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Class 6 Ganita Prakash · Chapter 7
Class 6 Ganita Prakash · Chapter 7
Fractions
Page-wise textbook solutions (pages 151–186) · Every step · Figures · Common mistakes · Tips
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✔ Understood 0/84
๐️ Key ideas of this chapter
- Fractional unit: one of the equal parts of a whole (\(\dfrac{1}{2}\), \(\dfrac{1}{3}\), \(\dfrac{1}{4}\) …). More parts → smaller unit: \(\dfrac{1}{9}\) < \(\dfrac{1}{5}\).
- \(\dfrac{3}{4}\) means 3 units of \(\dfrac{1}{4}\). Top = numerator, bottom = denominator.
- Numerator > denominator → more than 1 → write as a mixed number: \(\dfrac{7}{2}\) = \(3\dfrac{1}{2}\).
- Equivalent fractions: multiply or divide top and bottom by the same number: \(\dfrac{1}{2}\) = \(\dfrac{2}{4}\) = \(\dfrac{3}{6}\).
- Lowest terms: divide by the highest common factor: \(\dfrac{16}{20}\) = \(\dfrac{4}{5}\).
- Compare, add, subtract: first make the denominators the same (Brahmagupta’s method).
๐ Textbook Page 151
๐ Page 151Equal shares
One roti is shared equally by 2 children, and another by 4 children. Which is more: \(\dfrac{1}{2}\) roti or \(\dfrac{1}{4}\) roti?1 roti shared equally by 2 children — each gets 1/2 roti (one piece of every roti). 1 roti shared equally by 4 children — each gets 1/4 roti (one piece of every roti).
12 children: each gets \(\dfrac{1}{2}\) roti.
24 children: each gets \(\dfrac{1}{4}\) roti.
3More children share the same roti, so each piece is smaller.
✅ Answer: \(\dfrac{1}{2}\) roti is more than \(\dfrac{1}{4}\) roti.
๐ Textbook Page 152
๐ Page 152Which is greater?
Which fraction is greater: \(\dfrac{1}{5}\) or \(\dfrac{1}{9}\)?
1Think of one roti shared by 5 children and another by 9 children.
2Sharing among more people gives a smaller share.
3So \(\dfrac{1}{9}\) < \(\dfrac{1}{5}\).
✅ Answer: \(\dfrac{1}{5}\) is greater.
⚠️ Common mistake: Thinking \(\dfrac{1}{9}\) is bigger because 9 is bigger than 5.
๐ Page 152Figure it Out · Q1
Three guavas together weigh 1 kg and are about the same size. Each guava weighs ____ kg.
11 kg is shared equally by 3 guavas.
21 ÷ 3 = \(\dfrac{1}{3}\).
✅ Answer: \(\dfrac{1}{3}\) kg
๐ Page 152Figure it Out · Q2
1 kg of rice is packed in four packets of equal weight. Each packet weighs ____ kg.
11 ÷ 4 = \(\dfrac{1}{4}\).
✅ Answer: \(\dfrac{1}{4}\) kg
๐ Page 152Figure it Out · Q3
Four friends share 3 glasses of sugarcane juice equally. Each drinks ____ glass.
1Each glass is shared by 4 friends: \(\dfrac{1}{4}\) glass from each glass.
2From 3 glasses: \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) = \(\dfrac{3}{4}\).
✅ Answer: \(\dfrac{3}{4}\) glass
๐ Textbook Page 153
๐ Page 153Figure it Out · Q4
The big fish weighs \(\dfrac{1}{2}\) kg and the small one \(\dfrac{1}{4}\) kg. Together they weigh ____ kg.
1\(\dfrac{1}{2}\) kg = \(\dfrac{2}{4}\) kg.
2\(\dfrac{2}{4}\) + \(\dfrac{1}{4}\) = \(\dfrac{3}{4}\).
✅ Answer: \(\dfrac{3}{4}\) kg
๐ Page 153Words for fractions
Find the words used for fractions in your home language (half, quarter, three quarters, one and a quarter, one and a half, two and a half).
1Hindi: aadha (½), paav (¼), teen paav (¾), savaa (1¼), dedh (1½), dhaai (2½).
2Tamil: arai (½), kaal (¼), mukkaal (¾).
3Ask at home and add your own language.
✅ Answer: Activity — answers differ by language.
๐ Page 153Figure it Out · Q5
Arrange from smallest to biggest: one and a half, three quarters, one and a quarter, half, quarter, two and a half.
1Write as numbers: \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(\dfrac{3}{4}\), \(1\dfrac{1}{4}\), \(1\dfrac{1}{2}\), \(2\dfrac{1}{2}\).
2Quarters: \(\dfrac{1}{4}\) = 1 quarter, \(\dfrac{1}{2}\) = 2 quarters, \(\dfrac{3}{4}\) = 3, \(1\dfrac{1}{4}\) = 5, \(1\dfrac{1}{2}\) = 6, \(2\dfrac{1}{2}\) = 10 quarters.
✅ Answer: quarter < half < three quarters < one and a quarter < one and a half < two and a half
๐ Textbook Page 154
๐ Page 154Chikki pieces
A chikki is broken into 2 pieces. The small piece is \(\dfrac{1}{4}\) chikki. How much is the bigger piece?The top-right part is ¼; the rest is the bigger piece.
1The bigger piece is made of 3 pieces of size \(\dfrac{1}{4}\).
23 × \(\dfrac{1}{4}\) = \(\dfrac{3}{4}\).
✅ Answer: \(\dfrac{3}{4}\) chikki
๐ Page 154Math Talk · Same size?
A chikki is cut into 6 equal parts in two different ways, giving pieces of different shapes. Are the \(\dfrac{1}{6}\) pieces the same size?One way: 6 rectangles of 4 squares each.
1Each piece is one of 6 equal parts of the same whole.
2So each covers 24 ÷ 6 = 4 small squares, whatever its shape.
✅ Answer: Yes — same size (area), different shapes.
๐ Textbook Page 155
๐ Page 155Fractional unit 1/3
The chikki is broken into 3 equal pieces. What fractional unit is one piece?Cut into 3 equal pieces.
13 equal parts → each part is \(\dfrac{1}{3}\).
✅ Answer: \(\dfrac{1}{3}\) chikki
๐ Page 155Figure it Out
How much of a whole chikki (6 × 4 = 24 small squares) is each piece a–h?A whole chikki = 6 × 4 = 24 small squares. Pieces a–h of the same chikki (each small square is 1 of the 24 squares).
1Count small squares (two half-squares make one).
2a: 2 squares → \(\dfrac{2}{24}\) = \(\dfrac{1}{12}\). b: triangle half of 3 × 4 = 6 squares → \(\dfrac{6}{24}\) = \(\dfrac{1}{4}\).
3c: half of 3 × 2 = 3 squares → \(\dfrac{3}{24}\) = \(\dfrac{1}{8}\). d: 4 squares → \(\dfrac{4}{24}\) = \(\dfrac{1}{6}\).
4e: 3 squares → \(\dfrac{1}{8}\). f: half of 2 × 4 = 4 squares → \(\dfrac{1}{6}\).
5g: 1 square → \(\dfrac{1}{24}\). h: half of 1 × 2 = 1 square → \(\dfrac{1}{24}\).
✅ Answer: a \(\dfrac{1}{12}\) · b \(\dfrac{1}{4}\) · c \(\dfrac{1}{8}\) · d \(\dfrac{1}{6}\) · e \(\dfrac{1}{8}\) · f \(\dfrac{1}{6}\) · g \(\dfrac{1}{24}\) · h \(\dfrac{1}{24}\)
๐ก Tip: Compare each piece with the whole: how many such pieces make 24 squares?
๐ Textbook Page 156
๐ Page 156Folding a strip
Fold a 1-unit strip into 2 equal parts, then into 4. What are the lengths?
1One fold: two parts, each \(\dfrac{1}{2}\).
2Fold again: four parts, each \(\dfrac{1}{4}\).
32 × \(\dfrac{1}{4}\) = \(\dfrac{2}{4}\), 3 × \(\dfrac{1}{4}\) = \(\dfrac{3}{4}\), 4 × \(\dfrac{1}{4}\) = \(\dfrac{4}{4}\) = 1.
✅ Answer: \(\dfrac{1}{2}\) each; then \(\dfrac{1}{4}\) each, and \(\dfrac{2}{4}\), \(\dfrac{3}{4}\), \(\dfrac{4}{4}\) = 1.
๐ Textbook Page 157
๐ Page 157Do it once more!
Fold again into 8 equal parts and fill the boxes.
1Each part = \(\dfrac{1}{8}\).
22 times \(\dfrac{1}{8}\) = \(\dfrac{2}{8}\), 3 times = \(\dfrac{3}{8}\), 4 times = \(\dfrac{4}{8}\), 5 times = \(\dfrac{5}{8}\), 6 times = \(\dfrac{6}{8}\), 7 times = \(\dfrac{7}{8}\).
38 times \(\dfrac{1}{8}\) = \(\dfrac{8}{8}\) = 1.
✅ Answer: \(\dfrac{1}{8}\), \(\dfrac{2}{8}\), \(\dfrac{3}{8}\), \(\dfrac{4}{8}\), \(\dfrac{5}{8}\), \(\dfrac{6}{8}\), \(\dfrac{7}{8}\), \(\dfrac{8}{8}\) = 1
๐ Page 157Collecting halves
Describe 1, 2, 3, 4 and 5 halves of a roti.Half rotis collected: 1 half, 2 halves, … 5 halves.
1\(\dfrac{1}{2}\) = 1 half; \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) = 2 halves = 1 roti.
23 halves = \(\dfrac{3}{2}\); 4 halves = 2 rotis; 5 halves = \(\dfrac{5}{2}\).
✅ Answer: \(\dfrac{1}{2}\), 1, \(\dfrac{3}{2}\), 2, \(\dfrac{5}{2}\) rotis
๐ Textbook Page 158
๐ Page 158Figure it Out · Q1
Continue the table of \(\dfrac{1}{2}\) for 2 more steps.
16 halves: \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) = 6 times half = 3 rotis.
27 halves: 7 times half = \(\dfrac{7}{2}\) rotis.
✅ Answer: 6 times half = 3; 7 times half = \(\dfrac{7}{2}\)
๐ Page 158Figure it Out · Q2
Create a similar table for \(\dfrac{1}{4}\).
1\(\dfrac{1}{4}\) = 1 times quarter.
2\(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) = 2 times quarter = \(\dfrac{2}{4}\).
33 times = \(\dfrac{3}{4}\); 4 times = \(\dfrac{4}{4}\) = 1; 5 times = \(\dfrac{5}{4}\).
✅ Answer: \(\dfrac{1}{4}\), \(\dfrac{2}{4}\), \(\dfrac{3}{4}\), 1, \(\dfrac{5}{4}\), …
๐ Page 158Figure it Out · Q3
Make \(\dfrac{1}{3}\) using a paper strip. Can you use it to make \(\dfrac{1}{6}\)?
1Fold the strip into 3 equal parts (like a letter): each is \(\dfrac{1}{3}\).
2Fold the folded strip in half again: 6 equal parts.
3Each is half of \(\dfrac{1}{3}\) = \(\dfrac{1}{6}\).
✅ Answer: Yes — halve each \(\dfrac{1}{3}\) to get \(\dfrac{1}{6}\).
๐ Page 158Figure it Out · Q4
Draw and write an addition statement for (a) 5 times \(\dfrac{1}{4}\) of a roti (b) 9 times \(\dfrac{1}{4}\) of a roti.
1(a) \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) = \(\dfrac{5}{4}\) = 1 roti + \(\dfrac{1}{4}\) roti.
2(b) nine quarters = \(\dfrac{9}{4}\) = 2 rotis + \(\dfrac{1}{4}\) roti.
✅ Answer: (a) \(\dfrac{5}{4}\) = \(1\dfrac{1}{4}\) rotis (b) \(\dfrac{9}{4}\) = \(2\dfrac{1}{4}\) rotis
๐ Page 158Figure it Out · Q5
Match each fractional unit (\(\dfrac{1}{3}\), \(\dfrac{1}{5}\), \(\dfrac{1}{8}\), \(\dfrac{1}{6}\)) with the correct picture.Each circle has one part shaded.
1Count the equal parts in each circle; one part is shaded.
2(a) 8 parts → \(\dfrac{1}{8}\). (b) 6 parts → \(\dfrac{1}{6}\). (c) 3 parts → \(\dfrac{1}{3}\). (d) 5 parts → \(\dfrac{1}{5}\).
✅ Answer: \(\dfrac{1}{3}\) → (c), \(\dfrac{1}{5}\) → (d), \(\dfrac{1}{8}\) → (a), \(\dfrac{1}{6}\) → (b)
๐ Textbook Page 159
๐ Page 159Length of the blue line
Each unit is split into 2 equal parts. How long is the blue line?
10 to 1 is split into 2 equal parts; the blue line covers one part.
✅ Answer: \(\dfrac{1}{2}\) unit
๐ Page 159Number line · Q1
A unit is split into 3 equal parts. What is the length of the blue line?
1Each part = \(\dfrac{1}{3}\).
2The blue line covers 2 parts → \(\dfrac{2}{3}\).
✅ Answer: \(\dfrac{2}{3}\)
๐ Page 159Number line · Q2
A unit is split into 5 equal parts. Write the lengths of the two blue lines.
1Each part = \(\dfrac{1}{5}\).
2Short line: 2 parts → \(\dfrac{2}{5}\).
3Long line: 4 parts → \(\dfrac{4}{5}\).
✅ Answer: \(\dfrac{2}{5}\) and \(\dfrac{4}{5}\)
๐ Page 159Number line · Q3
A unit is split into 8 equal parts. Write the fractions for the marks from 0 to 1.
1Each part = \(\dfrac{1}{8}\).
✅ Answer: \(\dfrac{1}{8}\), \(\dfrac{2}{8}\), \(\dfrac{3}{8}\), \(\dfrac{4}{8}\), \(\dfrac{5}{8}\), \(\dfrac{6}{8}\), \(\dfrac{7}{8}\), \(\dfrac{8}{8}\) = 1
๐ Textbook Page 160
๐ Page 160Figure it Out · Q1
On a number line, draw lines of lengths \(\dfrac{1}{10}\), \(\dfrac{3}{10}\) and \(\dfrac{4}{5}\).
1Split 0 to 1 into 10 equal parts.
2\(\dfrac{1}{10}\) = 1 part, \(\dfrac{3}{10}\) = 3 parts.
3\(\dfrac{4}{5}\) = \(\dfrac{8}{10}\) = 8 parts.
✅ Answer: See the number line.
๐ Page 160Figure it Out · Q2
Write five more fractions of your choice and mark them on the number line.
1Example: \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(\dfrac{3}{4}\), \(\dfrac{5}{4}\), \(\dfrac{3}{2}\).
2Split each unit into 4 equal parts and count.
✅ Answer: Many answers, e.g. \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(\dfrac{3}{4}\), \(\dfrac{5}{4}\), \(\dfrac{3}{2}\).
๐ Page 160Figure it Out · Q3
How many fractions lie between 0 and 1?
1\(\dfrac{1}{2}\), \(\dfrac{1}{3}\), \(\dfrac{1}{4}\), … and \(\dfrac{1}{100}\), \(\dfrac{1}{1000}\) … are all between 0 and 1.
2Between any two fractions we can always find another one.
✅ Answer: Infinitely many — they never end.
๐ Page 160Figure it Out · Q4
The blue line is \(\dfrac{1}{2}\) unit long. How long is the black line?
1Each unit is split into 2 parts of \(\dfrac{1}{2}\).
2The black line covers 3 such parts.
✅ Answer: \(\dfrac{3}{2}\) units
๐ Page 160Figure it Out · Q5
Write the lengths of the four black lines (each unit split into 5 parts).
1Each part = \(\dfrac{1}{5}\).
2The lines cover 6, 7, 8 and 9 parts.
✅ Answer: \(\dfrac{6}{5}\), \(\dfrac{7}{5}\), \(\dfrac{8}{5}\), \(\dfrac{9}{5}\)
๐ Textbook Page 161
๐ Page 161Two groups
Sort the fractions you marked into lengths less than 1 unit and more than 1 unit. What is common to those more than 1?
| Less than 1 unit | More than 1 unit |
|---|---|
| \(\dfrac{1}{2}\), \(\dfrac{2}{3}\), \(\dfrac{2}{5}\), \(\dfrac{4}{5}\), \(\dfrac{1}{10}\), \(\dfrac{3}{10}\) | \(\dfrac{3}{2}\), \(\dfrac{6}{5}\), \(\dfrac{7}{5}\), \(\dfrac{8}{5}\), \(\dfrac{9}{5}\) |
1Less than 1: \(\dfrac{1}{2}\), \(\dfrac{2}{3}\), \(\dfrac{2}{5}\), \(\dfrac{4}{5}\), \(\dfrac{1}{10}\), \(\dfrac{3}{10}\), \(\dfrac{1}{8}\) …
2More than 1: \(\dfrac{3}{2}\), \(\dfrac{6}{5}\), \(\dfrac{7}{5}\), \(\dfrac{8}{5}\), \(\dfrac{9}{5}\).
3In fractions more than 1, the numerator is bigger than the denominator.
✅ Answer: More than 1 ⇔ numerator > denominator.
๐ Textbook Page 162
๐ Page 162Figure it Out · Q1
How many whole units are there in \(\dfrac{7}{2}\)?
1\(\dfrac{7}{2}\) = 7 halves; 2 halves make 1 whole.
27 halves = 3 wholes + 1 half.
✅ Answer: 3 whole units (\(\dfrac{7}{2}\) = \(3\dfrac{1}{2}\))
๐ Page 162Figure it Out · Q2
How many whole units are there in \(\dfrac{4}{3}\) and in \(\dfrac{7}{3}\)?
13 thirds make 1 whole.
2\(\dfrac{4}{3}\) = 1 + \(\dfrac{1}{3}\); \(\dfrac{7}{3}\) = 2 + \(\dfrac{1}{3}\).
✅ Answer: \(\dfrac{4}{3}\): 1 whole; \(\dfrac{7}{3}\): 2 wholes
๐ Page 162Whole units · Q1
Find the number of whole units in (a) \(\dfrac{8}{3}\) (b) \(\dfrac{11}{5}\) (c) \(\dfrac{9}{4}\).
1(a) 8 ÷ 3 = 2, remainder 2 → \(2\dfrac{2}{3}\).
2(b) 11 ÷ 5 = 2, remainder 1 → \(2\dfrac{1}{5}\).
3(c) 9 ÷ 4 = 2, remainder 1 → \(2\dfrac{1}{4}\).
✅ Answer: 2 in each
๐ Page 162Whole units · Q2
Can all fractions greater than 1 be written as mixed numbers?
1Divide the numerator by the denominator: quotient = whole part, remainder = new numerator.
2If the remainder is 0 (like \(\dfrac{6}{3}\) = 2), the answer is just a whole number.
✅ Answer: Yes — except that some become whole numbers with no fractional part (\(\dfrac{6}{3}\) = 2).
๐ Page 162Whole units · Q3
Write as mixed fractions: (a) \(\dfrac{9}{2}\) (b) \(\dfrac{9}{5}\) (c) \(\dfrac{21}{19}\) (d) \(\dfrac{47}{9}\) (e) \(\dfrac{12}{11}\) (f) \(\dfrac{19}{6}\).
1(a) 9 ÷ 2 = 4 remainder 1 → \(4\dfrac{1}{2}\)
2(b) 9 ÷ 5 = 1 remainder 4 → \(1\dfrac{4}{5}\)
3(c) 21 ÷ 19 = 1 remainder 2 → \(1\dfrac{2}{19}\)
4(d) 47 ÷ 9 = 5 remainder 2 → \(5\dfrac{2}{9}\)
5(e) 12 ÷ 11 = 1 remainder 1 → \(1\dfrac{1}{11}\)
6(f) 19 ÷ 6 = 3 remainder 1 → \(3\dfrac{1}{6}\)
✅ Answer: (a) \(4\dfrac{1}{2}\) · (b) \(1\dfrac{4}{5}\) · (c) \(1\dfrac{2}{19}\) · (d) \(5\dfrac{2}{9}\) · (e) \(1\dfrac{1}{11}\) · (f) \(3\dfrac{1}{6}\)
๐ก Tip: Numerator ÷ denominator: the quotient is the whole part, the remainder stays over the same denominator.
๐ Textbook Page 163
๐ Page 163Figure it Out
Write as fractions: (a) \(3\dfrac{1}{4}\) (b) \(7\dfrac{2}{3}\) (c) \(9\dfrac{4}{9}\) (d) \(3\dfrac{1}{6}\) (e) \(2\dfrac{3}{11}\) (f) \(3\dfrac{9}{10}\).
1Each whole = denominator pieces. Multiply the whole by the denominator and add the numerator.
2(a) 3 × 4 + 1 = 13 → \(\dfrac{13}{4}\)
3(b) 7 × 3 + 2 = 23 → \(\dfrac{23}{3}\)
4(c) 9 × 9 + 4 = 85 → \(\dfrac{85}{9}\)
5(d) 3 × 6 + 1 = 19 → \(\dfrac{19}{6}\)
6(e) 2 × 11 + 3 = 25 → \(\dfrac{25}{11}\)
7(f) 3 × 10 + 9 = 39 → \(\dfrac{39}{10}\)
✅ Answer: (a) \(\dfrac{13}{4}\) · (b) \(\dfrac{23}{3}\) · (c) \(\dfrac{85}{9}\) · (d) \(\dfrac{19}{6}\) · (e) \(\dfrac{25}{11}\) · (f) \(\dfrac{39}{10}\)
๐ Textbook Page 164
๐ Page 164Equal lengths
Are \(\dfrac{1}{2}\) and \(\dfrac{2}{4}\) equal? Are \(\dfrac{2}{4}\) and \(\dfrac{4}{8}\) equal? Check \(\dfrac{1}{3}\) and \(\dfrac{2}{6}\).
1Folding: the \(\dfrac{1}{2}\) strip covers exactly 2 quarters and 4 eighths.
2In the same way \(\dfrac{1}{3}\) covers exactly 2 sixths.
✅ Answer: Yes: \(\dfrac{1}{2}\) = \(\dfrac{2}{4}\) = \(\dfrac{4}{8}\) and \(\dfrac{1}{3}\) = \(\dfrac{2}{6}\).
๐ Page 164Fraction wall · Q1
Are \(\dfrac{1}{2}\) and \(\dfrac{3}{6}\) equal?Fraction wall up to 1/6 (each row is 1 unit).
1On the wall, three \(\dfrac{1}{6}\) pieces end exactly where \(\dfrac{1}{2}\) ends.
✅ Answer: Yes
๐ Page 164Fraction wall · Q2
Are \(\dfrac{2}{3}\) and \(\dfrac{4}{6}\) equivalent? Why?
1\(\dfrac{2}{3}\) and \(\dfrac{4}{6}\) end at the same point on the wall.
2Also 2 × 2 = 4 and 3 × 2 = 6.
✅ Answer: Yes — they show the same length.
๐ Page 164Fraction wall · Q3
How many pieces of length \(\dfrac{1}{6}\) make \(\dfrac{1}{2}\)?
1\(\dfrac{3}{6}\) = \(\dfrac{1}{2}\).
✅ Answer: 3
๐ Page 164Fraction wall · Q4
How many pieces of length \(\dfrac{1}{6}\) make \(\dfrac{1}{3}\)?
1\(\dfrac{2}{6}\) = \(\dfrac{1}{3}\).
✅ Answer: 2
๐ Textbook Page 165
๐ Page 165Figure it Out · Q1
Are \(\dfrac{3}{6}\), \(\dfrac{4}{8}\), \(\dfrac{5}{10}\) equivalent fractions? Why?
1In each, the numerator is half of the denominator.
2So each equals \(\dfrac{1}{2}\); on the wall they all end at the middle.
✅ Answer: Yes — each is \(\dfrac{1}{2}\).
๐ Page 165Figure it Out · Q2
Write two equivalent fractions for \(\dfrac{2}{6}\).
1Divide both by 2: \(\dfrac{1}{3}\).
2Multiply both by 2: \(\dfrac{4}{12}\).
✅ Answer: \(\dfrac{1}{3}\) and \(\dfrac{4}{12}\) (also \(\dfrac{6}{18}\) …)
๐ Page 165Figure it Out · Q3
\(\dfrac{4}{6}\) = ___ = ___ = ___ … (write as many as you can).
1Divide by 2: \(\dfrac{2}{3}\).
2Multiply \(\dfrac{2}{3}\) by 3, 4, 5, …: \(\dfrac{6}{9}\), \(\dfrac{8}{12}\), \(\dfrac{10}{15}\), \(\dfrac{12}{18}\) …
✅ Answer: \(\dfrac{2}{3}\) = \(\dfrac{8}{12}\) = \(\dfrac{12}{18}\) = \(\dfrac{16}{24}\) = … (also \(\dfrac{6}{9}\), \(\dfrac{10}{15}\))
๐ Textbook Page 166
๐ Page 166Facts for one roti
One roti is shared by 4 children. Write the division, addition and multiplication facts.
1Division: 1 ÷ 4 = \(\dfrac{1}{4}\).
2Addition: 1 = \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\).
3Multiplication: 1 = 4 × \(\dfrac{1}{4}\).
✅ Answer: Each child gets \(\dfrac{1}{4}\).
๐ Page 166Figure it Out · Q1
Three rotis are shared equally by four children. Show the division and write the facts.
1Cut each roti into 4 parts; each child takes one part of every roti → 3 pieces of \(\dfrac{1}{4}\).
2Division: 3 ÷ 4 = \(\dfrac{3}{4}\).
3Addition: 3 = \(\dfrac{3}{4}\) + \(\dfrac{3}{4}\) + \(\dfrac{3}{4}\) + \(\dfrac{3}{4}\).
4Multiplication: 3 = 4 × \(\dfrac{3}{4}\).
✅ Answer: Each child gets \(\dfrac{3}{4}\) roti.
๐ Page 166Figure it Out · Q2
Show how much each child gets when 2 rotis are shared equally by 4 children. Write the facts.
1Cut each roti into 4 parts; each child gets 2 parts → \(\dfrac{2}{4}\) = \(\dfrac{1}{2}\).
2Division: 2 ÷ 4 = \(\dfrac{2}{4}\) = \(\dfrac{1}{2}\).
3Addition: 2 = \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\).
4Multiplication: 2 = 4 × \(\dfrac{1}{2}\).
✅ Answer: \(\dfrac{1}{2}\) roti each
๐ Page 166Figure it Out · Q3
2 cakes are divided equally among 5 children. How much does Anil get? With 10 children, how many cakes give the same share?
12 ÷ 5 = \(\dfrac{2}{5}\) cake.
2For 10 children (twice as many), we need twice the cakes: 4 cakes.
34 ÷ 10 = \(\dfrac{4}{10}\) = \(\dfrac{2}{5}\).
✅ Answer: \(\dfrac{2}{5}\) cake; 4 cakes for 10 children (\(\dfrac{2}{5}\) = \(\dfrac{4}{10}\)).
๐ Textbook Page 167
๐ Page 167Same share
Compare the shares: 1 roti for 2 children, 2 rotis for 4, 3 rotis for 6.
11 ÷ 2 = \(\dfrac{1}{2}\); 2 ÷ 4 = \(\dfrac{2}{4}\); 3 ÷ 6 = \(\dfrac{3}{6}\).
2In each case one child gets half a roti.
✅ Answer: \(\dfrac{1}{2}\) = \(\dfrac{2}{4}\) = \(\dfrac{3}{6}\) — equivalent fractions.
๐ Textbook Page 168
๐ Page 168More equivalents of 1/2
Find more fractions equivalent to \(\dfrac{1}{2}\).
1Multiply top and bottom by the same number: × 4, 5, 6, 10.
✅ Answer: \(\dfrac{4}{8}\), \(\dfrac{5}{10}\), \(\dfrac{6}{12}\), \(\dfrac{10}{20}\) (and many more)
๐ Page 168Are the shares the same?
Share equally: 2 rotis among 3 children, 4 rotis among 6, 6 rotis among 9. Are the shares the same? Why?2 rotis shared equally by 3 children.
12 ÷ 3 = \(\dfrac{2}{3}\); 4 ÷ 6 = \(\dfrac{4}{6}\); 6 ÷ 9 = \(\dfrac{6}{9}\).
2Doubling or tripling both the rotis and the children keeps each share the same.
3\(\dfrac{4}{6}\) and \(\dfrac{6}{9}\) both simplify to \(\dfrac{2}{3}\).
✅ Answer: Yes — all are \(\dfrac{2}{3}\). The numerator and denominator are multiplied by the same number.
๐ Page 168Figure it Out · a
5 glasses of juice shared by 4 friends is the same as ___ glasses shared by 8 friends. So \(\dfrac{5}{4}\) = □/8.
1Friends doubled (4 → 8), so glasses double: 10.
2\(\dfrac{5}{4}\) = \(\dfrac{10}{8}\).
✅ Answer: 10 glasses; \(\dfrac{5}{4}\) = \(\dfrac{10}{8}\)
๐ Page 168Figure it Out · b
4 kg of potatoes in 3 bags is the same as 12 kg in ___ bags. So \(\dfrac{4}{3}\) = 12/□.
14 kg → 12 kg is × 3, so bags × 3: 9.
2\(\dfrac{4}{3}\) = \(\dfrac{12}{9}\).
✅ Answer: 9 bags; \(\dfrac{4}{3}\) = \(\dfrac{12}{9}\)
๐ Textbook Page 169
๐ Page 169Figure it Out · c
7 rotis among 5 children is the same as ___ rotis among ___ children. So \(\dfrac{7}{5}\) = □/□.
1Multiply both by 2: 14 rotis among 10 children.
✅ Answer: e.g. 14 among 10: \(\dfrac{7}{5}\) = \(\dfrac{14}{10}\)
๐ Page 169Which group gets more?
1 chikki between 2 children, or 5 chikkis among 8 children?
1Compare \(\dfrac{1}{2}\) and \(\dfrac{5}{8}\).
2\(\dfrac{1}{2}\) = \(\dfrac{4}{8}\) and \(\dfrac{4}{8}\) < \(\dfrac{5}{8}\).
✅ Answer: The second group (\(\dfrac{5}{8}\) each).
๐ Page 1691/2 or 4/7?
1 chikki between 2 children, or 4 chikkis among 7 children?
1\(\dfrac{1}{2}\) = \(\dfrac{4}{8}\).
24 chikkis among 7 is more than 4 chikkis among 8: \(\dfrac{4}{7}\) > \(\dfrac{4}{8}\).
✅ Answer: \(\dfrac{4}{7}\) > \(\dfrac{1}{2}\) — the second group gets more.
๐ก Tip: Same numerator? The smaller denominator gives the bigger fraction.
๐ Textbook Page 170
๐ Page 170More units, same children
If the number of children stays the same but more units are shared, what happens to each share? Explain \(\dfrac{1}{5}\) < \(\dfrac{2}{5}\), \(\dfrac{3}{7}\) < \(\dfrac{4}{7}\), \(\dfrac{1}{2}\) < \(\dfrac{5}{8}\).
1More things for the same number of people → each gets more.
2\(\dfrac{1}{5}\) < \(\dfrac{2}{5}\) and \(\dfrac{3}{7}\) < \(\dfrac{4}{7}\): same denominator, bigger numerator.
3\(\dfrac{1}{2}\) = \(\dfrac{4}{8}\) < \(\dfrac{5}{8}\).
✅ Answer: Each share becomes bigger.
๐ Page 170Bigger share · Group 1 or 2?
(1) 3 glasses among 4 children or 7 glasses among 10? (2) 4 glasses among 7 or 5 glasses among 7? Which pair was easier?
1(1) \(\dfrac{3}{4}\) = \(\dfrac{15}{20}\), \(\dfrac{7}{10}\) = \(\dfrac{14}{20}\) → Group 1 gets more.
2(2) \(\dfrac{4}{7}\) < \(\dfrac{5}{7}\) → Group 2 gets more.
3Pair (2) is easier: the denominators are already the same.
✅ Answer: (1) Group 1 (\(\dfrac{3}{4}\)) (2) Group 2 (\(\dfrac{5}{7}\)); the second pair was easier.
๐ Textbook Page 171
๐ Page 171Comparing 3/4 and 7/10
Compare \(\dfrac{3}{4}\) and \(\dfrac{7}{10}\) using equivalent fractions.
1Common denominator 20 (or 40).
2\(\dfrac{3}{4}\) = \(\dfrac{15}{20}\), \(\dfrac{7}{10}\) = \(\dfrac{14}{20}\).
315 > 14.
✅ Answer: \(\dfrac{3}{4}\) > \(\dfrac{7}{10}\)
๐ Textbook Page 172
๐ Page 172Same fractional units
Find equivalent fractions with the same fractional unit: (a) \(\dfrac{7}{2}\), \(\dfrac{3}{5}\) (b) \(\dfrac{8}{3}\), \(\dfrac{5}{6}\) (c) \(\dfrac{3}{4}\), \(\dfrac{3}{5}\) (d) \(\dfrac{6}{7}\), \(\dfrac{8}{5}\) (e) \(\dfrac{9}{4}\), \(\dfrac{5}{2}\) (f) \(\dfrac{1}{10}\), \(\dfrac{2}{9}\) (g) \(\dfrac{8}{3}\), \(\dfrac{11}{4}\) (h) \(\dfrac{13}{6}\), \(\dfrac{1}{9}\)
1Use a common multiple of the two denominators (the smallest one is easiest).
2(a) common denominator 10: \(\dfrac{7}{2}\) = \(\dfrac{35}{10}\), \(\dfrac{3}{5}\) = \(\dfrac{6}{10}\)
3(b) common denominator 6: \(\dfrac{8}{3}\) = \(\dfrac{16}{6}\), \(\dfrac{5}{6}\) = \(\dfrac{5}{6}\)
4(c) common denominator 20: \(\dfrac{3}{4}\) = \(\dfrac{15}{20}\), \(\dfrac{3}{5}\) = \(\dfrac{12}{20}\)
5(d) common denominator 35: \(\dfrac{6}{7}\) = \(\dfrac{30}{35}\), \(\dfrac{8}{5}\) = \(\dfrac{56}{35}\)
6(e) common denominator 4: \(\dfrac{9}{4}\) = \(\dfrac{9}{4}\), \(\dfrac{5}{2}\) = \(\dfrac{10}{4}\)
7(f) common denominator 90: \(\dfrac{1}{10}\) = \(\dfrac{9}{90}\), \(\dfrac{2}{9}\) = \(\dfrac{20}{90}\)
8(g) common denominator 12: \(\dfrac{8}{3}\) = \(\dfrac{32}{12}\), \(\dfrac{11}{4}\) = \(\dfrac{33}{12}\)
9(h) common denominator 18: \(\dfrac{13}{6}\) = \(\dfrac{39}{18}\), \(\dfrac{1}{9}\) = \(\dfrac{2}{18}\)
✅ Answer: (a) \(\dfrac{35}{10}\), \(\dfrac{6}{10}\) · (b) \(\dfrac{16}{6}\), \(\dfrac{5}{6}\) · (c) \(\dfrac{15}{20}\), \(\dfrac{12}{20}\) · (d) \(\dfrac{30}{35}\), \(\dfrac{56}{35}\) · (e) \(\dfrac{9}{4}\), \(\dfrac{10}{4}\) · (f) \(\dfrac{9}{90}\), \(\dfrac{20}{90}\) · (g) \(\dfrac{32}{12}\), \(\dfrac{33}{12}\) · (h) \(\dfrac{39}{18}\), \(\dfrac{2}{18}\)
๐ Page 172Lowest terms · Example
Is \(\dfrac{16}{20}\) in lowest terms? Reduce it.
14 is a common factor of 16 and 20.
216 ÷ 4 = 4, 20 ÷ 4 = 5.
34 and 5 have no common factor except 1.
✅ Answer: No; \(\dfrac{16}{20}\) = \(\dfrac{4}{5}\)
๐ Textbook Page 173
๐ Page 173Figure it Out
Express in lowest terms: (a) \(\dfrac{17}{51}\) (b) \(\dfrac{64}{144}\) (c) \(\dfrac{126}{147}\) (d) \(\dfrac{525}{112}\)
1(a) divide by 17: \(\dfrac{1}{3}\)
2(b) divide by 16: \(\dfrac{4}{9}\)
3(c) divide by 21: \(\dfrac{6}{7}\)
4(d) divide by 7: \(\dfrac{75}{16}\) (= \(4\dfrac{11}{16}\))
✅ Answer: (a) \(\dfrac{1}{3}\) · (b) \(\dfrac{4}{9}\) · (c) \(\dfrac{6}{7}\) · (d) \(\dfrac{75}{16}\)
๐ก Tip: Divide top and bottom by their highest common factor — or keep dividing by small common factors step by step.
๐ Page 1734/5 or 7/9?
Which is greater: \(\dfrac{4}{5}\) or \(\dfrac{7}{9}\)?
145 is a common multiple of 5 and 9.
2\(\dfrac{4}{5}\) = \(\dfrac{36}{45}\), \(\dfrac{7}{9}\) = \(\dfrac{35}{45}\).
✅ Answer: \(\dfrac{4}{5}\)
๐ Textbook Page 174
๐ Page 174Figure it Out · Q1
Compare and justify: (a) \(\dfrac{8}{3}\), \(\dfrac{5}{2}\) (b) \(\dfrac{4}{9}\), \(\dfrac{3}{7}\) (c) \(\dfrac{7}{10}\), \(\dfrac{9}{14}\) (d) \(\dfrac{12}{5}\), \(\dfrac{8}{5}\) (e) \(\dfrac{9}{4}\), \(\dfrac{5}{2}\)
1(a) \(\dfrac{8}{3}\) = \(\dfrac{16}{6}\), \(\dfrac{5}{2}\) = \(\dfrac{15}{6}\) → \(\dfrac{8}{3}\) > \(\dfrac{5}{2}\)
2(b) \(\dfrac{4}{9}\) = \(\dfrac{28}{63}\), \(\dfrac{3}{7}\) = \(\dfrac{27}{63}\) → \(\dfrac{4}{9}\) > \(\dfrac{3}{7}\)
3(c) \(\dfrac{7}{10}\) = \(\dfrac{49}{70}\), \(\dfrac{9}{14}\) = \(\dfrac{45}{70}\) → \(\dfrac{7}{10}\) > \(\dfrac{9}{14}\)
4(d) \(\dfrac{12}{5}\) = \(\dfrac{12}{5}\), \(\dfrac{8}{5}\) = \(\dfrac{8}{5}\) → \(\dfrac{12}{5}\) > \(\dfrac{8}{5}\)
5(e) \(\dfrac{9}{4}\) = \(\dfrac{9}{4}\), \(\dfrac{5}{2}\) = \(\dfrac{10}{4}\) → \(\dfrac{9}{4}\) < \(\dfrac{5}{2}\)
✅ Answer: (a) \(\dfrac{8}{3}\) > \(\dfrac{5}{2}\) · (b) \(\dfrac{4}{9}\) > \(\dfrac{3}{7}\) · (c) \(\dfrac{7}{10}\) > \(\dfrac{9}{14}\) · (d) \(\dfrac{12}{5}\) > \(\dfrac{8}{5}\) · (e) \(\dfrac{9}{4}\) < \(\dfrac{5}{2}\)
๐ Page 174Figure it Out · Q2
Write in ascending order: (a) \(\dfrac{7}{10}\), \(\dfrac{11}{15}\), \(\dfrac{2}{5}\) (b) \(\dfrac{19}{24}\), \(\dfrac{5}{6}\), \(\dfrac{7}{12}\)
1(a) Denominator 30: \(\dfrac{21}{30}\), \(\dfrac{22}{30}\), \(\dfrac{12}{30}\).
2(b) Denominator 24: \(\dfrac{19}{24}\), \(\dfrac{20}{24}\), \(\dfrac{14}{24}\).
✅ Answer: (a) \(\dfrac{2}{5}\) < \(\dfrac{7}{10}\) < \(\dfrac{11}{15}\) (b) \(\dfrac{7}{12}\) < \(\dfrac{19}{24}\) < \(\dfrac{5}{6}\)
๐ Page 174Figure it Out · Q3
Write in descending order: (a) \(\dfrac{25}{16}\), \(\dfrac{7}{8}\), \(\dfrac{13}{4}\), \(\dfrac{17}{32}\) (b) \(\dfrac{3}{4}\), \(\dfrac{12}{5}\), \(\dfrac{7}{12}\), \(\dfrac{5}{4}\)
1(a) Denominator 32: \(\dfrac{50}{32}\), \(\dfrac{28}{32}\), \(\dfrac{104}{32}\), \(\dfrac{17}{32}\).
2(b) Denominator 60: \(\dfrac{45}{60}\), \(\dfrac{144}{60}\), \(\dfrac{35}{60}\), \(\dfrac{75}{60}\).
✅ Answer: (a) \(\dfrac{13}{4}\) > \(\dfrac{25}{16}\) > \(\dfrac{7}{8}\) > \(\dfrac{17}{32}\) (b) \(\dfrac{12}{5}\) > \(\dfrac{5}{4}\) > \(\dfrac{3}{4}\) > \(\dfrac{7}{12}\)
๐ Textbook Page 175
๐ Page 175Meena’s chikki
Meena ate \(\dfrac{1}{2}\) of the chikki and her brother \(\dfrac{1}{4}\). How much did they eat together? How much is left?
1\(\dfrac{1}{2}\) = \(\dfrac{2}{4}\).
2\(\dfrac{2}{4}\) + \(\dfrac{1}{4}\) = \(\dfrac{3}{4}\) eaten.
3Left: 1 − \(\dfrac{3}{4}\) = \(\dfrac{1}{4}\).
✅ Answer: \(\dfrac{3}{4}\) eaten; \(\dfrac{1}{4}\) left
๐ Textbook Page 176
๐ Page 176Example · 2/5 + 1/5
Find \(\dfrac{2}{5}\) + \(\dfrac{1}{5}\).
1Same fractional unit \(\dfrac{1}{5}\): 2 units + 1 unit = 3 units.
✅ Answer: \(\dfrac{3}{5}\)
๐ Textbook Page 177
๐ Page 177Example · 4/7 + 6/7
Find \(\dfrac{4}{7}\) + \(\dfrac{6}{7}\). Try it on a number line too.
14 sevenths + 6 sevenths = 10 sevenths = \(\dfrac{10}{7}\).
2\(\dfrac{10}{7}\) = 1 + \(\dfrac{3}{7}\) = \(1\dfrac{3}{7}\).
3On the number line: start at \(\dfrac{4}{7}\), jump 6 sevenths more → land at \(\dfrac{10}{7}\).
✅ Answer: \(\dfrac{10}{7}\) = \(1\dfrac{3}{7}\); the number line gives the same answer.
๐ Textbook Page 178
๐ Page 178Example · 1/4 + 1/3
Find \(\dfrac{1}{4}\) + \(\dfrac{1}{3}\).
1Common denominator 12.
2\(\dfrac{1}{4}\) = \(\dfrac{3}{12}\), \(\dfrac{1}{3}\) = \(\dfrac{4}{12}\).
3\(\dfrac{3}{12}\) + \(\dfrac{4}{12}\) = \(\dfrac{7}{12}\).
✅ Answer: \(\dfrac{7}{12}\)
๐ Textbook Page 179
๐ Page 179Example · 2/3 + 1/5 and 1/6 + 1/3
Find \(\dfrac{2}{3}\) + \(\dfrac{1}{5}\) and \(\dfrac{1}{6}\) + \(\dfrac{1}{3}\).
1\(\dfrac{2}{3}\) + \(\dfrac{1}{5}\) = \(\dfrac{10}{15}\) + \(\dfrac{3}{15}\) = \(\dfrac{13}{15}\).
2\(\dfrac{1}{6}\) + \(\dfrac{1}{3}\) = \(\dfrac{1}{6}\) + \(\dfrac{2}{6}\) = \(\dfrac{3}{6}\) = \(\dfrac{1}{2}\).
✅ Answer: \(\dfrac{13}{15}\) and \(\dfrac{1}{2}\)
๐ Page 179Figure it Out · Q1
Add using Brahmagupta’s method: (a) \(\dfrac{2}{7}\)+\(\dfrac{5}{7}\)+\(\dfrac{6}{7}\) (b) \(\dfrac{3}{4}\)+\(\dfrac{1}{3}\) (c) \(\dfrac{2}{3}\)+\(\dfrac{5}{6}\) (d) \(\dfrac{2}{3}\)+\(\dfrac{2}{7}\) (e) \(\dfrac{3}{4}\)+\(\dfrac{1}{3}\)+\(\dfrac{1}{5}\) (f) \(\dfrac{2}{3}\)+\(\dfrac{4}{5}\) (g) \(\dfrac{4}{5}\)+\(\dfrac{2}{3}\) (h) \(\dfrac{3}{5}\)+\(\dfrac{5}{8}\) (i) \(\dfrac{9}{2}\)+\(\dfrac{5}{4}\) (j) \(\dfrac{8}{3}\)+\(\dfrac{2}{7}\) (k) \(\dfrac{3}{4}\)+\(\dfrac{1}{3}\)+\(\dfrac{1}{5}\) (l) \(\dfrac{2}{3}\)+\(\dfrac{4}{5}\)+\(\dfrac{3}{7}\) (m) \(\dfrac{9}{2}\)+\(\dfrac{5}{4}\)+\(\dfrac{7}{6}\)
1(a) LCM 7: \(\dfrac{2}{7}\) + \(\dfrac{5}{7}\) + \(\dfrac{6}{7}\) = \(\dfrac{13}{7}\) = \(1\dfrac{6}{7}\)
2(b) LCM 12: \(\dfrac{9}{12}\) + \(\dfrac{4}{12}\) = \(\dfrac{13}{12}\) = \(1\dfrac{1}{12}\)
3(c) LCM 6: \(\dfrac{4}{6}\) + \(\dfrac{5}{6}\) = \(\dfrac{9}{6}\) = \(\dfrac{3}{2}\) = \(1\dfrac{1}{2}\)
4(d) LCM 21: \(\dfrac{14}{21}\) + \(\dfrac{6}{21}\) = \(\dfrac{20}{21}\)
5(e) LCM 60: \(\dfrac{45}{60}\) + \(\dfrac{20}{60}\) + \(\dfrac{12}{60}\) = \(\dfrac{77}{60}\) = \(1\dfrac{17}{60}\)
6(f) LCM 15: \(\dfrac{10}{15}\) + \(\dfrac{12}{15}\) = \(\dfrac{22}{15}\) = \(1\dfrac{7}{15}\)
7(g) LCM 15: \(\dfrac{12}{15}\) + \(\dfrac{10}{15}\) = \(\dfrac{22}{15}\) = \(1\dfrac{7}{15}\)
8(h) LCM 40: \(\dfrac{24}{40}\) + \(\dfrac{25}{40}\) = \(\dfrac{49}{40}\) = \(1\dfrac{9}{40}\)
9(i) LCM 4: \(\dfrac{18}{4}\) + \(\dfrac{5}{4}\) = \(\dfrac{23}{4}\) = \(5\dfrac{3}{4}\)
10(j) LCM 21: \(\dfrac{56}{21}\) + \(\dfrac{6}{21}\) = \(\dfrac{62}{21}\) = \(2\dfrac{20}{21}\)
11(k) LCM 60: \(\dfrac{45}{60}\) + \(\dfrac{20}{60}\) + \(\dfrac{12}{60}\) = \(\dfrac{77}{60}\) = \(1\dfrac{17}{60}\)
12(l) LCM 105: \(\dfrac{70}{105}\) + \(\dfrac{84}{105}\) + \(\dfrac{45}{105}\) = \(\dfrac{199}{105}\) = \(1\dfrac{94}{105}\)
13(m) LCM 12: \(\dfrac{54}{12}\) + \(\dfrac{15}{12}\) + \(\dfrac{14}{12}\) = \(\dfrac{83}{12}\) = \(6\dfrac{11}{12}\)
14Note: (k) is the same as (e), and (g) is (f) in the other order — the sum does not change.
✅ Answer: (a) \(\dfrac{13}{7}\) · (b) \(\dfrac{13}{12}\) · (c) \(\dfrac{3}{2}\) · (d) \(\dfrac{20}{21}\) · (e) \(\dfrac{77}{60}\) · (f) \(\dfrac{22}{15}\) · (g) \(\dfrac{22}{15}\) · (h) \(\dfrac{49}{40}\) · (i) \(\dfrac{23}{4}\) · (j) \(\dfrac{62}{21}\) · (k) \(\dfrac{77}{60}\) · (l) \(\dfrac{199}{105}\) · (m) \(\dfrac{83}{12}\)
๐ Page 179Figure it Out · Q2
Rahim mixes \(\dfrac{2}{3}\) litre of yellow paint with \(\dfrac{3}{4}\) litre of blue paint. How much green paint does he make?
1\(\dfrac{2}{3}\) + \(\dfrac{3}{4}\) = \(\dfrac{8}{12}\) + \(\dfrac{9}{12}\) = \(\dfrac{17}{12}\).
2= \(1\dfrac{5}{12}\) litres.
✅ Answer: \(\dfrac{17}{12}\) = \(1\dfrac{5}{12}\) litres
๐ Page 179Figure it Out · Q3
Geeta bought \(\dfrac{2}{5}\) m and Shamim \(\dfrac{3}{4}\) m of lace for a border of perimeter 1 m. Total lace? Is it enough?
1\(\dfrac{2}{5}\) + \(\dfrac{3}{4}\) = \(\dfrac{8}{20}\) + \(\dfrac{15}{20}\) = \(\dfrac{23}{20}\) = \(1\dfrac{3}{20}\) m.
2\(1\dfrac{3}{20}\) m is more than 1 m.
✅ Answer: \(\dfrac{23}{20}\) m — yes, enough (\(\dfrac{3}{20}\) m extra).
๐ Textbook Page 180
๐ Page 1806/7 − 4/7
Find \(\dfrac{6}{7}\) − \(\dfrac{4}{7}\). Try it on the number line too.
1Same unit \(\dfrac{1}{7}\): 6 units − 4 units = 2 units.
2On the number line start at \(\dfrac{6}{7}\) and move back 4 sevenths to \(\dfrac{2}{7}\).
✅ Answer: \(\dfrac{2}{7}\)
๐ Textbook Page 181
๐ Page 181Figure it Out
Find (1) \(\dfrac{5}{8}\) − \(\dfrac{3}{8}\) (2) \(\dfrac{7}{9}\) − \(\dfrac{5}{9}\) (3) \(\dfrac{10}{27}\) − \(\dfrac{1}{27}\)
1(1) \(\dfrac{2}{8}\) = \(\dfrac{1}{4}\)
2(2) \(\dfrac{2}{9}\)
3(3) \(\dfrac{9}{27}\) = \(\dfrac{1}{3}\)
✅ Answer: (1) \(\dfrac{1}{4}\) (2) \(\dfrac{2}{9}\) (3) \(\dfrac{1}{3}\)
๐ Page 1813/4 − 2/3
Find \(\dfrac{3}{4}\) − \(\dfrac{2}{3}\). Why multiply \(\dfrac{3}{4}\) by 3 and \(\dfrac{2}{3}\) by 4?
1We need the same denominator: 12 is a common multiple of 4 and 3.
2To turn 4 into 12 multiply by 3; to turn 3 into 12 multiply by 4.
3\(\dfrac{9}{12}\) − \(\dfrac{8}{12}\) = \(\dfrac{1}{12}\).
✅ Answer: \(\dfrac{1}{12}\)
๐ Textbook Page 182
๐ Page 182Figure it Out · Q1
Subtract: (a) \(\dfrac{8}{15}\) − \(\dfrac{3}{15}\) (b) \(\dfrac{2}{5}\) − \(\dfrac{4}{15}\) (c) \(\dfrac{5}{6}\) − \(\dfrac{4}{9}\) (d) \(\dfrac{2}{3}\) − \(\dfrac{1}{2}\)
1(a) \(\dfrac{5}{15}\) = \(\dfrac{1}{3}\)
2(b) \(\dfrac{6}{15}\) − \(\dfrac{4}{15}\) = \(\dfrac{2}{15}\)
3(c) \(\dfrac{15}{18}\) − \(\dfrac{8}{18}\) = \(\dfrac{7}{18}\)
4(d) \(\dfrac{4}{6}\) − \(\dfrac{3}{6}\) = \(\dfrac{1}{6}\)
✅ Answer: (a) \(\dfrac{1}{3}\) (b) \(\dfrac{2}{15}\) (c) \(\dfrac{7}{18}\) (d) \(\dfrac{1}{6}\)
๐ Page 182Figure it Out · Q2
Subtract as indicated: (a) \(\dfrac{13}{4}\) from \(\dfrac{10}{3}\) (b) \(\dfrac{18}{5}\) from \(\dfrac{23}{3}\) (c) \(\dfrac{29}{7}\) from \(\dfrac{45}{7}\)
1“A from B” means B − A.
2(a) \(\dfrac{40}{12}\) − \(\dfrac{39}{12}\) = \(\dfrac{1}{12}\)
3(b) \(\dfrac{115}{15}\) − \(\dfrac{54}{15}\) = \(\dfrac{61}{15}\) = \(4\dfrac{1}{15}\)
4(c) \(\dfrac{16}{7}\) = \(2\dfrac{2}{7}\)
✅ Answer: (a) \(\dfrac{1}{12}\) (b) \(\dfrac{61}{15}\) (c) \(\dfrac{16}{7}\)
⚠️ Common mistake: Doing A − B for “A from B”. ‘Subtract 3 from 10’ means 10 − 3.
๐ Page 182Figure it Out · Q3a
Jaya’s school is \(\dfrac{7}{10}\) km from home. She goes \(\dfrac{1}{2}\) km by auto and walks the rest. How far does she walk?
1\(\dfrac{7}{10}\) − \(\dfrac{1}{2}\) = \(\dfrac{7}{10}\) − \(\dfrac{5}{10}\) = \(\dfrac{2}{10}\) = \(\dfrac{1}{5}\).
✅ Answer: \(\dfrac{1}{5}\) km
๐ Page 182Figure it Out · Q3b
Jeevika takes \(\dfrac{10}{3}\) minutes for a round of the park; Namit takes \(\dfrac{13}{4}\) minutes. Who is faster, and by how much?
1\(\dfrac{10}{3}\) = \(\dfrac{40}{12}\), \(\dfrac{13}{4}\) = \(\dfrac{39}{12}\).
2\(\dfrac{39}{12}\) < \(\dfrac{40}{12}\), so Namit takes less time.
3Difference = \(\dfrac{1}{12}\) minute (5 seconds).
✅ Answer: Namit, by \(\dfrac{1}{12}\) minute
๐ Textbook Page 183
๐ Page 183Egyptian fractions
Check that \(\dfrac{19}{24}\) = \(\dfrac{1}{2}\) + \(\dfrac{1}{6}\) + \(\dfrac{1}{8}\).
1\(\dfrac{12}{24}\) + \(\dfrac{4}{24}\) + \(\dfrac{3}{24}\) = \(\dfrac{19}{24}\) ✔
✅ Answer: True
๐ Textbook Page 184
๐ Page 184Puzzle · Q1
Find three different fractional units that add up to 1.
1Two different units cannot make 1 (\(\dfrac{1}{2}\) is the biggest unit).
2One unit must be \(\dfrac{1}{2}\); the next must be \(\dfrac{1}{3}\).
31 − \(\dfrac{1}{2}\) − \(\dfrac{1}{3}\) = \(\dfrac{1}{6}\).
✅ Answer: \(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) + \(\dfrac{1}{6}\) = 1 (the only answer)
๐ Textbook Page 185
๐ Page 185Puzzle · Q2
Find four different fractional units that add up to 1. (There are six answers.)
1Start with \(\dfrac{1}{2}\) (it must be used), then try \(\dfrac{1}{3}\) or \(\dfrac{1}{4}\) next.
2\(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) + \(\dfrac{1}{7}\) + \(\dfrac{1}{42}\); \(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) + \(\dfrac{1}{8}\) + \(\dfrac{1}{24}\); \(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) + \(\dfrac{1}{9}\) + \(\dfrac{1}{18}\)
3\(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) + \(\dfrac{1}{10}\) + \(\dfrac{1}{15}\); \(\dfrac{1}{2}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{5}\) + \(\dfrac{1}{20}\); \(\dfrac{1}{2}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{6}\) + \(\dfrac{1}{12}\)
✅ Answer: Six answers — e.g. \(\dfrac{1}{2}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{6}\) + \(\dfrac{1}{12}\) = 1
๐ Where do we use fractions?
- Sharing food, recipes (½ cup), money (quarter rupee long ago), time (quarter past), measuring cloth and lace.
๐ Link to higher classes
- Class 7: multiplying and dividing fractions, decimals. Class 8: rational numbers.
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