FractionsClass 6 Mathematics NCERT Solutions

Given: Total weight of 3 guavas = 1 kg The guavas are of the same size.

To Find: The weight of each guava.

Solution: Since 3 guavas weigh 1 kg, to find the weight of one guava, we divide the total weight by the number of guavas. Weight of one guava = $\frac{\text{Total weight}}{\text{Number of guavas}} = \frac{1}{3}$ kg.

Final Answer: Each guava will roughly weigh $\frac{1}{3}$ kg.

Given: Total weight of rice = 1 kg Number of packets = 4 The packets are of equal weight.

To Find: The weight of each packet.

Solution: To find the weight of one packet, we divide the total weight of rice by the number of packets. Weight of each packet = $\frac{\text{Total weight}}{\text{Number of packets}} = \frac{1}{4}$ kg.

Final Answer: The weight of each packet is $\frac{1}{4}$ kg.

Given: Number of glasses of sugarcane juice = 3 Number of friends = 4 The juice is shared equally.

To Find: The amount of juice each friend drank.

Solution: To find the share of each friend, we divide the total number of glasses of juice by the number of friends. Share per friend = $\frac{\text{Total glasses of juice}}{\text{Number of friends}} = \frac{3}{4}$ glass of sugarcane juice.

Final Answer: Each one drank $\frac{3}{4}$ glass of sugarcane juice.

Given: Weight of the big fish = $\frac{1}{2}$ kg Weight of the small fish = $\frac{1}{4}$ kg

To Find: The total weight of both fish.

Solution: To find the total weight, we add the weights of the two fish. Total weight = Weight of big fish + Weight of small fish $= \frac{1}{2} + \frac{1}{4}$ To add these fractions, we need a common denominator, which is 4. $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$ So, the total weight is: $= \frac{2}{4} + \frac{1}{4} = \frac{2+1}{4} = \frac{3}{4} \text{ kg}$

Final Answer: Together they weigh $\frac{3}{4}$ kg.

Given: The fraction words: One and a half, three quarters, one and a quarter, half, quarter, two and a half.

To Find: Arrange these in order from smallest to biggest.

Solution: First, let's convert each word form into a numerical fraction:

Now, we compare these numerical values. $\frac{1}{4} = 0.25$ $\frac{1}{2} = 0.5$ $\frac{3}{4} = 0.75$ $1\frac{1}{4} = 1.25$ $1\frac{1}{2} = 1.5$ $2\frac{1}{2} = 2.5$

Arranging these numbers in ascending order: $0.25 < 0.5 < 0.75 < 1.25 < 1.5 < 2.5$

This corresponds to the following order of fractions: $\frac{1}{4} < \frac{1}{2} < \frac{3}{4} < 1\frac{1}{4} < 1\frac{1}{2} < 2\frac{1}{2}$

Final Answer: The correct order from smallest to biggest is: quarter, half, three quarters, one and a quarter, one and a half, two and a half.

To Find: The fraction of the whole chikki that each piece represents.

Solution: The fraction is determined by how many equal parts the whole is divided into. If a whole is divided into 'n' equal parts, each part is $\frac{1}{n}$ of the whole.

a. The chikki is divided into 12 equal parts. One part is $\frac{1}{12}$ of the whole. b. The chikki is divided into 4 equal parts. One part is $\frac{1}{4}$ of the whole. c. The chikki is divided into 8 equal parts. One part is $\frac{1}{8}$ of the whole. d. The chikki is divided into 6 equal parts. One part is $\frac{1}{6}$ of the whole. e. The chikki is divided into 8 equal parts. One part is $\frac{1}{8}$ of the whole. f. The chikki is divided into 6 equal parts. One part is $\frac{1}{6}$ of the whole. g. The chikki is divided into 24 equal parts. One part is $\frac{1}{24}$ of the whole. h. The chikki is divided into 24 equal parts. One part is $\frac{1}{24}$ of the whole.

Final Answer: a. $\frac{1}{12}$ b. $\frac{1}{4}$ c. $\frac{1}{8}$ d. $\frac{1}{6}$ e. $\frac{1}{8}$ f. $\frac{1}{6}$ g. $\frac{1}{24}$ h. $\frac{1}{24}$

Given: A table showing multiples of $\frac{1}{2}$. The last entry is 5 times half.

To Find: The next two steps in the table.

Solution: The pattern is adding $\frac{1}{2}$ in each step. The step after '5 times half' is '6 times half'. Addition statement: $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2} = 6 \times \frac{1}{2} = \frac{6}{2} = 3$

The next step is '7 times half'. Addition statement: $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2} = 7 \times \frac{1}{2} = \frac{7}{2} = 3\frac{1}{2}$

Final Answer: The next two steps are:

To Do: Create a table for multiples of $\frac{1}{4}$ similar to the one for $\frac{1}{2}$.

Solution: We can create a table showing the cumulative addition of the fractional unit $\frac{1}{4}$.

Final Answer: Yes, a similar table for $\frac{1}{4}$ can be created as shown above.

To Do: Describe how to make $\frac{1}{6}$ from a paper strip already folded into thirds.

Final Answer: Yes, you can make $\frac{1}{6}$ from a $\frac{1}{3}$ strip by folding the $\frac{1}{3}$ section in half.

To Do: Describe a picture and write an addition statement for the given fractional quantities.

Solution: a. 5 times $\frac{1}{4}$ of a roti

Final Answer: a. Addition statement: $\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{5}{4}$. b. Addition statement: $\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{9}{4}$.

To Do: Match each fraction to the picture that represents it.

Solution: A fraction $\frac{1}{n}$ represents one part of a whole that has been divided into 'n' equal parts.

To Do: Describe how to mark the given fractional lengths on a number line.

Final Answer: The lengths are marked on a number line by dividing the unit length into 10 equal parts and marking the 1st part for $\frac{1}{10}$, the 3rd part for $\frac{3}{10}$, and the 8th part for $\frac{4}{5}$.

To Do: Choose five fractions and describe how to mark them on a number line.

Solution: Let's choose the fractions: $\frac{1}{2}, \frac{3}{4}, \frac{2}{5}, \frac{5}{8}, \frac{7}{10}$.

Final Answer: An example set of five fractions is $\frac{1}{2}, \frac{3}{4}, \frac{2}{5}, \frac{5}{8}, \frac{7}{10}$. Each can be marked by dividing the unit length on a number line into the number of parts indicated by the denominator and counting the number of parts indicated by the numerator from 0.

To Find: The number of fractions between 0 and 1.

Solution: Between any two distinct numbers (like 0 and 1), we can always find another number. For example, $\frac{1}{2}$ is between 0 and 1. Then, $\frac{1}{4}$ is between 0 and $\frac{1}{2}$. Then, $\frac{1}{8}$ is between 0 and $\frac{1}{4}$, and so on. We can continue this process infinitely. We can also find fractions like $\frac{1}{3}, \frac{1}{5}, \frac{2}{3}, \frac{3}{5}$, etc. This shows that there is no end to the number of fractions we can find between 0 and 1.

Final Answer: There are an infinite (uncountable) number of fractions that lie between 0 and 1.

Given: A number line where the distance between whole numbers is divided into two equal parts. Each part is $\frac{1}{2}$ unit. The blue line goes from 0 to the first mark, so its length is $\frac{1}{2}$. The black line goes from 0 to the third mark (one mark past 1).

To Find: The length of the black line.

Solution: Since each part is $\frac{1}{2}$ unit long, we can count the number of parts from 0 to the end of the black line. The black line covers 3 parts. So, the length of the black line is $3 \times \frac{1}{2} = \frac{3}{2}$. Alternatively, the line goes to 1 and then one more part of $\frac{1}{2}$. So, the length is $1 + \frac{1}{2} = 1\frac{1}{2} = \frac{3}{2}$.

Final Answer: The fraction that gives the length of the black line is $\frac{3}{2}$.

Given: A number line where the distance between whole numbers is divided into five equal parts. Each part is $\frac{1}{5}$ unit. Black lines are shown ending at various points beyond 1.

To Find: The lengths of the black lines.

Solution: The point 1 is at the 5th mark from 0. The black lines end at points after this.

Final Answer: The lengths of the black lines are $\frac{6}{5}, \frac{7}{5}, \frac{8}{5}$, and $\frac{9}{5}$ respectively.

Given: The fraction $\frac{7}{2}$.

To Find: The number of whole units in $\frac{7}{2}$.

Solution: To find the number of whole units, we perform the division of the numerator by the denominator. $7 \div 2 = 3 \text{ with a remainder of } 1$ This means that $\frac{7}{2}$ is equal to 3 whole units and a remainder of 1 part of size $\frac{1}{2}$. So, $\frac{7}{2} = 3\frac{1}{2}$. The whole number part is 3.

Final Answer: There are 3 whole units in $\frac{7}{2}$.

Given: The fractions $\frac{4}{3}$ and $\frac{7}{3}$.

To Find: The number of whole units in each fraction.

Solution: For $\frac{4}{3}$: $4 \div 3 = 1 \text{ with a remainder of } 1$ So, $\frac{4}{3} = 1\frac{1}{3}$. The whole number part is 1.

For $\frac{7}{3}$: $7 \div 3 = 2 \text{ with a remainder of } 1$ So, $\frac{7}{3} = 2\frac{1}{3}$. The whole number part is 2.

Final Answer: There is 1 whole unit in $\frac{4}{3}$ and there are 2 whole units in $\frac{7}{3}$.

To Find: The number of whole units in each given fraction.

Solution: a. For $\frac{8}{3}$: $8 \div 3 = 2$ with a remainder of 2. So there are 2 whole units.

b. For $\frac{11}{5}$: $11 \div 5 = 2$ with a remainder of 1. So there are 2 whole units.

c. For $\frac{9}{4}$: $9 \div 4 = 2$ with a remainder of 1. So there are 2 whole units.

Final Answer: a. 2 b. 2 c. 2

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

Solution: No. A fraction greater than 1 can be written as a mixed number only if the numerator is not a multiple of the denominator. If the numerator is a multiple of the denominator, the fraction simplifies to a whole number, not a mixed number. For example, $\frac{7}{3}$ is greater than 1 and can be written as the mixed number $2\frac{1}{3}$. However, $\frac{8}{4}$ is also greater than 1, but it simplifies to the whole number 2, which does not have a fractional part.

Final Answer: No. Fractions greater than 1 where the numerator is a multiple of the denominator simplify to a whole number, not a mixed number.

To Do: Convert the given improper fractions into mixed fractions.

Solution: To convert an improper fraction to a mixed fraction, divide the numerator by the denominator. The quotient is the whole number part, the remainder is the new numerator, and the denominator stays the same.

a. $\frac{9}{2}$: $9 \div 2 = 4$ with remainder 1. So, $\frac{9}{2} = 4\frac{1}{2}$.

b. $\frac{9}{5}$: $9 \div 5 = 1$ with remainder 4. So, $\frac{9}{5} = 1\frac{4}{5}$.

c. $\frac{21}{19}$: $21 \div 19 = 1$ with remainder 2. So, $\frac{21}{19} = 1\frac{2}{19}$.

d. $\frac{47}{9}$: $47 \div 9 = 5$ with remainder 2. So, $\frac{47}{9} = 5\frac{2}{9}$.

e. $\frac{12}{11}$: $12 \div 11 = 1$ with remainder 1. So, $\frac{12}{11} = 1\frac{1}{11}$.

f. $\frac{19}{6}$: $19 \div 6 = 3$ with remainder 1. So, $\frac{19}{6} = 3\frac{1}{6}$.

Final Answer: a. $4\frac{1}{2}$ b. $1\frac{4}{5}$ c. $1\frac{2}{19}$ d. $5\frac{2}{9}$ e. $1\frac{1}{11}$ f. $3\frac{1}{6}$

To Do: Convert the given mixed numbers into improper fractions.

Solution: To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. Formula: $a\frac{b}{c} = \frac{a \times c + b}{c}$.

a. $3\frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}$.

b. $7\frac{2}{3} = \frac{(7 \times 3) + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3}$.

c. $9\frac{4}{9} = \frac{(9 \times 9) + 4}{9} = \frac{81 + 4}{9} = \frac{85}{9}$.

d. $3\frac{1}{6} = \frac{(3 \times 6) + 1}{6} = \frac{18 + 1}{6} = \frac{19}{6}$.

e. $2\frac{3}{11} = \frac{(2 \times 11) + 3}{11} = \frac{22 + 3}{11} = \frac{25}{11}$.

f. $3\frac{9}{10} = \frac{(3 \times 10) + 9}{10} = \frac{30 + 9}{10} = \frac{39}{10}$.

Final Answer: a. $\frac{13}{4}$ b. $\frac{23}{3}$ c. $\frac{85}{9}$ d. $\frac{19}{6}$ e. $\frac{25}{11}$ f. $\frac{39}{10}$

To Find: Whether $\frac{1}{2}$ and $\frac{3}{6}$ are equal.

Solution: To check if two fractions are equal, we can simplify them to their lowest terms. The fraction $\frac{1}{2}$ is already in its lowest terms. For the fraction $\frac{3}{6}$, we can divide both the numerator and the denominator by their greatest common divisor, which is 3. $\frac{3 \div 3}{6 \div 3} = \frac{1}{2}$ Since both fractions simplify to $\frac{1}{2}$, they are equal.

Final Answer: Yes, the lengths $\frac{1}{2}$ and $\frac{3}{6}$ are equal.

To Find: Whether $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent fractions and why.

Solution: Equivalent fractions represent the same value. We can check this by simplifying $\frac{4}{6}$ to its lowest terms. The greatest common divisor of 4 and 6 is 2. $\frac{4 \div 2}{6 \div 2} = \frac{2}{3}$ Since $\frac{4}{6}$ simplifies to $\frac{2}{3}$, the two fractions are equivalent. They are equivalent because they represent the same portion of a whole.

Final Answer: Yes, $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent fractions because $\frac{4}{6}$ simplifies to $\frac{2}{3}$.

To Find: The number of $\frac{1}{6}$ pieces needed to make a length of $\frac{1}{2}$.

Solution: This is a division problem: We need to find how many times $\frac{1}{6}$ fits into $\frac{1}{2}$. $\text{Number of pieces} = \frac{1}{2} \div \frac{1}{6}$ To divide by a fraction, we multiply by its reciprocal: $= \frac{1}{2} \times \frac{6}{1} = \frac{6}{2} = 3$ Alternatively, we can use a common denominator. $\frac{1}{2} = \frac{3}{6}$. This shows that a length of $\frac{1}{2}$ is made up of 3 pieces of length $\frac{1}{6}$.

Final Answer: 3 pieces of length $\frac{1}{6}$ will make a length of $\frac{1}{2}$.

To Find: The number of $\frac{1}{6}$ pieces needed to make a length of $\frac{1}{3}$.

Solution: We need to calculate $\frac{1}{3} \div \frac{1}{6}$. $= \frac{1}{3} \times \frac{6}{1} = \frac{6}{3} = 2$ Alternatively, using a common denominator: $\frac{1}{3} = \frac{2}{6}$. This shows that a length of $\frac{1}{3}$ is made up of 2 pieces of length $\frac{1}{6}$.

Final Answer: 2 pieces of length $\frac{1}{6}$ will make a length of $\frac{1}{3}$.

To Find: Which fraction is greater: $\frac{1}{2}$ or $\frac{5}{8}$.

Solution: To compare the fractions, we need to express them with a common denominator. The least common multiple of 2 and 8 is 8. Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 8: $\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$ Now we compare $\frac{4}{8}$ and $\frac{5}{8}$. Since the denominators are the same, we compare the numerators. $5 > 4$, so $\frac{5}{8} > \frac{4}{8}$. Therefore, $\frac{5}{8} > \frac{1}{2}$.

Final Answer: The children in the group where 5 chikkis are divided among 8 will get more chikki each.

To Find: Which fraction is greater: $\frac{1}{2}$ or $\frac{4}{7}$.

Solution: We can compare these fractions by finding a common denominator. The least common multiple of 2 and 7 is 14. Convert both fractions to have a denominator of 14: $\frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14}$ $\frac{4}{7} = \frac{4 \times 2}{7 \times 2} = \frac{8}{14}$ Now we compare $\frac{7}{14}$ and $\frac{8}{14}$. Since $8 > 7$, we have $\frac{8}{14} > \frac{7}{14}$. Therefore, $\frac{4}{7} > \frac{1}{2}$.

Final Answer: The children in the group where 4 chikkis are divided among 7 will get more chikki each.

To Find: Which group gets a larger share in each case, and which comparison was easier.

Solution: 1. Comparing Group 1 ($\rac{3}{4}$) and Group 2 ($\rac{7}{10}$): We need to find a common denominator for 4 and 10. The least common multiple is 20.

2. Comparing Group 1 ($\rac{4}{7}$) and Group 2 ($\rac{5}{7}$): The denominators are already the same (7). We can directly compare the numerators. Since $5 > 4$, we have $\frac{5}{7} > \frac{4}{7}$. So, Group 2 gets a larger share.

Which was easier? The second comparison was easier because the fractions already had a common denominator (the number of children was the same). This allowed for a direct comparison of the numerators (the number of glasses of juice).

To Find: Whether the fractions $\frac{3}{6}, \frac{4}{8}, \frac{5}{10}$ are equivalent.

Solution: To check if they are equivalent, we simplify each fraction to its lowest terms.

Since all three fractions simplify to $\frac{1}{2}$, they are equivalent.

Final Answer: Yes, they are equivalent fractions because they all represent the same value, $\frac{1}{2}$.

Given: The fraction $\frac{2}{6}$.

To Find: Two equivalent fractions for $\frac{2}{6}$.

Solution: We can find equivalent fractions by either simplifying the given fraction or by multiplying its numerator and denominator by the same non-zero number.

Final Answer: Two equivalent fractions for $\frac{2}{6}$ are $\frac{1}{3}$ and $\frac{4}{12}$.

Given: The fraction $\frac{4}{6}$.

To Find: As many equivalent fractions for $\frac{4}{6}$ as possible.

Solution: First, we can simplify $\frac{4}{6}$ to its lowest terms by dividing the numerator and denominator by their greatest common divisor, 2. $\frac{4 \div 2}{6 \div 2} = \frac{2}{3}$ Now, we can find more equivalent fractions by multiplying the numerator and denominator of $\frac{2}{3}$ by the same integer (2, 3, 4, etc.).

Final Answer: $\frac{4}{6} = \frac{2}{3} = \frac{6}{9} = \frac{8}{12} = \frac{10}{15} = \ldots$

Given: 3 rotis are shared equally among 4 children.

To Find: The fraction of roti each child gets, and the corresponding division, addition, and multiplication facts.

Final Answer: Fraction of roti each child gets is $\frac{3}{4}$. Division fact: $3 \div 4 = \frac{3}{4}$ Addition fact: $\frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{3}{4} = 3$ Multiplication fact: $4 \times \frac{3}{4} = 3$

Given: 2 rotis are shared equally among 4 children.

To Find: The fraction of roti each child gets, and the corresponding division, addition, and multiplication facts.

Final Answer: Fraction of roti each child gets is $\frac{1}{2}$. Division fact: $2 \div 4 = \frac{1}{2}$ Addition fact: $\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = 2$ Multiplication fact: $4 \times \frac{1}{2} = 2$

Given: 2 cakes are divided equally among 5 children.

To Find: The amount of cake Anil would get.

Solution: To find the share for one child, we divide the total number of cakes by the number of children. Share per child = $\frac{\text{Total cakes}}{\text{Number of children}} = \frac{2}{5}$. Anil is one of the children, so he would get this share.

Final Answer: Anil would get $\frac{2}{5}$ of a cake.

To Find: The missing numbers in the equivalent fraction problems.

Solution: a. We have the equation $\frac{5}{4} = \frac{x}{8}$. To get from the denominator 4 to 8, we multiply by 2. To keep the fraction equivalent, we must also multiply the numerator by 2. $x = 5 \times 2 = 10$ So, 5 glasses for 4 friends is the same share as 10 glasses for 8 friends.

b. We have the equation $\frac{4}{3} = \frac{12}{y}$. To get from the numerator 4 to 12, we multiply by 3. To keep the fraction equivalent, we must also multiply the denominator by 3. $y = 3 \times 3 = 9$ So, 4 kg in 3 bags is the same ratio as 12 kg in 9 bags.

c. We need to find an equivalent fraction for $\frac{7}{5}$. We can multiply the numerator and denominator by any integer. Let's choose 2. $\frac{7 \times 2}{5 \times 2} = \frac{14}{10}$ So, 7 rotis for 5 children is the same share as 14 rotis for 10 children. (Other answers are possible.)

Final Answer: a. 10; So, $\frac{5}{4} = \frac{10}{8}$. b. 9; So, $\frac{4}{3} = \frac{12}{9}$. c. 14, 10 (example); So, $\frac{7}{5} = \frac{14}{10}$.

To Do: Simplify the given fractions to their lowest terms.

Solution: To express a fraction in lowest terms, we divide the numerator and the denominator by their highest common factor (HCF).

a. $\frac{17}{51}$: The factors of 17 are 1, 17. Since $51 = 3 \times 17$, the HCF is 17. $\frac{17 \div 17}{51 \div 17} = \frac{1}{3}$

b. $\frac{64}{144}$: We can simplify in steps by dividing by common factors like 2 or 4. $\frac{64}{144} = \frac{32}{72} = \frac{16}{36} = \frac{8}{18} = \frac{4}{9}$ Alternatively, the HCF of 64 and 144 is 16. $\frac{64 \div 16}{144 \div 16} = \frac{4}{9}$.

c. $\frac{126}{147}$: Both numbers are divisible by 3 (sum of digits is divisible by 3) and 7. So they are divisible by 21. HCF is 21. $\frac{126 \div 21}{147 \div 21} = \frac{6}{7}$

d. $\frac{525}{112}$: The sum of digits of 525 is 12, so it is divisible by 3. 112 is not. Both end in a number that suggests divisibility by 7. $525 = 7 \times 75$, $112 = 7 \times 16$. HCF is 7. $\frac{525 \div 7}{112 \div 7} = \frac{75}{16}$ 75 and 16 have no common factors other than 1, so this is the lowest term.

Final Answer: a. $\frac{1}{3}$ b. $\frac{4}{9}$ c. $\frac{6}{7}$ d. $\frac{75}{16}$

To Do: For each pair of fractions, find equivalent fractions with a common denominator.

Solution: We find the least common multiple (LCM) of the denominators for each pair.

a. $\frac{7}{2}$ and $\frac{3}{5}$. LCM of 2 and 5 is 10. $\frac{7}{2} = \frac{7 \times 5}{2 \times 5} = \frac{35}{10}$; $\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}$.

b. $\frac{8}{3}$ and $\frac{5}{6}$. LCM of 3 and 6 is 6. $\frac{8}{3} = \frac{8 \times 2}{3 \times 2} = \frac{16}{6}$; $\frac{5}{6}$ remains $\frac{5}{6}$.

c. $\frac{3}{4}$ and $\frac{3}{5}$. LCM of 4 and 5 is 20. $\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$; $\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}$.

d. $\frac{6}{7}$ and $\frac{8}{5}$. LCM of 7 and 5 is 35. $\frac{6}{7} = \frac{6 \times 5}{7 \times 5} = \frac{30}{35}$; $\frac{8}{5} = \frac{8 \times 7}{5 \times 7} = \frac{56}{35}$.

e. $\frac{9}{4}$ and $\frac{5}{2}$. LCM of 4 and 2 is 4. $\frac{9}{4}$ remains $\frac{9}{4}$; $\frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4}$.

f. $\frac{1}{10}$ and $\frac{2}{9}$. LCM of 10 and 9 is 90. $\frac{1}{10} = \frac{1 \times 9}{10 \times 9} = \frac{9}{90}$; $\frac{2}{9} = \frac{2 \times 10}{9 \times 10} = \frac{20}{90}$.

g. $\frac{8}{3}$ and $\frac{11}{4}$. LCM of 3 and 4 is 12. $\frac{8}{3} = \frac{8 \times 4}{3 \times 4} = \frac{32}{12}$; $\frac{11}{4} = \frac{11 \times 3}{4 \times 3} = \frac{33}{12}$.

h. $\frac{13}{6}$ and $\frac{1}{9}$. LCM of 6 and 9 is 18. $\frac{13}{6} = \frac{13 \times 3}{6 \times 3} = \frac{39}{18}$; $\frac{1}{9} = \frac{1 \times 2}{9 \times 2} = \frac{2}{18}$.

Final Answer: a. $\frac{35}{10}$ and $\frac{6}{10}$ b. $\frac{16}{6}$ and $\frac{5}{6}$ c. $\frac{15}{20}$ and $\frac{12}{20}$ d. $\frac{30}{35}$ and $\frac{56}{35}$ e. $\frac{9}{4}$ and $\frac{10}{4}$ f. $\frac{9}{90}$ and $\frac{20}{90}$ g. $\frac{32}{12}$ and $\frac{33}{12}$ h. $\frac{39}{18}$ and $\frac{2}{18}$

To Do: Compare the pairs of fractions.

Solution: We use the method of finding a common denominator.

a. $\frac{8}{3}$ and $\frac{5}{2}$: LCM of 3 and 2 is 6. $\frac{8}{3} = \frac{16}{6}$; $\frac{5}{2} = \frac{15}{6}$. Since $16 > 15$, $\frac{16}{6} > \frac{15}{6}$, so $\frac{8}{3} > \frac{5}{2}$.

b. $\frac{4}{9}$ and $\frac{3}{7}$: LCM of 9 and 7 is 63. $\frac{4}{9} = \frac{28}{63}$; $\frac{3}{7} = \frac{27}{63}$. Since $28 > 27$, $\frac{28}{63} > \frac{27}{63}$, so $\frac{4}{9} > \frac{3}{7}$.

c. $\frac{7}{10}$ and $\frac{9}{14}$: LCM of 10 and 14 is 70. $\frac{7}{10} = \frac{49}{70}$; $\frac{9}{14} = \frac{45}{70}$. Since $49 > 45$, $\frac{49}{70} > \frac{45}{70}$, so $\frac{7}{10} > \frac{9}{14}$.

d. $\frac{12}{5}$ and $\frac{8}{5}$: The denominators are the same. We compare the numerators. Since $12 > 8$, $\frac{12}{5} > \frac{8}{5}$.

e. $\frac{9}{4}$ and $\frac{5}{2}$: LCM of 4 and 2 is 4. $\frac{9}{4}$ stays the same; $\frac{5}{2} = \frac{10}{4}$. Since $9 < 10$, $\frac{9}{4} < \frac{10}{4}$, so $\frac{9}{4} < \frac{5}{2}$.

Final Answer: a. $\frac{8}{3} > \frac{5}{2}$ b. $\frac{4}{9} > \frac{3}{7}$ c. $\frac{7}{10} > \frac{9}{14}$ d. $\frac{12}{5} > \frac{8}{5}$ e. $\frac{9}{4} < \frac{5}{2}$

To Do: Arrange the given sets of fractions in ascending (smallest to largest) order.

Solution: a. $\frac{7}{10}, \frac{11}{15}, \frac{2}{5}$ The LCM of 10, 15, and 5 is 30. $\frac{7}{10} = \frac{21}{30}$; $\frac{11}{15} = \frac{22}{30}$; $\frac{2}{5} = \frac{12}{30}$. Comparing the numerators: $12 < 21 < 22$. So the order is $\frac{12}{30}, \frac{21}{30}, \frac{22}{30}$. The ascending order is $\frac{2}{5}, \frac{7}{10}, \frac{11}{15}$.

b. $\frac{19}{24}, \frac{5}{6}, \frac{7}{12}$ The LCM of 24, 6, and 12 is 24. $\frac{19}{24}$ stays the same; $\frac{5}{6} = \frac{20}{24}$; $\frac{7}{12} = \frac{14}{24}$. Comparing the numerators: $14 < 19 < 20$. So the order is $\frac{14}{24}, \frac{19}{24}, \frac{20}{24}$. The ascending order is $\frac{7}{12}, \frac{19}{24}, \frac{5}{6}$.

Final Answer: a. $\frac{2}{5}, \frac{7}{10}, \frac{11}{15}$ b. $\frac{7}{12}, \frac{19}{24}, \frac{5}{6}$

To Do: Arrange the given sets of fractions in descending (largest to smallest) order.

Solution: a. $\frac{25}{16}, \frac{7}{8}, \frac{13}{4}, \frac{17}{32}$ The LCM of 16, 8, 4, and 32 is 32. $\frac{25}{16} = \frac{50}{32}$; $\frac{7}{8} = \frac{28}{32}$; $\frac{13}{4} = \frac{104}{32}$; $\frac{17}{32}$ stays the same. Comparing the numerators: $104 > 50 > 28 > 17$. So the order is $\frac{104}{32}, \frac{50}{32}, \frac{28}{32}, \frac{17}{32}$. The descending order is $\frac{13}{4}, \frac{25}{16}, \frac{7}{8}, \frac{17}{32}$.

b. $\frac{3}{4}, \frac{12}{5}, \frac{7}{12}, \frac{5}{4}$ The LCM of 4, 5, and 12 is 60. $\frac{3}{4} = \frac{45}{60}$; $\frac{12}{5} = \frac{144}{60}$; $\frac{7}{12} = \frac{35}{60}$; $\frac{5}{4} = \frac{75}{60}$. Comparing the numerators: $144 > 75 > 45 > 35$. So the order is $\frac{144}{60}, \frac{75}{60}, \frac{45}{60}, \frac{35}{60}$. The descending order is $\frac{12}{5}, \frac{5}{4}, \frac{3}{4}, \frac{7}{12}$.