .

Divide the unit interval into 10 equal parts where each part is . Mark at the first tick, at the third tick, and at the eighth tick.

2. Write five more fractions of your choice and mark them on the number line.

For example . Each can be marked by counting units from 0.

3. How many fractions lie between 0 and 1? Think, discuss with your classmates, and write your answer.

Infinitely many. Between any two fractions you can always find another, so there are countless fractions between 0 and 1.

4. What is the length of the blue line and black line shown below? The distance between 0 and 1 is 1 unit long, and it is divided into two equal parts. The length of each part is . So the blue line is units long. Write the fraction that gives the length of the black line in the box.

The black line goes up to plus one half, so its length is .

5. Write the fraction that gives the lengths of the black lines in the respective boxes.

With fifths, the successive black lines are units long.

Figure it Out

1. How many whole units are there in ?

, so there are 3 whole units and a half.

2. How many whole units are there in and in ?

has 1 whole unit, has 2 whole units.

Figure it Out

1. Figure out the number of whole units in each of the following fractions: a.

, so 2 whole units.

1. Figure out the number of whole units in each of the following fractions: b.

, so 2 whole units.

1. Figure out the number of whole units in each of the following fractions: c.

, so 2 whole units.

2. Can all fractions greater than 1 be written as such mixed numbers?

No. Fractions like are whole numbers with no fractional part, so they are not mixed numbers.

3. Write the following fractions as mixed fractions (e.g., ): a.

3. Write the following fractions as mixed fractions (e.g., ): b.

3. Write the following fractions as mixed fractions (e.g., ): c.

3. Write the following fractions as mixed fractions (e.g., ): d.

3. Write the following fractions as mixed fractions (e.g., ): e.

3. Write the following fractions as mixed fractions (e.g., ): f.

Figure it Out

Write the following mixed numbers as fractions: a.

because plus .

Write the following mixed numbers as fractions: b.

Write the following mixed numbers as fractions: c.

Write the following mixed numbers as fractions: d.

Write the following mixed numbers as fractions: e.

Write the following mixed numbers as fractions: f.

Answer the following questions after looking at the fraction wall:

1. Are the lengths and equal?

Yes. so they have the same length.

2. Are and equivalent fractions? Why?

Yes. They mark the same length on the wall because .

3. How many pieces of length will make a length of ?

Three pieces, because .

4. How many pieces of length will make a length of ?

Two pieces, because .

Figure it Out

1. Are , , equivalent fractions? Why?

Yes. All equal and occupy the same length on the fraction wall.

2. Write two equivalent fractions for .

For example and (any fraction equivalent to like ).

3. = = = ............ (Write as many as you can)

by multiplying numerator and denominator by the same number.

Figure it Out

1. Three rotis are shared equally by four children. Show the division in the picture and write a fraction for how much each child gets. Also, write the corresponding division facts, addition facts, and multiplication facts. Fraction of roti each child gets is ______.

Each child gets roti. Division fact: . Addition fact: . Multiplication fact: .

2. Draw a picture to show how much each child gets when 2 rotis are shared equally by 4 children. Also, write the corresponding division facts, addition facts, and multiplication facts.

Each child gets roti. Division: . Addition: . Multiplication: .

3. Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get?

Anil gets of a cake.

Figure it Out

Find the missing numbers: a. 5 glasses of juice shared equally among 4 friends is the same as ____ glasses of juice shared equally among 8 friends. So, .

10 glasses among 8 friends, because .

Find the missing numbers: b. 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided equally in ___ bags. So, .

9 bags, because .

Find the missing numbers: c. 7 rotis divided among 5 children is the same as rotis divided among _ children. So, .

One possibility is 14 rotis among 10 children, . Any equivalent fraction like also works.

Figure it Out

Suppose the number of children is kept the same, but the number of units that are being shared is increased? What can you say about each child’s share now? Why? Discuss how your reasoning explains , , and .

Share increases because more is divided among the same number of people. Numerators increase while denominator stays same, so the fraction grows.

Now, decide in which of the two groups will each child get a larger share: 1. Group 1 : 3 glasses of sugarcane juice divided equally among 4 children. Group 2: 7 glasses of sugarcane juice divided equally among 10 children.

Group 1 gives , Group 2 gives . So Group 1 gets slightly more .

Now, decide in which of the two groups will each child get a larger share: 2. Group 1 : 4 glasses of sugarcane juice divided equally among 7 children. Group 2: 5 glasses of sugarcane juice divided equally among 7 children. Which groups were easier to compare? Why?

Group 2 gives . The second pair was easier because denominators are already same, so we just compare numerators.

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same.

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. a. and

, .

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. b. and

, .

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. c. and

, .

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. d. and

, .

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. e. and

, .

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. f. and

, .

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. g. and

, .

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. h. and

, .

Figure it Out

Express the following fractions in lowest terms: a.

after dividing by 17.

Express the following fractions in lowest terms: b.

after dividing by 16.

Express the following fractions in lowest terms: c.

after dividing by 21.

Express the following fractions in lowest terms: d.

after dividing by 7.

Figure it Out

1. Compare the following fractions and justify your answers: a. ,

, so .

1. Compare the following fractions and justify your answers: b. ,

, so .

1. Compare the following fractions and justify your answers: c. ,

, so .

1. Compare the following fractions and justify your answers: d. ,

Same denominator, so .

1. Compare the following fractions and justify your answers: e. ,

, so .

2. Write the following fractions in ascending order. a. , ,

, so ascending is .

2. Write the following fractions in ascending order. b. , ,

, so .

3. Write the following fractions in descending order. a. , , ,

, so descending is .

3. Write the following fractions in descending order. b. , , ,

, so .

Figure it Out

1. Add the following fractions using Brahmagupta’s method: a.

Same denominator: .

1. Add the following fractions using Brahmagupta’s method: b.

, sum .

1. Add the following fractions using Brahmagupta’s method: c.

1. Add the following fractions using Brahmagupta’s method: d.

, sum .

1. Add the following fractions using Brahmagupta’s method: e.

Common denominator 60: .

1. Add the following fractions using Brahmagupta’s method: f.

1. Add the following fractions using Brahmagupta’s method: g.

Same as f: .

1. Add the following fractions using Brahmagupta’s method: h.

1. Add the following fractions using Brahmagupta’s method: i.

1. Add the following fractions using Brahmagupta’s method: j.

1. Add the following fractions using Brahmagupta’s method: k.

Same as e: .

1. Add the following fractions using Brahmagupta’s method: l.

Common denominator 105: .

1. Add the following fractions using Brahmagupta’s method: m.

Common denominator 12: .

2. Rahim mixes litres of yellow paint with litres of blue paint to make green paint. What is the volume of green paint he has made?

litres.

3. Geeta bought meter of lace and Shamim bought meter of the same lace to put a complete border on a table cloth whose perimeter is 1 meter long. Find the total length of the lace they both have bought. Will the lace be sufficient to cover the whole border?

Total m, which is more than 1 m, so it is sufficient and m will remain.

Figure it Out

1.

2.

3.

Figure it Out

1. Carry out the following subtractions using Brahmagupta’s method: a.

1. Carry out the following subtractions using Brahmagupta’s method: b.

1. Carry out the following subtractions using Brahmagupta’s method: c.

1. Carry out the following subtractions using Brahmagupta’s method: d.

2. Subtract as indicated: a. from

2. Subtract as indicated: b. from

2. Subtract as indicated: c. from

3. Solve the following problems: a. Jaya’s school is km from her home. She takes an auto for km from her home daily, and then walks the remaining distance to reach her school. How much does she walk daily to reach the school?

km.

3. Solve the following problems: b. Jeevika takes minutes to take a complete round of the park and her friend Namit takes minutes to do the same. Who takes less time and by how much?

, so Namit is faster by minute.