Class 6 Maths (Ganita Prakash) - Chapter 7 - Fractions
Figure it Out 152
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.
* Answer: 1/3
* Explanation: The total weight (1 kg) is divided equally among 3 guavas. So, the weight of one guava is 1 kg ÷ 3 = 1/3 kg.
2. A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is ___ kg.
* Answer: 1/4
* Explanation: The total weight (1 kg) is divided equally into 4 packets. So, the weight of one packet is 1 kg ÷ 4 = 1/4 kg.
3. Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank ____ glass of sugarcane juice.
* Answer: 3/4
* Explanation: The total juice (3 glasses) is shared equally among 4 friends. So, each friend gets 3 glasses ÷ 4 = 3/4 of a glass.
4. The big fish weighs 1/2 kg. The small one weighs 1/4 kg. Together they weigh ____ kg.
* Answer: 3/4
* Explanation: To find the total weight, add the fractions: 1/2 + 1/4. First, convert 1/2 to 2/4. So, 2/4 + 1/4 = 3/4 kg.
5. Arrange these fraction words in order of size from the smallest to the biggest...
* Answer: quarter, half, three quarters, one and a quarter, one and a half, two and a half
* Explanation: First, convert the words to numbers or decimal equivalents for easy comparison.
* Quarter = 1/4 = 0.25
* Half = 1/2 = 0.5
* Three quarters = 3/4 = 0.75
* One and a quarter = 1 1/4 = 1.25
* One and a half = 1 1/2 = 1.5
* Two and a half = 2 1/2 = 2.5
* Arranging these from smallest to largest gives the answer above.
Figure it Out 158
1. Continue this table of 1/2 for 2 more steps.
* Answer:
* 1/2 of 1/2 is 1/4
* 1/2 of 1/4 is 1/8
* 1/2 of 1/8 is 1/16
* Explanation: To find half of a fraction, you multiply it by 1/2. So, 1/2 × 1/8 = 1/16.
2. Create a similar table for 1/4?
* Answer:
* 1/4 of 1/4 is 1/16
* 1/4 of 1/16 is 1/64
* 1/4 of 1/64 is 1/256
* Explanation: To find a quarter of a fraction, you multiply it by 1/4. So, 1/4 × 1/64 = 1/256.
3. Make 1/3 using a paper strip. Can you use this to also make 1/6?
* Answer: Yes.
* Explanation: If you fold a paper strip into 3 equal parts, each part is 1/3. If you then fold each of these 1/3 parts in half, you will have 6 equal parts in total. Each of these smaller parts is 1/6 of the whole strip. This shows that 1/6 is half of 1/3.
4. Draw a picture and write an addition statement...
* a. 5 times 1/4 of a roti
Addition: 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 5/4
* Multiplication: 5 × (1/4) = 5/4
Draw a roti divided into 4 equal quarters. Shade 5 of these quarters.
*
* * b. 9 times 1/4 of a roti
* Draw two rotis, each divided into 4 equal quarters. Shade all 8 quarters of the two rotis and shade one more quarter from a third roti, making 9 shaded quarters total.
* Addition: 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 9/4
* Multiplication: 9 × (1/4) = 9/4
Figure it Out 160
1. On a number line, draw lines of lengths 1/10, 3/10, and 4/5.
* Explanation: Draw a number line from 0 to 1. Divide it into 10 equal parts.
* 1/10 is the first mark from 0.
* 3/10 is the third mark from 0.
* 4/5 is equivalent to 8/10, which is the eighth mark from 0.
2. Write five more fractions of your choice and mark them on the number line.
* Answer: (Example) 1/5, 2/5, 3/5, 7/10, 9/10
* Explanation: Mark these on the number line divided into tenths. 1/5 = 2/10, 2/5 = 4/10, 3/5 = 6/10.
3. How many fractions lie between 0 and 1?
* Answer: Infinitely many.
* Explanation: You can always find a new fraction between any two fractions by averaging them (e.g., between 1/10 and 2/10 is 1.5/10 or 3/20). This process can be repeated forever.
4. What is the length of the blue line and black line... The blue line is 1/2 units long. Write the fraction that gives the length of the black line...
* Answer: 1/4
* Explanation: The distance from 0 to 1 is divided into two equal parts by the blue line (1/2). The black line is half the length of the blue section. So, 1/2 of 1/2 = 1/4.
5. Write the fraction that gives the lengths of the black lines...
* From 0 to a: 1/8
* From 0 to b: 3/8
* Explanation: The segment from 0 to 1 is divided into 8 equal parts. Line 'a' covers 1 part, so it's 1/8. Line 'b' covers 3 parts, so it's 3/8.
Figure it Out 162
1. How many whole units are there in 7/2?
* Answer: 3
* Explanation: 7/2 is 7 halves. 2 halves make 1 whole. So, 7 halves make 7 ÷ 2 = 3 whole units with 1 half left over (3 1/2).
2. How many whole units are there in 4/3 and in 7/3?
* Answer: 4/3 has 1 whole unit. 7/3 has 2 whole units.
* Explanation:
* 4/3: 3 thirds make 1 whole. 4 thirds make 4 ÷ 3 = 1 whole with 1 third left over (1 1/3).
* 7/3: 3 thirds make 1 whole. 7 thirds make 7 ÷ 3 = 2 wholes with 1 third left over (2 1/3).
1. The number of whole units in each of the following fractions:
* (a) 8/3: 2 wholes (8 ÷ 3 = 2 remainder 2, so 2 2/3)
* (b) 11/5: 2 wholes (11 ÷ 5 = 2 remainder 1, so 2 1/5)
* (c) 9/4: 2 wholes (9 ÷ 4 = 2 remainder 1, so 2 1/4)
2. Can all fractions greater than 1 be written as such mixed numbers?
* Answer: Yes.
* Explanation: Any fraction where the numerator is greater than the denominator (an improper fraction) can be divided to find how many whole units are in it, with the remainder written as a proper fraction. This is the definition of a mixed number.
Figure it Out 163
Write the following mixed numbers as fractions:
* Explanation: To convert a mixed number to a fraction: (Whole number × Denominator) + Numerator, all over the original denominator.
* 2 1/3 = (2 × 3 + 1)/3 = (6 + 1)/3 = 7/3
* 3 2/5 = (3 × 5 + 2)/5 = (15 + 2)/5 = 17/5
* 1 3/4 = (1 × 4 + 3)/4 = (4 + 3)/4 = 7/4
* 2 5/6 = (2 × 6 + 5)/6 = (12 + 5)/6 = 17/6
Figure it Out 166
1. Three rotis are shared equally by four children.
* Fraction of roti each child gets: 3/4
* Division fact: 3 ÷ 4 = 3/4
* Addition fact: 1/4 + 1/4 + 1/4 = 3/4 (Each child gets a quarter from each of the three rotis)
* Multiplication fact: 3 × (1/4) = 3/4
* Picture: Draw three circles (rotis). Divide each circle into 4 equal parts. Show that each of the 4 children gets one part from each roti, totaling 3 parts. Since 4 parts make a whole roti, 3 parts are 3/4 of a roti.
2. Two rotis are shared equally by 4 children.
* Fraction of roti each child gets: 1/2
* Division fact: 2 ÷ 4 = 2/4 = 1/2
* Addition fact: 1/4 + 1/4 = 2/4 = 1/2 (Each child gets a quarter from each of the two rotis)
* Multiplication fact: 2 × (1/4) = 2/4 = 1/2
* Picture: Draw two circles (rotis). Divide each circle into 4 equal parts. Show that each of the 4 children gets one part from each roti, totaling 2 parts. Since 4 parts make a whole roti, 2 parts are 2/4 or 1/2 of a roti.
Figure it Out 168
1. Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get?
* Answer: 2/5 of a cake.
* Explanation: Total cake ÷ Number of children = 2 ÷ 5 = 2/5.
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.
* Answer: 10
* Explanation: Each of the 4 friends gets 5/4 glass. We need to find an equivalent fraction where the denominator is 8. 5/4 = (5×2)/(4×2) = 10/8. This means 10 glasses shared among 8 friends gives the same share (10/8 = 5/4).
* b. 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided equally in ____ bags.
* Answer: 9
* Explanation: Each bag holds 4/3 kg. We need to find how many bags (x) will hold 12/x kg, which is equal to 4/3. So, 12/x = 4/3. Cross-multiplying: 4x = 36, so x = 9.
* c. 7 rotis divided among 5 children is the same as ____ rotis divided among _____ children.
* Answer: (Multiple answers possible) Example: 14 rotis divided among 10 children.
* Explanation: Each child gets 7/5 rotis. Find an equivalent fraction by multiplying the numerator and denominator by the same number. (7×2)/(5×2) = 14/10. So, 14 rotis among 10 children gives the same share.
Figure it Out 172
Find equivalent fractions for the given pairs of fractions such that the fractional units are the same.
* Explanation: Find a common denominator for each pair.
* 1/2 and 1/3
* Common Denominator: 6
* 1/2 = 3/6 and 1/3 = 2/6
* 1/2 and 2/3
* Common Denominator: 6
* 1/2 = 3/6 and 2/3 = 4/6
* 1/2 and 1/4
* Common Denominator: 4
* 1/2 = 2/4 and 1/4 = 1/4
* 1/2 and 3/4
* Common Denominator: 4
* 1/2 = 2/4 and 3/4 = 3/4
* 1/4 and 1/3
* Common Denominator: 12
* 1/4 = 3/12 and 1/3 = 4/12
Figure it Out 179
1. Add the following fractions using Brahmagupta's method:
* Explanation: Brahmagupta's method is to find the LCM of the denominators (the "smallest common fractional unit") and then add.
* 1/2 + 1/3
* LCM of 2 and 3 is 6.
* 1/2 = 3/6, 1/3 = 2/6
* 3/6 + 2/6 = 5/6
* 1/4 + 1/6
* LCM of 4 and 6 is 12.
* 1/4 = 3/12, 1/6 = 2/12
* 3/12 + 2/12 = 5/12
* 1/6 + 1/2
* LCM of 6 and 2 is 6.
* 1/6 = 1/6, 1/2 = 3/6
* 1/6 + 3/6 = 4/6 = 2/3
2. Rahim mixes 2/3 litres of yellow paint with 3/4 litres of blue paint...
* Answer: 1 5/12 litres
* Explanation:
* Total volume = 2/3 + 3/4
* LCM of 3 and 4 is 12.
* 2/3 = 8/12, 3/4 = 9/12
* 8/12 + 9/12 = 17/12 = 1 5/12 litres.
3. Geeta bought 2/5 meter of lace and Shamim bought 3/4 meter... Find the total length... Will the lace be sufficient...?
* Answer: Total length = 1 3/20 meters. No, it is not sufficient.
* Explanation:
* Total lace = 2/5 + 3/4
* LCM of 5 and 4 is 20.
* 2/5 = 8/20, 3/4 = 15/20
* 8/20 + 15/20 = 23/20 = 1 3/20 meters.
* The perimeter of the cloth is 1 meter. 1 3/20 meters is greater than 1 meter (1 3/20 = 1.15). Yes, the lace will be sufficient.
* *Correction: The problem states the perimeter is 1 meter. 1 3/20 > 1, so the lace IS sufficient.*
Figure it Out 183
3. Solve the following problems:
a. Jaya's school is 7/10 km from her home. She takes an auto for 1/2 km from her home daily, and then walks the remaining distance...
* Answer: 1/5 km
* Explanation: To find the remaining distance, subtract the auto distance from the total distance.
* 7/10 - 1/2
* Convert 1/2 to 5/10.
* 7/10 - 5/10 = 2/10 = 1/5 km.
b. Jeevika takes 10/3 minutes to take a complete round of the park and her friend Namit takes 13/4 minutes to do the same. Who takes less time and by how much?
* Answer: Namit takes less time by 1/12 of a minute.
* Explanation:
* Convert these improper fractions to make comparison easier.
* Jeevika: 10/3 = 3 1/3 minutes.
* Namit: 13/4 = 3 1/4 minutes.
* To compare 1/3 and 1/4, find a common denominator (12). 1/3 = 4/12, 1/4 = 3/12. Since 3/12 is less than 4/12, Namit is faster.
* Difference: 10/3 - 13/4
* LCM of 3 and 4 is 12.
* 10/3 = 40/12, 13/4 = 39/12
* 40/12 - 39/12 = 1/12 minute.
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