6ನೇ ತರಗತಿ ಗಣಿತ (ಗಣಿತ ಪ್ರಕಾಶ) ಭಾಗ-2ರ 7ನೇ ಅಧ್ಯಾಯ “ಭಿನ್ನರಾಶಿಗಳು” (Fractions) ಪಾಠದ ಸರಳ ಇಂಗ್ಲಿಷ್ ಟಿಪ್ಪಣಿಗಳು ಇಲ್ಲಿವೆ. ಭಿನ್ನರಾಶಿ ಘಟಕಗಳು, ಸಮಾನ ಪಾಲು, ಸಂಖ್ಯಾರೇಖೆಯ ಮೇಲೆ ಭಿನ್ನರಾಶಿಗಳು, ಮಿಶ್ರ ಭಿನ್ನರಾಶಿಗಳು, ಸಮಾನ ಭಿನ್ನರಾಶಿಗಳು, ಕನಿಷ್ಠ ರೂಪ, ಭಿನ್ನರಾಶಿಗಳ ಹೋಲಿಕೆ ಹಾಗೂ ಬ್ರಹ್ಮಗುಪ್ತನ ವಿಧಾನದಲ್ಲಿ ಕೂಡುವಿಕೆ-ಕಳೆಯುವಿಕೆಯನ್ನು ಪಠ್ಯಪುಸ್ತಕದ ಚಿತ್ರಗಳೊಂದಿಗೆ ವಿವರಿಸಲಾಗಿದೆ. ಎಲ್ಲಾ “Figure it Out” ಪ್ರಶ್ನೆಗಳಿಗೆ ಉತ್ತರಗಳು, ಆಟಗಳು, ಉಚಿತ PDF ಗಳು ಮತ್ತು ಶಾಲೆಯ ಹೆಸರಿನಲ್ಲಿ ಬಳಸಬಹುದಾದ ಪ್ರಶ್ನೆಪತ್ರಿಕೆ (Word) ಲಭ್ಯವಿದೆ.
Fractions
The full lesson in simple words with textbook maps and pictures, animations, games and quizzes, followed by answers to every question and activity.
Questions and answers: all Figure it Out questions, puzzles and activities.
One roti shared by 4 children
Fraction wallThe Lesson
When things are shared equally, fractions tell us how much each share is. Let us learn about fractions.
Start Fractions as equal shares
If one roti is divided equally between two children, each child gets half a roti. The fraction ‘one half’ is written as 1⁄2. We also read it as ‘one upon two’.
If one roti is shared equally among 4 children, each child gets 1⁄4 roti.
When 4 children share 1 roti, more children share the same roti, so each child gets a smaller share. So 1⁄2 roti is more than 1⁄4 roti.
One roti shared equally by 4 children: each gets ¼7.1 Fractional units and equal shares
Which is greater, 1⁄5 or 1⁄9? 9 is bigger than 5, but if one roti is shared among 9 children each gets less than when it is shared among 5 children. If I share with more people, I will get less. So 1⁄9 < 1⁄5. Also, 1⁄100 is bigger than 1⁄200!
When one unit is divided into several equal parts, each part is called a fractional unit: 1⁄2, 1⁄3, 1⁄4, 1⁄5, … These are also called unit fractions.
Fractions have been used and named in India since ancient times. In the Rig Veda, the fraction 3⁄4 is called tri-pada. This has the same meaning as the words for 3⁄4 in many Indian languages today, e.g., ‘teen paav’ in colloquial Hindi and ‘mukkaal’ in Tamil.
7.2 Fractional units as parts of a whole
A chikki broken into 2 pieces: the small piece is 1⁄4 chikki. The bigger piece has 3 pieces of 1⁄4 in it, so the bigger piece is 3⁄4 chikki.
A chikki can be divided into 6 equal parts in different ways. We get 1⁄6 chikki pieces of different shapes but they are all the same size.
If we break the chikki into 3 equal pieces, each piece is 1⁄3 chikki.
A chikki broken into ¼ and ¾
A chikki cut into 6 equal pieces in two ways7.3 Measuring using fractional units
Take a paper strip one unit long. Fold it into two equal parts: each part is 1⁄2. Fold again: you get four equal parts, each 1⁄4. Then 2 times 1⁄4 = 2⁄4, 3 times 1⁄4 = 3⁄4 and 4 times 1⁄4 = 4⁄4 = 1.
Fractional quantities can be measured using fractional units. For example, ½ + ½ + ½ = 3 times half.
Reading fractions: 3⁄4 is read ‘three quarters’ or ‘three upon four’, but reading it as ‘3 times 1⁄4’ shows the fractional unit (1⁄4) and how many such units (3) there are.
In the fraction 5⁄6, 5 is the numerator and 6 is the denominator.
Folding a strip into 4 equal parts
Collecting halves of a roti7.4 Marking fraction lengths on the number line
The distance between 0 and 1 is one unit. If it is divided into 2 equal parts, each part is 1⁄2 unit, so a line reaching the first mark is 1⁄2 unit long. In the same way we can mark thirds, fifths, eighths and so on.
Every fraction has a point on the number line.
Number line: unit divided into 27.5 Mixed fractions
In fractions less than 1, the numerator is smaller than the denominator. In fractions more than 1, the numerator is larger than the denominator.
3⁄2 = 1⁄2 + 1⁄2 + 1⁄2 = 1 + 1⁄2 and 5⁄2 = 1 + 1 + 1⁄2 = 2 + 1⁄2.
8⁄3 = 2 + 2⁄3 is also called ‘two and two thirds’ and written as 22⁄3.
A mixed number or mixed fraction contains a whole number (the whole part) and a fraction less than 1 (the fractional part).
Mixed number to fraction: 3 + 3⁄4 = 1 + 1 + 1 + 3⁄4. Since 1 = 4 quarters, 3 + 3⁄4 = (4 × 1⁄4) + (4 × 1⁄4) + (4 × 1⁄4) + (3 × 1⁄4) = 15⁄4.
7.6 Equivalent fractions
Using paper strips: 1⁄2 = 2⁄4 = 4⁄8. These are equivalent fractions: they show the same length but use different fractional units. A fraction wall shows many of them.
½ = 2⁄4 = 4⁄8 on paper stripsFraction wall up to tenthsEquivalent fractions using equal shares
One roti shared equally by four children: each gets 1⁄4. Division fact: 1 ÷ 4 = 1⁄4. Addition fact: 1 = 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4. Multiplication fact: 1 = 4 × 1⁄4.
1 roti for 2 children, 2 rotis for 4 children and 3 rotis for 6 children give the same share to each child. So 1⁄2 = 2⁄4 = 3⁄6. Fractions where the shares are equal are called equivalent fractions.
2⁄3 is the simplest form of 4⁄6 and of 6⁄9.
½ = 2⁄4 = 3⁄6 by sharing rotisWhich share is more?
1 chikki for 2 children or 5 chikkis for 8 children? 1⁄2 = 4⁄8 and 4⁄8 < 5⁄8, so the second group gets more.
If the number of units shared is the same but the number of children is more, the share is less: 4⁄7 > 4⁄8, and 4⁄8 = 1⁄2, so 4⁄7 > 1⁄2.
To compare 3⁄4 and 7⁄10, make the denominators the same (use 4 × 10 = 40): 3⁄4 = 30⁄40 and 7⁄10 = 28⁄40. Since 30⁄40 > 28⁄40, 3⁄4 > 7⁄10.
Lowest terms (simplest form)
If the numerator and denominator have no common factor except 1, the fraction is in lowest terms. Divide both by their highest common factor: 16⁄20 = 4⁄5.
It can also be done in steps: 36⁄60 = 18⁄30 = 9⁄15 = 3⁄5 (divide by 2, by 2, then by 3).
7.7 Comparing fractions
Which is greater, 4⁄5 or 7⁄9? 45 is a common multiple of 5 and 9: 4⁄5 = 36⁄45, 7⁄9 = 35⁄45. So 4⁄5 > 7⁄9.
- Step 1: Change the fractions to equivalent fractions with the same denominator (same fractional unit).
- Step 2: Compare the numerators.
7.8 Addition and subtraction of fractions
Meena ate 1⁄2 of a chikki and her brother ate 1⁄4. Together: 1⁄2 + 1⁄4 = 1⁄4 + 1⁄4 + 1⁄4 = 3 × 1⁄4 = 3⁄4.
Same fractional unit: just add the number of fractional units: 2⁄5 + 1⁄5 = 3⁄5; 4⁄7 + 6⁄7 = 10⁄7 = 13⁄7.
Different fractional units: first make equivalent fractions with the same denominator: 1⁄4 + 1⁄3 = 3⁄12 + 4⁄12 = 7⁄12.
Total chikki eaten by Meena and her brotherThis method was first described in general by Brahmagupta in 628 CE. Example: 2⁄3 + 1⁄5 = 10⁄15 + 3⁄15 = 13⁄15.
Subtraction works the same way: 6⁄7 − 4⁄7 = 2⁄7; 3⁄4 − 2⁄3 = 9⁄12 − 8⁄12 = 1⁄12.
6⁄7 − 4⁄7: removing 4 shaded parts7.9 A pinch of history
A fraction was called bhinna (‘broken’) in Sanskrit, and also bhaga or ansha (‘part’ or ‘piece’).
The way we write fractions today originated in India. The Bakshali manuscript (around 300 CE) wrote ½ as 1 above 2. This was used by Aryabhata (499 CE), Brahmagupta (628 CE), Sridharacharya (c. 750 CE) and Mahaviracharya (c. 850 CE). The line between numerator and denominator was later introduced by the Moroccan mathematician Al-Hassar (12th century).
Ancient Egyptians and Babylonians mainly used fractional units; sums of fractional units are called ‘Egyptian fractions’, e.g., 19⁄24 = 1⁄2 + 1⁄6 + 1⁄8.
General fractions and their rules were first introduced in India; the Sulba-sutras show this even in Vedic times. Brahmagupta’s methods are still used today and reached Europe via the Arabs, coming into general use there around the 17th century.
★ Summary
- Fraction as equal share: when a whole number of units is shared equally, a fraction results.
- Fractional unit: each equal part of one whole.
- In 5⁄6, 5 is the numerator and 6 is the denominator.
- Mixed fractions have a whole part and a fractional part.
- Every fraction has a point on the number line.
- Equivalent fractions represent the same share or number.
- Lowest terms: numerator and denominator have no common factor other than 1.
- Brahmagupta’s method: make the denominators the same, then add or subtract the numerators.
Game 1: Less than, equal to or more than 1?
Read the fraction and tap the answer. 8 cards!
Game 2: Quick fractions
- Which is greater: 1⁄5 or 1⁄9?
- 3⁄6 is equal to
- 2⁄5 + 1⁄5 =
- 1⁄4 + 1⁄3 =
- 9⁄2 as a mixed fraction
- 16⁄20 in lowest terms
Game 3: Mixed number to fraction
Questions & Answers
Figure it Out and in-text questions pages 24–57
7.1 Figure it Out (pages 24–25)
Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh ____ kg.
Page 241⁄3 kg
A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is ____ kg.
Page 241⁄4 kg
Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank ____ glass of sugarcane juice.
Page 243⁄4 glass (3 ÷ 4)
The big fish weighs 1⁄2 kg. The small one weighs 1⁄4 kg. Together they weigh ____ kg.
Page 251⁄2 + 1⁄4 = 1⁄4 + 1⁄4 + 1⁄4 = 3⁄4 kg
Find out the words for fractions used in your language: one and a half, three quarters, one and a quarter, half, quarter and two and a half.
Page 25In Kannada: half – ಅರ್ಧ; quarter – ಕಾಲು; three quarters – ಮುಕ್ಕಾಲು; one and a quarter – ಒಂದೂ ಕಾಲು; one and a half – ಒಂದೂವರೆ; two and a half – ಎರಡೂವರೆ. (Write the words used in your home.)
Arrange these fraction words from the smallest to the biggest: one and a half, three quarters, one and a quarter, half, quarter, two and a half.
Page 25quarter < half < three quarters < one and a quarter < one and a half < two and a half
7.2 Fractional units as parts of a whole (pages 26–27)
By dividing the whole chikki into 6 equal parts in different ways, we get 1⁄6 chikki pieces of different shapes. Are they of the same size?
Page 26Yes. Each piece is 1⁄6 of the same whole chikki, so all the pieces are the same size (amount), even though their shapes are different.
The figures show different fractional units of a whole chikki. How much of a whole chikki is each piece?
Page 27
Pieces a to h of a whole chikkiThe whole chikki has 24 small squares (6 × 4).
- a. 2 squares → 1⁄12
- b. 6 squares → 1⁄4
- c. 3 squares → 1⁄8
- d. 4 squares → 1⁄6
- e. 3 squares → 1⁄8
- f. 4 squares → 1⁄6
- g. 1 square → 1⁄24
- h. 1 square → 1⁄24
7.3 Measuring using fractional units (pages 28–30)
Fold the strip into 8 equal parts and fill in the blank boxes.
Page 29
Folding a strip into 8 equal parts1⁄8; 2 times 1⁄8 = 2⁄8; 3 times 1⁄8 = 3⁄8; 4 times 1⁄8 = 4⁄8; 5 times 1⁄8 = 5⁄8; 6 times 1⁄8 = 6⁄8; 7 times 1⁄8 = 7⁄8; 8 times 1⁄8 = 8⁄8 = 1.
Continue this table of 1⁄2 for 2 more steps.
Page 30- ½ + ½ + ½ + ½ + ½ + ½ = 6 times half = 6⁄2 = 3
- ½ + ½ + ½ + ½ + ½ + ½ + ½ = 7 times half = 7⁄2
Can you create a similar table for 1⁄4?
Page 30Yes: 1⁄4 = 1 times quarter; 1⁄4 + 1⁄4 = 2 times quarter = 2⁄4; 1⁄4 + 1⁄4 + 1⁄4 = 3 times quarter = 3⁄4; 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 = 4 times quarter = 4⁄4 = 1; and so on.
Make 1⁄3 using a paper strip. Can you use this to also make 1⁄6?
Page 30Fold the strip into 3 equal parts: each part is 1⁄3. Yes, fold it once more into two equal parts: each third is halved and the strip has 6 equal parts, each 1⁄6.
Draw a picture and write an addition statement to show: a. 5 times 1⁄4 of a roti, b. 9 times 1⁄4 of a roti.
Page 30- a. 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 = 5⁄4 (one full roti and one quarter piece)
- b. 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 = 9⁄4 (two full rotis and one quarter piece)
Match each fractional unit with the correct picture.
Page 30
Match the fractional unit with the pictureCount the equal parts in each circle: first picture (8 parts) → 1⁄8, second picture (6 parts) → 1⁄6, third picture (3 parts) → 1⁄3, fourth picture (5 parts) → 1⁄5.
7.4 Number line (pages 31–32)
A unit is divided into three equal parts. Write the fraction that gives the length of the blue line.
Page 31
Number line: unit divided into 32⁄3
A unit is divided into 5 equal parts. Write the fractions for the lengths of the blue lines.
Page 31
Number line: unit divided into 5Short blue line: 2⁄5; long blue line: 4⁄5.
Now, a unit is divided into 8 equal parts. Write the appropriate fractions.
Page 311⁄8, 2⁄8, 3⁄8, 4⁄8, 5⁄8, 6⁄8, 7⁄8, 8⁄8 = 1
On a number line, draw lines of lengths 1⁄10, 3⁄10 and 4⁄5.
Page 32Divide 0 to 1 into 10 equal parts. The 1⁄10 line reaches the 1st mark and the 3⁄10 line the 3rd mark. For 4⁄5, divide 0 to 1 into 5 equal parts and go to the 4th mark (same as 8⁄10).
Write five more fractions of your choice and mark them on the number line.
Page 32Example: 1⁄4, 3⁄4, 5⁄6, 7⁄8, 3⁄2. Divide each unit into the number of parts given by the denominator and count the numerator parts from 0.
How many fractions lie between 0 and 1?
Page 32Infinitely many (endless). We can always divide the unit into more and more equal parts, e.g., 1⁄2, 1⁄3, 1⁄100, … all lie between 0 and 1.
What is the length of the blue line and black line?
Page 32
Blue and black linesBlue line: 1⁄2; black line: 3⁄2 (3 halves).
Write the fractions that give the lengths of the black lines.
Page 32
Black lines in fifthsFrom shortest to longest: 6⁄5, 7⁄5, 8⁄5, 9⁄5
7.5 Mixed fractions (pages 33–35)
Classify the fractions marked on the number line into lengths less than 1 unit and more than 1 unit. What is common in fractions greater than 1?
Page 33| Less than 1 unit | More than 1 unit |
|---|---|
| 1⁄2, 2⁄3, 2⁄5, 4⁄5, 1⁄10, 3⁄10 | 3⁄2, 6⁄5, 7⁄5, 8⁄5, 9⁄5 |
In fractions greater than 1, the numerator is larger than the denominator.
How many whole units are there in 7⁄2?
Page 343 whole units (7⁄2 = 3 + 1⁄2)
How many whole units are there in 4⁄3 and in 7⁄3?
Page 344⁄3: 1 whole unit (1 + 1⁄3). 7⁄3: 2 whole units (2 + 1⁄3).
Figure out the number of whole units in: a. 8⁄3 b. 11⁄5 c. 9⁄4
Page 34- a. 2 (8⁄3 = 2 + 2⁄3)
- b. 2 (11⁄5 = 2 + 1⁄5)
- c. 2 (9⁄4 = 2 + 1⁄4)
Can all fractions greater than 1 be written as such mixed numbers?
Page 34Yes. Take out as many whole units as possible; what is left is a fraction less than 1. (If nothing is left, the fraction is a whole number, e.g., 6⁄3 = 2.)
Write as mixed fractions: a. 9⁄2 b. 9⁄5 c. 21⁄19 d. 47⁄9 e. 12⁄11 f. 19⁄6
Page 34- a. 41⁄2
- b. 14⁄5
- c. 12⁄19
- d. 52⁄9
- e. 11⁄11
- f. 31⁄6
Write the mixed numbers as fractions: a. 31⁄4 b. 72⁄3 c. 94⁄9 d. 31⁄6 e. 23⁄11 f. 39⁄10
Page 35- a. 13⁄4
- b. 23⁄3
- c. 85⁄9
- d. 19⁄6
- e. 25⁄11
- f. 39⁄10
(Multiply the whole part by the denominator and add the numerator, e.g., 3 × 4 + 1 = 13.)
7.6 Equivalent fractions (pages 36–41)
Are the lengths 1⁄2 and 2⁄4 equal? Are 2⁄4 and 4⁄8 equal? Check whether 1⁄3 and 2⁄6 are equivalent.
Page 36Yes, 1⁄2 = 2⁄4 = 4⁄8. And yes, 1⁄3 and 2⁄6 are equivalent: folding a third in half gives two sixths of the same length.
Are the lengths 1⁄2 and 3⁄6 equal?
Page 36
Fraction wall up to sixthsYes, on the fraction wall 1⁄2 and 3⁄6 end at the same place.
Are 2⁄3 and 4⁄6 equivalent fractions? Why?
Page 36Yes, they have the same length on the fraction wall (two thirds = four sixths).
How many pieces of length 1⁄6 will make a length of 1⁄2?
Page 363 pieces (3⁄6 = 1⁄2)
How many pieces of length 1⁄6 will make a length of 1⁄3?
Page 362 pieces (2⁄6 = 1⁄3)
Are 3⁄6, 4⁄8, 5⁄10 equivalent fractions? Why?
Page 37Yes. On the fraction wall all three have the same length as 1⁄2.
Write two equivalent fractions for 2⁄6.
Page 371⁄3 and 4⁄12 (also 3⁄9).
4⁄6 = ☐ = ☐ = ☐ = … (Write as many as you can)
Page 374⁄6 = 2⁄3 = 6⁄9 = 8⁄12 = 10⁄15 = 12⁄18 = …
Three rotis are shared equally by four children. Write the fraction each child gets and the division, addition and multiplication facts.
Page 38
Three rotis shared by four children- Each child gets 3⁄4 roti.
- Division fact: 3 ÷ 4 = 3⁄4
- Addition fact: 3 = 3⁄4 + 3⁄4 + 3⁄4 + 3⁄4
- Multiplication fact: 3 = 4 × 3⁄4
Draw a picture to show how much each child gets when 2 rotis are shared equally by 4 children. Write the facts.
Page 38- Each child gets 2⁄4 (= 1⁄2) roti.
- Division fact: 2 ÷ 4 = 2⁄4
- Addition fact: 2 = 2⁄4 + 2⁄4 + 2⁄4 + 2⁄4
- Multiplication fact: 2 = 4 × 2⁄4
Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get? If there are 10 children, how many cakes are needed for the same share?
Page 38
2 cakes for 5 children and 4 cakes for 10 childrenAnil gets 2⁄5 cake. For 10 children, 4 cakes are needed, so 2⁄5 = 4⁄10.
Find some more fractions equivalent to 1⁄2.
Page 404⁄8, 5⁄10, 6⁄12, 10⁄20, …
Equally divide the rotis: 2 rotis among 3 children, 4 rotis among 6 children, 6 rotis among 9 children. Are the shares the same? Why?
Page 40
2 rotis for 3, 4 rotis for 6, 6 rotis for 9 childrenEach child gets 2⁄3, 4⁄6 and 6⁄9. Yes, the shares are the same: 2⁄3 = 4⁄6 = 6⁄9. In each case the numerator and denominator of 2⁄3 are multiplied by the same number (2 or 3).
Find the missing numbers: a. 5 glasses of juice among 4 friends = ____ glasses among 8 friends. b. 4 kg of potatoes in 3 bags = 12 kg in ____ bags. c. 7 rotis among 5 children = ____ rotis among ____ children.
Page 40- a. 10 glasses; 5⁄4 = 10⁄8
- b. 9 bags; 4⁄3 = 12⁄9
- c. e.g., 14 rotis among 10 children; 7⁄5 = 14⁄10
In which group will each child get more chikki: 1 chikki between 2 children or 4 chikkis among 7 children?
Page 411⁄2 = 4⁄8 and 4⁄7 > 4⁄8 (4 shared among fewer children). So 4⁄7 > 1⁄2: the children of the second group get more.
If the number of children is the same but the number of units shared is increased, what happens to each child’s share? Explain 1⁄5 < 2⁄5, 3⁄7 < 4⁄7 and 1⁄2 < 5⁄8.
Page 42Each child’s share increases, because more units are shared among the same number of children. So 1⁄5 < 2⁄5 and 3⁄7 < 4⁄7. Also 1⁄2 = 4⁄8 < 5⁄8.
In which group will each child get a larger share? 1. Group 1: 3 glasses among 4 children; Group 2: 7 glasses among 10 children. 2. Group 1: 4 glasses among 7 children; Group 2: 5 glasses among 7 children. Which were easier to compare?
Page 42- 1. 3⁄4 = 30⁄40, 7⁄10 = 28⁄40. Group 1 gets more.
- 2. 4⁄7 < 5⁄7, so Group 2 gets more.
- The second pair was easier because the number of children (denominator) was the same.
Find equivalent fractions with the same fractional units: a. 7⁄2, 3⁄5 b. 8⁄3, 5⁄6 c. 3⁄4, 3⁄5 d. 6⁄7, 8⁄5 e. 9⁄4, 5⁄2 f. 1⁄10, 2⁄9 g. 8⁄3, 11⁄4 h. 13⁄6, 1⁄9
Page 44- a. 35⁄10, 6⁄10
- b. 16⁄6, 5⁄6
- c. 15⁄20, 12⁄20
- d. 30⁄35, 56⁄35
- e. 9⁄4, 10⁄4
- f. 9⁄90, 20⁄90
- g. 32⁄12, 33⁄12
- h. 39⁄18, 2⁄18
Express in lowest terms: a. 17⁄51 b. 64⁄144 c. 126⁄147 d. 525⁄112
Page 45- a. ÷17 → 1⁄3
- b. ÷16 → 4⁄9
- c. ÷21 → 6⁄7
- d. ÷7 → 75⁄16
7.7 Comparing fractions (page 46)
Compare and justify: a. 8⁄3, 5⁄2 b. 4⁄9, 3⁄7 c. 7⁄10, 9⁄14 d. 12⁄5, 8⁄5 e. 9⁄4, 5⁄2
Page 46- a. 16⁄6, 15⁄6 → 8⁄3 > 5⁄2
- b. 28⁄63, 27⁄63 → 4⁄9 > 3⁄7
- c. 49⁄70, 45⁄70 → 7⁄10 > 9⁄14
- d. same denominator, 12 > 8 → 12⁄5 > 8⁄5
- e. 9⁄4, 10⁄4 → 9⁄4 < 5⁄2
Write in ascending order: a. 7⁄10, 11⁄15, 2⁄5 b. 19⁄24, 5⁄6, 7⁄12
Page 46- a. (21⁄30, 22⁄30, 12⁄30) → 2⁄5 < 7⁄10 < 11⁄15
- b. (19⁄24, 20⁄24, 14⁄24) → 7⁄12 < 19⁄24 < 5⁄6
Write in descending order: a. 25⁄16, 7⁄8, 13⁄4, 17⁄32 b. 3⁄4, 12⁄5, 7⁄12, 5⁄4
Page 46- a. (50⁄32, 28⁄32, 104⁄32, 17⁄32) → 13⁄4 > 25⁄16 > 7⁄8 > 17⁄32
- b. (45⁄60, 144⁄60, 35⁄60, 75⁄60) → 12⁄5 > 5⁄4 > 3⁄4 > 7⁄12
7.8 Addition and subtraction (pages 47–54)
How much of the total chikki is remaining (Meena ate 1⁄2 and her brother 1⁄4)?
Page 471 − 3⁄4 = 1⁄4 of the chikki is remaining.
Try adding 4⁄7 + 6⁄7 using a number line. Do you get the same answer?
Page 49Mark sevenths on the number line, move 4 steps and then 6 more steps: we reach 10⁄7 = 13⁄7. Yes, the same answer.
Add using Brahmagupta’s method: a. 2⁄7 + 5⁄7 + 6⁄7; b. 3⁄4 + 1⁄3; c. 2⁄3 + 5⁄6; d. 2⁄3 + 2⁄7; e. 3⁄4 + 1⁄3 + 1⁄5; f. 2⁄3 + 4⁄5; g. 4⁄5 + 2⁄3; h. 3⁄5 + 5⁄8; i. 9⁄2 + 5⁄4; j. 8⁄3 + 2⁄7; k. 3⁄4 + 1⁄3 + 1⁄5; l. 2⁄3 + 4⁄5 + 3⁄7; m. 9⁄2 + 5⁄4 + 7⁄6
Page 51- a. 13⁄7 = 16⁄7
- b. 9⁄12 + 4⁄12 = 13⁄12 = 11⁄12
- c. 4⁄6 + 5⁄6 = 9⁄6 = 3⁄2 = 11⁄2
- d. 14⁄21 + 6⁄21 = 20⁄21
- e. 45⁄60 + 20⁄60 + 12⁄60 = 77⁄60 = 117⁄60
- f. 10⁄15 + 12⁄15 = 22⁄15 = 17⁄15
- g. 12⁄15 + 10⁄15 = 22⁄15 = 17⁄15
- h. 24⁄40 + 25⁄40 = 49⁄40 = 19⁄40
- i. 18⁄4 + 5⁄4 = 23⁄4 = 53⁄4
- j. 56⁄21 + 6⁄21 = 62⁄21 = 220⁄21
- k. 77⁄60 = 117⁄60
- l. 70⁄105 + 84⁄105 + 45⁄105 = 199⁄105 = 194⁄105
- m. 54⁄12 + 15⁄12 + 14⁄12 = 83⁄12 = 611⁄12
Rahim mixes 2⁄3 litres of yellow paint with 3⁄4 litres of blue paint to make green paint. What is the volume of green paint?
Page 512⁄3 + 3⁄4 = 8⁄12 + 9⁄12 = 17⁄12 = 15⁄12 litres
Geeta bought 2⁄5 m of lace and Shamim bought 3⁄4 m to put a border on a tablecloth of perimeter 1 m. Find the total length. Is it sufficient?
Page 512⁄5 + 3⁄4 = 8⁄20 + 15⁄20 = 23⁄20 = 13⁄20 m. This is more than 1 m, so yes, the lace is sufficient.
Subtract: 1. 5⁄8 − 3⁄8 2. 7⁄9 − 5⁄9 3. 10⁄27 − 1⁄27
Page 53- 1. 2⁄8 = 1⁄4
- 2. 2⁄9
- 3. 9⁄27 = 1⁄3
Subtract using Brahmagupta’s method: a. 8⁄15 − 3⁄15 b. 2⁄5 − 4⁄15 c. 5⁄6 − 4⁄9 d. 2⁄3 − 1⁄2
Page 54- a. 5⁄15 = 1⁄3
- b. 6⁄15 − 4⁄15 = 2⁄15
- c. 15⁄18 − 8⁄18 = 7⁄18
- d. 4⁄6 − 3⁄6 = 1⁄6
Subtract as indicated: a. 13⁄4 from 10⁄3 b. 18⁄5 from 23⁄3 c. 29⁄7 from 45⁄7
Page 54- a. 10⁄3 − 13⁄4 = 40⁄12 − 39⁄12 = 1⁄12
- b. 23⁄3 − 18⁄5 = 115⁄15 − 54⁄15 = 61⁄15 = 41⁄15
- c. 45⁄7 − 29⁄7 = 16⁄7 = 22⁄7
Jaya’s school is 7⁄10 km from her home. She takes an auto for 1⁄2 km and walks the rest. How much does she walk daily?
Page 547⁄10 − 1⁄2 = 7⁄10 − 5⁄10 = 2⁄10 = 1⁄5 km
Jeevika takes 10⁄3 minutes to go round the park and Namit takes 13⁄4 minutes. Who takes less time and by how much?
Page 5410⁄3 = 40⁄12 and 13⁄4 = 39⁄12. Namit takes less time, by 40⁄12 − 39⁄12 = 1⁄12 minute.
Puzzle (pages 56–57)
Can you find three different fractional units that add up to 1?
Page 561⁄2 + 1⁄3 + 1⁄6 = 1 (3⁄6 + 2⁄6 + 1⁄6 = 6⁄6). This is the only solution.
Can you find four different fractional units that add up to 1? (There are six solutions.)
Page 57- 1⁄2 + 1⁄3 + 1⁄7 + 1⁄42
- 1⁄2 + 1⁄3 + 1⁄8 + 1⁄24
- 1⁄2 + 1⁄3 + 1⁄9 + 1⁄18
- 1⁄2 + 1⁄3 + 1⁄10 + 1⁄15
- 1⁄2 + 1⁄4 + 1⁄5 + 1⁄20
- 1⁄2 + 1⁄4 + 1⁄6 + 1⁄12
