Chapter 7 : Fractions (NCERT Solution)

In-text

Math Talk & Discussion Questions

These are the “think and discuss” prompts that appear inside the lessons — the reasoning ones between the worked examples.

7.1 Units 7.2 Whole 7.4 Number line 7.6 Fraction wall 7.6 Shares 7.6 Same units 7.8 On the line Puzzle

7.1

Fractional Units & Equal Shares

Key idea: a unit fraction like \(\tfrac1n\) is “1 shared among \(n\).” The more people share one thing, the smaller each share — so a bigger denominator means a smaller fraction.

Which is more — \(\tfrac12\) roti or \(\tfrac14\) roti?

When 2 children share 1 roti, each gets \(\tfrac12\) roti. When 4 children share 1 roti, each gets \(\tfrac14\) roti.

In the second case more children share the same one roti, so each child gets a smaller piece.

\(\tfrac12 \;>\; \tfrac14\) — half a roti is more.

Which fraction is greater — \(\tfrac15\) or \(\tfrac19\)?

Think of these as shares. \(\tfrac15\) means 1 roti shared among 5 children; \(\tfrac19\) means 1 roti shared among 9 children.

Sharing with more people gives each person less. (A common mistake is to think 9 > 5 makes \(\tfrac19\) bigger — it is the opposite!)

\(\tfrac19 \;<\; \tfrac15\) — so \(\tfrac15\) is greater.

7.2

Fractional Units as Parts of a Whole

By dividing the whole chikki into 6 equal parts in different ways, we get \(\tfrac16\) pieces of different shapes. Are they the same size?

Yes — they are exactly the same size. Whatever the shape of the cut, if a whole is split into 6 equal parts, every part is \(\tfrac16\) of the whole.

“Same size” means same area, not same shape. A long thin sixth and a triangular sixth both cover the same amount of chikki.

All \(\tfrac16\) pieces are equal in size ✓

7.4

Marking Fraction Lengths on the Number Line

Key idea: to place a fraction, split the gap from 0 to 1 into as many equal parts as the denominator, then count numerator parts from 0.

Intro

What is the length of the blue line? (The gap 0→1 is split into two equal parts.)

0 to 1 is one unit. It is divided into 2 equal parts, so each part is \(\tfrac12\) unit. The blue line covers one such part.

Blue line \(=\tfrac12\) unit

The unit is split into three equal parts. Write the length of the blue line.

Each part is \(\tfrac13\) unit. The blue line covers 2 of these parts.

Blue line \(=\tfrac23\) unit

The unit is split into 5 equal parts. Write the lengths of the two blue lines.

Each part is \(\tfrac15\) unit. The short line covers 2 parts and the long line covers 4 parts.

\(\tfrac25\) and \(\tfrac45\)

The unit is divided into 8 equal parts. Write the appropriate fractions.

Each part is \(\tfrac18\). Counting from 0, the marks are:

\(\tfrac18,\ \tfrac28,\ \tfrac38,\ \tfrac48,\ \tfrac58,\ \tfrac68,\ \tfrac78\)

7.6

Equivalent Fractions — using the Fraction Wall

Key idea: two fractions are equivalent if they mark the same length. On the fraction wall, equal lengths line up perfectly.

Are the lengths \(\tfrac12\) and \(\tfrac36\) equal?

On the wall, three \(\tfrac16\) pieces reach exactly to the \(\tfrac12\) mark.

Yes, \(\tfrac12=\tfrac36\)

Are \(\tfrac23\) and \(\tfrac46\) equivalent fractions? Why?

Yes. On the fraction wall they cover the same length — four \(\tfrac16\) pieces line up with two \(\tfrac13\) pieces.

Yes, \(\tfrac23=\tfrac46\)

How many pieces of length \(\tfrac16\) make a length of \(\tfrac12\)?

Since \(\tfrac12=\tfrac36\), we need three \(\tfrac16\) pieces.

3 pieces

How many pieces of length \(\tfrac16\) make a length of \(\tfrac13\)?

Since \(\tfrac13=\tfrac26\), we need two \(\tfrac16\) pieces.

2 pieces

7.6

Comparing Shares by Reasoning

Keep the number of children the same but increase the number of units shared. What happens to each child’s share? Explain \(\tfrac15<\tfrac25\), \(\tfrac37<\tfrac47\), \(\tfrac12<\tfrac58\).

Each child gets a larger share. If the same children share more units, everyone gets more.

That is why, with the same fractional unit (same denominator), a bigger numerator means a bigger fraction:

\(\tfrac15<\tfrac25,\quad \tfrac37<\tfrac47,\quad \tfrac48\!\left(=\tfrac12\right)<\tfrac58\)

In which group does each child get a larger share?1. Group 1: 3 glasses among 4 children • Group 2: 7 glasses among 10 children2. Group 1: 4 glasses among 7 children • Group 2: 5 glasses among 7 children

Pair 1 — compare \(\tfrac34\) and \(\tfrac{7}{10}\). Make same denominator 20:

\(\dfrac34=\dfrac{3\times5}{4\times5}=\dfrac{15}{20}\), \(\dfrac{7}{10}=\dfrac{7\times2}{10\times2}=\dfrac{14}{20}\)

\(\tfrac{15}{20}>\tfrac{14}{20}\), so Group 1 children get more.

Pair 2 — same number of children (7), so just compare numerators:

\(\tfrac47\) vs \(\tfrac57\Rightarrow \tfrac57>\tfrac47\), so Group 2 children get more.

Pair 1 → Group 1 • Pair 2 → Group 2

Which pair of groups was easier to compare, and why?

The second pair was easier — both groups shared among the same number of children (7), so the fractions already had the same denominator. We only had to compare the numerators \(4\) and \(5\).

Same denominator ⇒ compare numerators directly

7.6

Rewriting Pairs with the Same Fractional Unit

Find equivalent fractions for each pair so that both use the same fractional unit (same denominator).

For each pair, use a common multiple of the two denominators, then rescale each fraction.

a. \(\tfrac72,\tfrac35\)
LCD \(=10\): \(\;\tfrac{35}{10}\) and \(\tfrac{6}{10}\)

b. \(\tfrac83,\tfrac56\)
LCD \(=6\): \(\;\tfrac{16}{6}\) and \(\tfrac{5}{6}\)

c. \(\tfrac34,\tfrac35\)
LCD \(=20\): \(\;\tfrac{15}{20}\) and \(\tfrac{12}{20}\)

d. \(\tfrac67,\tfrac85\)
LCD \(=35\): \(\;\tfrac{30}{35}\) and \(\tfrac{56}{35}\)

e. \(\tfrac94,\tfrac52\)
LCD \(=4\): \(\;\tfrac{9}{4}\) and \(\tfrac{10}{4}\)

f. \(\tfrac{1}{10},\tfrac29\)
LCD \(=90\): \(\;\tfrac{9}{90}\) and \(\tfrac{20}{90}\)

g. \(\tfrac83,\tfrac{11}{4}\)
LCD \(=12\): \(\;\tfrac{32}{12}\) and \(\tfrac{33}{12}\)

h. \(\tfrac{13}{6},\tfrac19\)
LCD \(=18\): \(\;\tfrac{39}{18}\) and \(\tfrac{2}{18}\)

7.8

Adding on the Number Line

Try adding \(\tfrac47+\tfrac67\) using a number line. Do you get the same answer?

Start at 0 and take 4 hops of \(\tfrac17\) to reach \(\tfrac47\). From there, take 6 more hops of \(\tfrac17\).

→

Total hops \(=4+6=10\) hops of \(\tfrac17\).

\[\tfrac47+\tfrac67=\tfrac{10}{7}=1+\tfrac37=1\tfrac37\]

Yes — the number line gives the same answer, \(1\tfrac37\) ✓

★

Puzzle — Egyptian Fractions

Idea: a unit fraction has 1 on top. Two different unit fractions can never add to 1 (the biggest is \(\tfrac12\), and \(\tfrac12+\tfrac12\) already uses two equal halves).

Can you find three different unit fractions that add up to 1?

Start from \(\tfrac13+\tfrac13+\tfrac13=1\). To make them different, one \(\tfrac13\) must grow — the only larger unit fraction is \(\tfrac12\). So \(\tfrac12\) must be used.

Now \(\tfrac12+\tfrac14+\tfrac14=1\); one \(\tfrac14\) must grow, and the only larger option (below \(\tfrac12\)) is \(\tfrac13\). That fixes \(\tfrac12\) and \(\tfrac13\); the last piece must be \(1-\tfrac12-\tfrac13=\tfrac16\).

\(\tfrac12+\tfrac13+\tfrac16=1\) — the only solution

Can you find four different unit fractions that add up to 1? (There are six solutions!)

Using similar reasoning, all six ways to write 1 as four different unit fractions are:

\(\tfrac12+\tfrac13+\tfrac{1}{10}+\tfrac{1}{15}=1\)

\(\tfrac12+\tfrac13+\tfrac19+\tfrac{1}{18}=1\)

\(\tfrac12+\tfrac13+\tfrac18+\tfrac{1}{24}=1\)

\(\tfrac12+\tfrac13+\tfrac17+\tfrac{1}{42}=1\)

\(\tfrac12+\tfrac14+\tfrac15+\tfrac{1}{20}=1\)

\(\tfrac12+\tfrac14+\tfrac16+\tfrac{1}{12}=1\)

Tip: check any one by making all denominators the same — the numerators should add to the denominator.

Figure it Out

All Exercises — Worked Solutions

Every numbered Figure it Out question, solved step by step.

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

7.1

Fractional Units & Equal Shares

Fill in the blanks with fractions.

Three guavas together weigh 1 kg. If they are roughly the same size, each guava weighs ____ kg.

1 kg is shared equally by 3 guavas: \(1\div 3=\tfrac13\).

\(\tfrac13\) kg

1 kg of rice is packed into four equal packets. Each packet weighs ____ kg.

\(1\div 4=\tfrac14\).

\(\tfrac14\) kg

Four friends share 3 glasses of sugarcane juice equally. Each one drank ____ glass.

3 glasses shared by 4 friends: \(3\div 4=\tfrac34\).

\(\tfrac34\) glass

A big fish weighs \(\tfrac12\) kg and a small one weighs \(\tfrac14\) kg. Together they weigh ____ kg.

Same unit \(\tfrac14\): \(\;\tfrac12=\tfrac24\).

\(\tfrac24+\tfrac14=\tfrac34\).

\(\tfrac34\) kg

Arrange in order of size, smallest to biggest: one and a half, three quarters, one and a quarter, half, quarter, two and a half.

Writing each as a number: quarter \(=\tfrac14\), half \(=\tfrac12\), three quarters \(=\tfrac34\), one and a quarter \(=1\tfrac14\), one and a half \(=1\tfrac12\), two and a half \(=2\tfrac12\).

\(\tfrac14,\ \tfrac12,\ \tfrac34,\ 1\tfrac14,\ 1\tfrac12,\ 2\tfrac12\)

7.2

Fractional Units as Parts of a Whole

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

Each answer is \(\tfrac1n\), where \(n\) is how many copies of that piece exactly cover one whole chikki.

a. \(\tfrac{1}{12}\)

b. \(\tfrac{1}{4}\)

c. \(\tfrac{1}{8}\)

d. \(\tfrac{1}{6}\)

e. \(\tfrac{1}{8}\)

f. \(\tfrac{1}{6}\)

g. \(\tfrac{1}{24}\)

h. \(\tfrac{1}{24}\)

7.3

Measuring Using Fractional Units

Continue this table of \(\tfrac12\) for 2 more steps.

Each step adds one more \(\tfrac12\):

\[\underbrace{\tfrac12+\tfrac12+\tfrac12+\tfrac12+\tfrac12+\tfrac12}_{6\text{ times}}=6\times\tfrac12=\tfrac62=3\]

\[\underbrace{\tfrac12+\cdots+\tfrac12}_{7\text{ times}}=7\times\tfrac12=\tfrac72\]

6 times \(\tfrac12\), then 7 times \(\tfrac12\)

Can you create a similar table for \(\tfrac14\)?

Add one more quarter each step:

\(\tfrac14=\) 1 time quarter

\(\tfrac14+\tfrac14=\tfrac24=\) 2 times quarter

\(\tfrac14+\tfrac14+\tfrac14=\tfrac34=\) 3 times

\(\tfrac14+\tfrac14+\tfrac14+\tfrac14=\tfrac44=1=\) 4 times

Make \(\tfrac13\) using a paper strip. Can you use this to also make \(\tfrac16\)?

Yes. Take the \(\tfrac13\) strip and fold it into two equal halves. Each half is half of \(\tfrac13\):

\[\tfrac12\times\tfrac13=\tfrac16\]

Folding a \(\tfrac13\) strip in half gives \(\tfrac16\) ✓

Draw a picture and write an addition statement to show: (a) 5 times \(\tfrac14\) of a roti, (b) 9 times \(\tfrac14\) of a roti.

(a) Five quarter-rotis. Four quarters make one whole roti, and one quarter is left over:

\[\tfrac14+\tfrac14+\tfrac14+\tfrac14+\tfrac14=\tfrac54=1\tfrac14\]

(b) Nine quarter-rotis. Eight quarters make two whole rotis, with one quarter left over:

\[\underbrace{\tfrac14+\cdots+\tfrac14}_{9\text{ times}}=\tfrac94=2\tfrac14\]

(a) \(\tfrac54=1\tfrac14\) • (b) \(\tfrac94=2\tfrac14\)

Match each fractional unit with the correct picture: \(\tfrac13,\ \tfrac15,\ \tfrac18,\ \tfrac16\).

Count the equal parts in each circle — the number of parts is the denominator.

\(\tfrac13\) → circle split into 3 parts

\(\tfrac15\) → circle split into 5 parts

\(\tfrac16\) → circle split into 6 parts

\(\tfrac18\) → circle split into 8 parts

7.4

Fraction Lengths on the Number Line

On a number line, draw lines of lengths \(\tfrac{1}{10},\ \tfrac{3}{10},\ \tfrac45\).

Split 0→1 into 10 equal parts (each \(\tfrac{1}{10}\)). Note \(\tfrac45=\tfrac{8}{10}\), so it reaches the 8th mark.

Marks at \(\tfrac{1}{10},\ \tfrac{3}{10}\) and \(\tfrac45=\tfrac{8}{10}\)

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

Any correct fractions work. For example \(\tfrac14,\ \tfrac25,\ \tfrac12,\ \tfrac34,\ \tfrac78\), placed between 0 and 1 by splitting the unit into 4, 5, 2, 4 and 8 parts respectively.

Sample: \(\tfrac14,\tfrac25,\tfrac12,\tfrac34,\tfrac78\)

How many fractions lie between 0 and 1?

Between any two fractions we can always squeeze another (for example, halfway between them). So the fractions never run out.

Infinitely many (uncountable)

The gap 0→1 is split into two halves, so the blue line is \(\tfrac12\). Write the length of the black line.

Each part is \(\tfrac12\). The black line stretches past 1, covering 3 half-parts.

Black line \(=\tfrac32\)

Write the fractions that give the lengths of the black lines.

Here the unit is split into fifths. The four black lines end at the marks just after 1:

\(\tfrac65,\ \tfrac75,\ \tfrac85,\ \tfrac95\)

7.5

Mixed Fractions

How many whole units are there in \(\tfrac72\)?

\(\tfrac72=\tfrac22+\tfrac22+\tfrac22+\tfrac12=1+1+1+\tfrac12\).

3 whole units

How many whole units are there in \(\tfrac43\) and in \(\tfrac73\)?

\(\tfrac43=1+\tfrac13\) and \(\tfrac73=2+\tfrac13\).

\(\tfrac43\): 1 whole • \(\tfrac73\): 2 wholes

Figure out the number of whole units in each: (a) \(\tfrac83\), (b) \(\tfrac{11}{5}\), (c) \(\tfrac94\).

a. \(\tfrac83\) \(=2+\tfrac23\) → 2

b. \(\tfrac{11}{5}\) \(=2+\tfrac15\) → 2

c. \(\tfrac94\) \(=2+\tfrac14\) → 2

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

No. A mixed number needs a fractional part that is less than 1. If the fraction is a whole number, there is no leftover fractional part.

Example: \(\tfrac84=2\), which is just a whole number.

No — e.g. \(\tfrac84=2\) is not a mixed number

Write the following as mixed fractions.

Divide the numerator by the denominator; the quotient is the whole part and the remainder stays over the denominator.

a. \(\tfrac92\) \(=4\tfrac12\)

b. \(\tfrac95\) \(=1\tfrac45\)

c. \(\tfrac{21}{19}\) \(=1\tfrac{2}{19}\)

d. \(\tfrac{47}{9}\) \(=5\tfrac29\)

e. \(\tfrac{12}{11}\) \(=1\tfrac{1}{11}\)

f. \(\tfrac{19}{6}\) \(=3\tfrac16\)

Write the following mixed numbers as (improper) fractions.

Use \(\;\text{whole}\,\tfrac{a}{b}=\dfrac{(\text{whole}\times b)+a}{b}\).

a. \(3\tfrac14\) \(=\tfrac{3\times4+1}{4}=\tfrac{13}{4}\)

b. \(7\tfrac23\) \(=\tfrac{21+2}{3}=\tfrac{23}{3}\)

c. \(9\tfrac49\) \(=\tfrac{81+4}{9}=\tfrac{85}{9}\)

d. \(3\tfrac16\) \(=\tfrac{18+1}{6}=\tfrac{19}{6}\)

e. \(2\tfrac{3}{11}\) \(=\tfrac{22+3}{11}=\tfrac{25}{11}\)

f. \(3\tfrac{9}{10}\) \(=\tfrac{30+9}{10}=\tfrac{39}{10}\)

7.6

Equivalent Fractions & Lowest Terms

Are \(\tfrac36,\ \tfrac48,\ \tfrac{5}{10}\) equivalent fractions? Why?

In each one the numerator is exactly half the denominator, so each equals \(\tfrac12\). On the fraction wall their lengths line up.

Yes — all equal \(\tfrac12\)

Write two equivalent fractions for \(\tfrac26\).

Divide top and bottom by 2 to get \(\tfrac13\); multiply top and bottom by useful numbers for more:

\(\tfrac13\) and \(\tfrac39\) (also \(\tfrac{4}{12},\ \tfrac{5}{15},\dots\))

\(\tfrac46=\ \square=\ \square=\ \square=\dots\) (write as many as you can)

\[\tfrac46=\tfrac23=\tfrac69=\tfrac{8}{12}=\tfrac{10}{15}=\cdots\]

Multiply numerator and denominator by the same number to keep making equivalents.

Three rotis are shared equally by four children. Give the fraction each child gets, and the division, addition and multiplication facts.

Cut each roti into 4 quarters; every child gets 1 quarter from each roti, i.e. 3 quarters.

Division fact: \(3\div4=\tfrac34\)

Addition fact: \(3=\tfrac34+\tfrac34+\tfrac34+\tfrac34\)

Multiplication fact: \(3=4\times\tfrac34\)

Each child gets \(\tfrac34\) roti

2 rotis are shared equally by 4 children. Draw the sharing and write the division, addition and multiplication facts.

Each roti is halved; each of the 4 children gets one half.

Division fact: \(2\div4=\tfrac24=\tfrac12\)

Addition fact: \(2=\tfrac12+\tfrac12+\tfrac12+\tfrac12\)

Multiplication fact: \(2=4\times\tfrac12\)

Each child gets \(\tfrac24=\tfrac12\) roti

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

\(2\div5=\tfrac25\).

\(\tfrac25\) cake

Find the missing numbers.a. \(\tfrac54=\tfrac{\square}{8}\) • b. \(\tfrac43=\tfrac{12}{\square}\) • c. \(\tfrac75=\tfrac{\square}{\square}\)

Denominator \(4\to8\) means \(\times2\), so numerator \(5\times2=10\): \(\;\tfrac54=\tfrac{10}{8}\). → 10

Numerator \(4\to12\) means \(\times3\), so denominator \(3\times3=9\): \(\;\tfrac43=\tfrac{12}{9}\). → 9

Multiply top and bottom by the same number, e.g. \(\times2\): \(\;\tfrac75=\tfrac{14}{10}\). → 14 rotis, 10 children (other multiples also work)

Express the following fractions in lowest terms: (a) \(\tfrac{17}{51}\), (b) \(\tfrac{64}{144}\), (c) \(\tfrac{126}{147}\), (d) \(\tfrac{525}{112}\).

Divide numerator and denominator by their highest common factor.

a. \(\tfrac{17}{51}=\tfrac{17\div17}{51\div17}=\tfrac13\)

b. \(\tfrac{64}{144}=\tfrac{64\div16}{144\div16}=\tfrac49\)

c. \(\tfrac{126}{147}=\tfrac{126\div21}{147\div21}=\tfrac67\)

d. \(\tfrac{525}{112}=\tfrac{525\div7}{112\div7}=\tfrac{75}{16}\)

7.7

Comparing Fractions

Method: rewrite the fractions with the same denominator (a common multiple), then just compare the numerators.

Compare the following fractions and justify your answers.

\(\tfrac83,\tfrac52\): LCD 6 → \(\tfrac{16}{6}\) vs \(\tfrac{15}{6}\) ⇒ \(\tfrac83>\tfrac52\)

\(\tfrac49,\tfrac37\): LCD 63 → \(\tfrac{28}{63}\) vs \(\tfrac{27}{63}\) ⇒ \(\tfrac49>\tfrac37\)

\(\tfrac{7}{10},\tfrac{9}{14}\): LCD 70 → \(\tfrac{49}{70}\) vs \(\tfrac{45}{70}\) ⇒ \(\tfrac{7}{10}>\tfrac{9}{14}\)

\(\tfrac{12}{5},\tfrac85\): same denominator ⇒ \(\tfrac{12}{5}>\tfrac85\)

\(\tfrac94,\tfrac52\): LCD 4 → \(\tfrac94\) vs \(\tfrac{10}{4}\) ⇒ \(\tfrac94<\tfrac52\)

Write in ascending order. (a) \(\tfrac{7}{10},\tfrac{11}{15},\tfrac25\) (b) \(\tfrac{19}{24},\tfrac56,\tfrac{7}{12}\)

LCD 30 → \(\tfrac{21}{30},\tfrac{22}{30},\tfrac{12}{30}\) ⇒ \(\;\tfrac25<\tfrac{7}{10}<\tfrac{11}{15}\)

LCD 24 → \(\tfrac{19}{24},\tfrac{20}{24},\tfrac{14}{24}\) ⇒ \(\;\tfrac{7}{12}<\tfrac{19}{24}<\tfrac56\)

Write in descending order. (a) \(\tfrac{25}{16},\tfrac78,\tfrac{13}{4},\tfrac{17}{32}\) (b) \(\tfrac34,\tfrac{12}{5},\tfrac{7}{12},\tfrac54\)

LCD 32 → \(\tfrac{50}{32},\tfrac{28}{32},\tfrac{104}{32},\tfrac{17}{32}\) ⇒ \(\;\tfrac{13}{4}>\tfrac{25}{16}>\tfrac78>\tfrac{17}{32}\)

LCD 60 → \(\tfrac{45}{60},\tfrac{144}{60},\tfrac{35}{60},\tfrac{75}{60}\) ⇒ \(\;\tfrac{12}{5}>\tfrac54>\tfrac34>\tfrac{7}{12}\)

7.8

Addition & Subtraction of Fractions

Brahmagupta’s method: make the fractional units the same (common denominator), then add or subtract the numerators; simplify at the end.

Add using Brahmagupta’s method.

a. \(\tfrac27+\tfrac57+\tfrac67=\tfrac{13}{7}\)

b. \(\tfrac34+\tfrac13=\tfrac{9}{12}+\tfrac{4}{12}=\tfrac{13}{12}\)

c. \(\tfrac23+\tfrac56=\tfrac{4}{6}+\tfrac{5}{6}=\tfrac96=\tfrac32\)

d. \(\tfrac23+\tfrac27=\tfrac{14}{21}+\tfrac{6}{21}=\tfrac{20}{21}\)

e. \(\tfrac34+\tfrac13+\tfrac15=\tfrac{45+20+12}{60}=\tfrac{77}{60}\)

f. \(\tfrac23+\tfrac45=\tfrac{10}{15}+\tfrac{12}{15}=\tfrac{22}{15}\)

g. \(\tfrac45+\tfrac23=\tfrac{12}{15}+\tfrac{10}{15}=\tfrac{22}{15}\)

h. \(\tfrac35+\tfrac58=\tfrac{24}{40}+\tfrac{25}{40}=\tfrac{49}{40}\)

i. \(\tfrac92+\tfrac54=\tfrac{18}{4}+\tfrac{5}{4}=\tfrac{23}{4}\)

j. \(\tfrac83+\tfrac27=\tfrac{56}{21}+\tfrac{6}{21}=\tfrac{62}{21}\)

k. \(\tfrac34+\tfrac13+\tfrac15=\tfrac{77}{60}\)

l. \(\tfrac23+\tfrac45+\tfrac37=\tfrac{70+84+45}{105}=\tfrac{199}{105}\)

m. \(\tfrac92+\tfrac54+\tfrac76=\tfrac{54+15+14}{12}=\tfrac{83}{12}\)

Rahim mixes \(\tfrac23\) litre of yellow paint with \(\tfrac34\) litre of blue paint. What volume of green paint did he make?

LCD of 3 and 4 is 12: \(\tfrac23=\tfrac{8}{12}\), \(\tfrac34=\tfrac{9}{12}\).

\(\tfrac{8}{12}+\tfrac{9}{12}=\tfrac{17}{12}=1\tfrac{5}{12}\).

\(1\tfrac{5}{12}\) litres of green paint

Geeta bought \(\tfrac25\) m of lace and Shamim bought \(\tfrac34\) m of the same lace to border a table cloth of perimeter 1 m. Find the total lace. Is it enough?

LCD of 5 and 4 is 20: \(\tfrac25=\tfrac{8}{20}\), \(\tfrac34=\tfrac{15}{20}\).

Total \(=\tfrac{8}{20}+\tfrac{15}{20}=\tfrac{23}{20}=1\tfrac{3}{20}\) m.

\(1\tfrac{3}{20}>1\) m, so it covers the whole 1 m border with lace to spare.

\(1\tfrac{3}{20}\) m — yes, it is enough ✓

Subtract (same denominator): (1) \(\tfrac58-\tfrac38\) (2) \(\tfrac79-\tfrac59\) (3) \(\tfrac{10}{27}-\tfrac{1}{27}\)

1. \(\tfrac58-\tfrac38=\tfrac28=\tfrac14\)

2. \(\tfrac79-\tfrac59=\tfrac29\)

3. \(\tfrac{10}{27}-\tfrac{1}{27}=\tfrac{9}{27}=\tfrac13\)

Carry out the subtractions using Brahmagupta’s method.

a. \(\tfrac{8}{15}-\tfrac{3}{15}=\tfrac{5}{15}=\tfrac13\)

b. \(\tfrac25-\tfrac{4}{15}=\tfrac{6}{15}-\tfrac{4}{15}=\tfrac{2}{15}\)

c. \(\tfrac56-\tfrac49=\tfrac{15}{18}-\tfrac{8}{18}=\tfrac{7}{18}\)

d. \(\tfrac23-\tfrac12=\tfrac{4}{6}-\tfrac{3}{6}=\tfrac16\)

Subtract as indicated. (a) \(\tfrac{13}{4}\) from \(\tfrac{10}{3}\) (b) \(\tfrac{18}{5}\) from \(\tfrac{23}{3}\) (c) \(\tfrac{29}{7}\) from \(\tfrac{45}{7}\)

\(\tfrac{10}{3}-\tfrac{13}{4}=\tfrac{40}{12}-\tfrac{39}{12}=\tfrac{1}{12}\)

\(\tfrac{23}{3}-\tfrac{18}{5}=\tfrac{115}{15}-\tfrac{54}{15}=\tfrac{61}{15}=4\tfrac{1}{15}\)

\(\tfrac{45}{7}-\tfrac{29}{7}=\tfrac{16}{7}=2\tfrac27\)

Q3a

Jaya’s school is \(\tfrac{7}{10}\) km from home. She takes an auto for \(\tfrac12\) km, then walks the rest. How far does she walk daily?

Walk \(=\tfrac{7}{10}-\tfrac12=\tfrac{7}{10}-\tfrac{5}{10}=\tfrac{2}{10}\).

Simplify: \(\tfrac{2}{10}=\tfrac15\).

She walks \(\tfrac15\) km daily

Q3b

Jeevika takes \(\tfrac{10}{3}\) minutes for one round of the park; Namit takes \(\tfrac{13}{4}\) minutes. Who is faster and by how much?

LCD 12: \(\tfrac{10}{3}=\tfrac{40}{12}\), \(\tfrac{13}{4}=\tfrac{39}{12}\).

\(\tfrac{39}{12}<\tfrac{40}{12}\), so Namit takes less time.

Difference \(=\tfrac{40}{12}-\tfrac{39}{12}=\tfrac{1}{12}\) minute.

Namit is faster, by \(\tfrac{1}{12}\) minute

7.9 · A Pinch of History

In ancient India a fraction was called bhinna (“broken”) in Sanskrit, also bhaga or ansha meaning “part.” The way we write fractions today began in India — the Bakshali manuscript (around 300 CE) already wrote them much like we do now.

General fractions and the rules for adding, subtracting, multiplying and dividing them were first set out formally by Brahmagupta in 628 CE. His methods for adding and subtracting fractions — make a common denominator, then work with the numerators — are still exactly what we use today.

These Indian ideas reached Europe through Arab scholars over later centuries and came into common European use around the 17th century, before spreading worldwide.