Key Concept: Fractional Units

b) $\frac{1}{2}$
[Solution Description]
Total chocolates = 5. Total children = 10. Each child gets $\frac{5}{10}$ of a chocolate. Simplifying $\frac{5}{10}$ gives $\frac{1}{2}$. So, each child gets $\frac{1}{2}$ chocolate.

Key Concept: Comparing Fractional Units

b) $\frac{1}{6}$
[Solution Description]
When comparing unit fractions, the one with the larger denominator is smaller. Here, 6 > 4, so $\frac{1}{6} < \frac{1}{4}$.

Key Concept: Comparing Fractions

a) $\frac{1}{2}$
[Solution Description]
To compare $\frac{1}{2}$ and $\frac{1}{5}$, observe that when a whole is divided among fewer people, each share is larger. Here, sharing between 2 people ($\frac{1}{2}$) gives a larger share than sharing among 5 people ($\frac{1}{5}$). Thus, $\frac{1}{2} > \frac{1}{5}$.

Key Concept: Equivalent Fractions, Simplest Form

b) $\frac{1}{2}$
[Solution Description]
Simplify $\frac{4}{8}$ by dividing numerator and denominator by 4 to get $\frac{1}{2}$. This is the simplest form.

Key Concept: Comparison of Unit Fractions

d) $\frac{1}{2}$
[Solution Description]
For unit fractions, the smaller the denominator, the larger the fraction. Among $\frac{1}{5}, \frac{1}{8}, \frac{1}{10}, \frac{1}{2}$, the smallest denominator is 2 in $\frac{1}{2}$. Therefore, $\frac{1}{2}$ is the largest fraction.

Key Concept: Unit Fractions - Equal Sharing

c) $\frac{1}{5}$
[Solution Description]
When a whole pizza is divided equally among 5 friends, each friend gets an equal share of $\frac{1}{5}$ of the pizza.
Therefore, each friend receives $\frac{1}{5}$ of the pizza.

Key Concept: Comparing Unit Fractions

a) $\frac{1}{6}$
[Solution Description]
Comparing $\frac{1}{6}$ and $\frac{1}{8}$: The denominators are 6 and 8. Since 6 is less than 8, $\frac{1}{6}$ is greater than $\frac{1}{8}$ because a smaller denominator means larger fractional units when divided equally.

Key Concept: Comparing Unit Fractions

d) $\frac{1}{9}$
[Solution Description]
To compare unit fractions, the larger the denominator, the smaller the fraction. Among the choices $\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}$, the largest denominator is 9, making $\frac{1}{9}$ the smallest fraction.

Key Concept: Fractional Units as Parts of a Whole

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that dividing a whole chikki into 6 equal parts means each part is $\frac{1}{6}$ of the chikki, regardless of its shape. This is correct because fractional units like $\frac{1}{6}$ represent equal divisions of a whole. The Reason explains that fractional units denote equal parts of a whole, regardless of their physical form. This is also true and directly explains why the Assertion is correct.

Key Concept: Fractional Units as Parts of a Whole

b) $\frac{2}{3}$
[Solution Description]
Total chocolates = 2. Number of friends = 3. Each friend gets $\frac{2}{3}$ of the total chocolate.

Key Concept: Visualizing Fractions

c) $\frac{1}{6}$
[Solution Description]
The whole chikki is divided into 6 equal parts. Each part represents $\frac{1}{6}$ of the chikki.

Key Concept: Visualizing Fractions Using Chikkis

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that dividing a whole chikki into 6 equal parts gives each part as $\frac{1}{6}$ of the chikki. This is true because fractions represent equal parts of a whole. The reason explains that the size of each part remains $\frac{1}{6}$ even if the shapes of the pieces differ, as long as they are equal in size. Both statements are correct, and the reason correctly explains why the assertion holds true irrespective of the shape of the pieces.

Key Concept: Equivalent Fractions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if $\frac{1}{3}$ and $\frac{2}{6}$ are equivalent, we can simplify $\frac{2}{6}$. Dividing numerator and denominator by 2 gives $\frac{1}{3}$, which matches the first fraction. This shows both fractions represent the same quantity. The reason correctly explains the assertion because equivalent fractions express the same value in different forms of fractional units.

Key Concept: Fractional Units

b) $\frac{1}{2}$
[Solution Description]
Total slices = 6. Slices eaten = 3. Fraction eaten = $\frac{3}{6} = \frac{1}{2}$. So, the fraction of the pizza eaten is $\frac{1}{2}$ when measured in half units.

Key Concept: Reading Fractions

b) Five sixths
[Solution Description]
The fraction $\frac{5}{6}$ can be read as "five sixths" or "five upon six."

Key Concept: Identifying fractions from visuals

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]

Key Concept: Reading Fractions

c) 3 times $\frac{1}{8}$
[Solution Description]
To emphasize the fractional units, $\frac{3}{8}$ is read as "3 times $\frac{1}{8}$" because it consists of three parts of $\frac{1}{8}$ each.

Key Concept: Measuring Using Fractional Units

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that folding a unit-length strip into two equal parts gives each part a length of $\frac{1}{2}$ units. This is correct because dividing 1 unit into two equal parts means each part is $\frac{1}{2}$ units long. The reason explains why this happens: folding the strip into two equal parts divides it into two halves. The reason correctly explains the assertion, as dividing the unit into two equal parts results in two parts, each being $\frac{1}{2}$ of the original unit.

Key Concept: Fractional units, Paper folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let's understand the initial assertion:
- The whole strip is considered as 1 unit.
- When folded into three equal parts, each part represents $\frac{1}{3}$ of the strip.
- Now, if we take one of these $\frac{1}{3}$ parts and fold it into two equal parts, each of these smaller parts will be half of $\frac{1}{3}$.
Mathematically:
$\text{New part size} = \frac{1}{3} \div 2 = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}.$
This confirms that the Assertion (A) is true.
Next, let's examine the Reason (R):
- The Reason states the mathematical operation explicitly, showing why the Assertion holds true.
- It correctly explains that dividing $\frac{1}{3}$ by 2 yields $\frac{1}{6}$, which directly supports the Assertion.
Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Equivalent fractions

b) $\frac{3}{6}$
[Solution Description]
From the syllabus, we know that $\frac{1}{2}$ is equivalent to $\frac{2}{4}$ and $\frac{4}{8}$. Check each option to see which matches these equivalents.

Key Concept: Fractional Measurement

b) $\frac{5}{8}$
[Solution Description]
The whole rope is represented as $\frac{8}{8}$. If 3 parts are used, the fraction used is $\frac{3}{8}$. Subtract the used fraction from the whole to get the unused fraction: $\frac{8}{8} - \frac{3}{8} = \frac{5}{8}$.
The unused part of the rope is $\frac{5}{8}$.

Key Concept: Equivalent fractions

b) $\frac{4}{6}$
[Solution Description]
To find a fraction equivalent to $\frac{2}{3}$, we can multiply numerator and denominator by the same number. Here, multiplying both by 2 gives $\frac{4}{6}$, which is equivalent to $\frac{2}{3}$.

Key Concept: Fractional units on number line

c) At $\frac{5}{6}$
[Solution Description]
The number line is divided into 6 equal parts, so each part represents $\frac{1}{6}$. To find the position for $5 \times \frac{1}{6}$, multiply $\frac{1}{6}$ by 5: $5 \times \frac{1}{6} = \frac{5}{6}$. This means the fifth mark from zero will represent $\frac{5}{6}$.

Key Concept: Fractional Units, Measurement, Number Line

c) At $\frac{5}{8}$ units
[Solution Description]
The number line is divided into 8 equal parts, so each part represents $\frac{1}{8}$ unit. To find where you land after moving $\frac{5}{8}$ units, multiply the number of parts by the size of each part: $5 \times \frac{1}{8} = \frac{5}{8}$. Thus, you will land exactly $\frac{5}{8}$ units from the starting point.

Key Concept: Numerator and Denominator

b) 5
[Solution Description]
A fraction has two main parts: numerator and denominator.
For the fraction $\frac{3}{5}$:
The numerator is the top number (3).
The denominator is the bottom number (5), which represents the total number of equal parts in the whole.
Thus, the denominator is 5.

Key Concept: Reading Fractions

b) 3
[Solution Description]
The numerator is the top part of a fraction. In $\frac{3}{4}$, 3 is the numerator.

Key Concept: Fractions on Number Line, Comparing Fractions

b) $\frac{3}{8}$
[Solution Description]
Convert $\frac{1}{4}$ to eighths: $\frac{1}{4} = \frac{2}{8}$. Subtract the starting point from the ending point: $\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$. Thus, the length of the red line is $\frac{3}{8}$.

Key Concept: Marking Fractions on the Number Line

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that when a unit length is divided into 4 equal parts, each part's length is $\frac{1}{4}$ unit. This is true because dividing the unit equally means each part has the same length, which is $\frac{1}{4}$ unit. The Reason explains the general case: dividing any unit length into $n$ equal parts gives each part as $\frac{1}{n}$ unit, which correctly justifies the Assertion. Hence, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Fraction subtraction, equivalent fractions

b) $\frac{7}{10}$
[Solution Description]
Convert $\frac{4}{5}$ to tenths: $\frac{4}{5} = \frac{8}{10}$. Moving back by $\frac{1}{10}$ gives $\frac{8}{10} - \frac{1}{10} = \frac{7}{10}$.

Key Concept: Marking Fractions on the Number Line

b) $\frac{1}{4}$
[Solution Description]
The distance between 0 and 1 is divided into 4 equal parts. Each part represents an equal fraction of the unit length. So, the length of the first part from 0 is $\frac{1}{4}$.

Key Concept: Identifying fractions on the number line

c) $\frac{5}{6}$
[Solution Description]
When the unit length is divided into 6 equal parts, each division represents $\frac{1}{6}$. The fifth division from 0 would therefore be $\frac{5}{6}$.

Key Concept: Fraction Representation on Number Line

b) $\frac{1}{3}$
[Solution Description]
The distance between 0 and 1 is 1 unit. It is divided into 3 equal parts. So, the length of each part is $\frac{1}{3}$ unit. Therefore, the length of the first part from 0 is $\frac{1}{3}$.

Key Concept: Fraction Placement

c) At the fifth mark from 0
[Solution Description] Dividing the interval from 0 to 1 into 8 equal parts means each part represents $\frac{1}{8}$. $\frac{5}{8}$ is the fifth mark from 0.

Key Concept: Visualizing Fractions

b) On the 6th mark
[Solution Description]
To mark $\frac{3}{4}$ on a number line divided into 8 equal parts:
1. Recognize that $\frac{3}{4} = \frac{6}{8}$ when converted to eighths.
2. Count 6 segments from 0 since each segment represents $\frac{1}{8}$.
3. The sixth mark corresponds to $\frac{6}{8}$ or $\frac{3}{4}$.

Key Concept: Mixed Fractions, Conversion between mixed numbers and improper fractions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's verify the assertion first. To convert $4 \frac{3}{8}$ to an improper fraction, follow these steps:
Step 1: Multiply the whole number (4) by the denominator of the fractional part (8).
$4 \times 8 = 32$
Step 2: Add the numerator (3) to this product.
$32 + 3 = 35$
Step 3: Place the result over the original denominator (8).
$\frac{35}{8}$
This matches the assertion, so the assertion is true.
Now, let's check the reason. The reason correctly describes the method for converting a mixed number to an improper fraction. Therefore, the reason is also true.
Furthermore, the reason logically explains the assertion because it outlines the exact process used to derive $\frac{35}{8}$ from $4 \frac{3}{8}$.

Key Concept: Mixed Fractions and Improper Fractions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $\frac{9}{4}$ can be written as a mixed number. To verify this, we perform the division of 9 by 4:
- 4 goes into 9 two times (4 × 2 = 8), leaving a remainder of 1.
- Therefore, $\frac{9}{4} = 2 + \frac{1}{4} = 2\frac{1}{4}$. This confirms that the assertion is true.
The reason explains what a mixed number is—a combination of a whole number and a proper fraction (where the numerator is less than the denominator). Since $\frac{1}{4}$ is a proper fraction (1 < 4), the reason is also true.
Additionally, the reason directly explains why $\frac{9}{4}$ can be expressed as a mixed number because it fits the definition provided. Thus, the reason correctly explains the assertion.

Key Concept: Identifying whole units in a fraction

b) 2
[Solution Description]
To find the number of whole units in $\frac{11}{5}$, divide the numerator by the denominator.
1. Divide 11 by 5: $11 \div 5 = 2$ with a remainder of 1.
2. The quotient 2 represents the whole units, and the remainder $\frac{1}{5}$ is the fractional part.
So, $\frac{11}{5} = 2 \frac{1}{5}$ has 2 whole units.

Key Concept: Mixed Fractions, Fractions greater than 1

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To convert $\frac{47}{9}$ into a mixed number:
1. Divide the numerator by the denominator: $47 \div 9 = 5$ with a remainder of $2$.
2. The quotient ($5$) becomes the whole number part.
3. The remainder ($2$) becomes the numerator of the fractional part, and the denominator remains $9$.
Thus, $\frac{47}{9} = 5 \frac{2}{9}$.
The assertion is true because $5 \frac{2}{9}$ correctly represents $\frac{47}{9}$ as a mixed number. The reason is also true because a mixed number indeed consists of a whole number and a proper fraction (where the numerator is less than the denominator). Moreover, the reason correctly explains why $\frac{47}{9}$ can be written as $5 \frac{2}{9}$.

Key Concept: Multi-concept: Mixed Fractions, Addition

b) 4$\frac{1}{6}$ cups
[Solution Description]
1. Convert $2\frac{1}{3}$ to improper fraction: $2 \times 3 + 1 = 7$ → $\frac{7}{3}$.
2. Convert $1\frac{5}{6}$ to improper fraction: $1 \times 6 + 5 = 11$ → $\frac{11}{6}$.
3. Find a common denominator (6): $\frac{14}{6} + \frac{11}{6} = \frac{25}{6}$.
4. Convert back to mixed number: $25 \div 6 = 4$ with remainder 1 → $4\frac{1}{6}$.

Key Concept: Converting improper fraction to mixed number

b) 2
[Solution Description]
To find the whole units in $\frac{11}{4}$, we divide 11 by 4.
4 goes into 11 two times (since $4 \times 2 = 8$).
The remainder is $11 - 8 = 3$.
So, $\frac{11}{4} = 2 + \frac{3}{4}$.
Therefore, there are 2 whole units in $\frac{11}{4}$.

Key Concept: Converting mixed number to improper fraction

d) $\frac{13}{4}$
[Solution Description]
To convert $3 \frac{1}{4}$ to an improper fraction, multiply the whole number by the denominator and add the numerator: $(3 \times 4) + 1 = 12 + 1 = 13$. The new numerator is $13$, so the improper fraction is $\frac{13}{4}$.

Key Concept: Converting improper fractions to mixed numbers, Multi-step problem

b) $\frac{5}{6}$
[Solution Description]
To convert $\frac{29}{6}$ to a mixed number:
Step 1: Divide 29 by 6. 6 goes into 29 four times (since $6 \times 4 = 24$).
Step 2: Subtract 24 from 29 to get the remainder: $29 - 24 = 5$.
Step 3: The whole part is 4 and the fractional part is $\frac{5}{6}$. Thus, the mixed number is $4 \frac{5}{6}$.

Key Concept: Converting mixed numbers to improper fractions, Combining concepts

a) $\frac{48}{8}$
[Solution Description]
Step 1: Convert both mixed numbers to improper fractions.
For $2 \frac{3}{8}$: $2 \times 8 + 3 = 19$, so $\frac{19}{8}$.
For $3 \frac{5}{8}$: $3 \times 8 + 5 = 29$, so $\frac{29}{8}$.
Step 2: Add the two fractions: $\frac{19}{8} + \frac{29}{8} = \frac{48}{8}$.
Step 3: Simplify $\frac{48}{8} = 6$.

Key Concept: Converting mixed fractions to improper fractions, conceptual complexity

b) $\frac{25}{9}$
[Solution Description]
To convert $2 \frac{7}{9}$ to an improper fraction:
Step 1: Multiply the whole number by the denominator: $2 \times 9 = 18$.
Step 2: Add the numerator to the result: $18 + 7 = 25$.
Step 3: Write the result over the original denominator: $\frac{25}{9}$.

Key Concept: Mixed to Improper Fraction Conversion

b) $\frac{13}{5}$
[Solution Description]
To convert $2 \frac{3}{5}$ to an improper fraction, follow these steps:
1. Multiply the whole number part (2) by the denominator of the fractional part (5):
$2 \times 5 = 10$.
2. Add the result to the numerator of the fractional part (3):
$10 + 3 = 13$.
3. Place this sum over the original denominator (5):
$\frac{13}{5}$.
Therefore, $2 \frac{3}{5} = \frac{13}{5}$.

Key Concept: Mixed to Improper Fraction Conversion

b) $\frac{25}{6}$
[Solution Description]
To convert $4 \frac{1}{6}$ to an improper fraction, follow these steps:
1. Multiply the whole number part (4) by the denominator of the fractional part (6):
$4 \times 6 = 24$.
2. Add the result to the numerator of the fractional part (1):
$24 + 1 = 25$.
3. Place this sum over the original denominator (6):
$\frac{25}{6}$.
Therefore, $4 \frac{1}{6} = \frac{25}{6}$.

Key Concept: Equivalent Fractions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that $\frac{3}{4}$ and $\frac{9}{12}$ are equivalent fractions. To verify this, we can simplify $\frac{9}{12}$ to its lowest terms:
Step 1: Find the greatest common divisor (GCD) of 9 and 12, which is 3.
Step 2: Divide both the numerator and denominator by 3:
$\frac{9 \div 3}{12 \div 3} = \frac{3}{4}$.
Since $\frac{9}{12}$ simplifies to $\frac{3}{4}$, they are indeed equivalent fractions.
The Reason states that multiplying the numerator and denominator of a fraction by the same non-zero number results in an equivalent fraction. This is a fundamental property of equivalent fractions and explains why $\frac{3}{4}$ and $\frac{9}{12}$ are equivalent (since $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$).
Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Comparing Fractions

c) $\frac{4}{5} > \frac{7}{9}$
[Solution Description]
Find equivalent fractions with the same denominator to compare $\frac{4}{5}$ and $\frac{7}{9}$. The least common denominator (LCD) is 45. Thus, $\frac{4}{5} = \frac{36}{45}$ and $\frac{7}{9} = \frac{35}{45}$. Since $36 > 35$, $\frac{4}{5} > \frac{7}{9}$.

Key Concept: Equivalent Fractions, Simplest Form

c) 3
[Solution Description]
First, simplify $\frac{24}{36}$ to its simplest form by dividing numerator and denominator by their greatest common divisor (GCD). GCD of 24 and 36 is 12. So, $\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}$. Thus, $\frac{c}{d} = \frac{2}{3}$. Now, since $\frac{a}{b}$ is equivalent to $\frac{24}{36}$, it means $\frac{a}{b} = \frac{k \times 24}{k \times 36} = \frac{2k}{3k}$ for some integer $k$. The difference between $b$ and $d$ is $3k - 3 = 3(k - 1)$. The smallest possible positive difference occurs when $k = 2$, giving $3(2 - 1) = 3$.

Key Concept: Equivalent Fractions, Fraction Walls

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if two fractions have equal lengths on a fraction wall, they are equivalent fractions. This is true because equivalent fractions denote the same quantity using different fractional units, which is visually represented by equal lengths on a fraction wall. The reason explains why this is the case: equivalent fractions represent the same portion of the whole but with different denominators (fractional units). Therefore, the reason correctly explains the assertion.

Key Concept: Fractional lengths, Equivalent fractions

b) 6
[Solution Description]
First, find how many $\frac{1}{12}$ pieces make 1 unit: $12$ pieces.
Next, since $\frac{1}{2}$ is half of 1 unit, the number of $\frac{1}{12}$ pieces in $\frac{1}{2}$ is $\frac{12}{2} = 6$.
So, 6 pieces of $\frac{1}{12}$ make $\frac{1}{2}$.

Key Concept: Creating Equivalent Fractions

d) $\frac{5}{10}$
[Solution Description]
To find which fraction is not equivalent to $\frac{2}{3}$, we can simplify each option or find a common denominator.
Step 1: Simplify each option:
- $\frac{4}{6} = \frac{2}{3}$
- $\frac{6}{9} = \frac{2}{3}$
- $\frac{8}{12} = \frac{2}{3}$
- $\frac{5}{10} = \frac{1}{2}$
Step 2: Compare simplified forms to $\frac{2}{3}$. Only $\frac{5}{10}$ simplifies to $\frac{1}{2}$, which is not equal to $\frac{2}{3}$.
The correct answer is $\frac{5}{10}$.

Key Concept: Equivalent Fractions, Real-life Comparisons

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let's verify the assertion. When 5 rotis are divided equally among 10 children, each child gets $\frac{5}{10}$ of a roti. Simplifying $\frac{5}{10}$ by dividing numerator and denominator by 5 gives $\frac{1}{2}$. So, the assertion is true. Now, checking the reason: $\frac{5}{10}$ simplifies to $\frac{1}{2}$, proving they are equivalent fractions representing equal shares. The reason correctly explains the assertion.

Key Concept: Comparing Fractions

a) Group 1 provides a larger share per child
[Solution Description]
First, find the share per child for both groups. For Group 1, it's $\frac{3}{4}$ cookie per child. For Group 2, it's $\frac{5}{8}$ cookie per child. To compare these fractions, convert them to equivalent fractions with a common denominator. The least common denominator (LCD) of 4 and 8 is 8. Convert $\frac{3}{4}$ to $\frac{6}{8}$. Now compare $\frac{6}{8}$ and $\frac{5}{8}$. Clearly, $\frac{6}{8} > \frac{5}{8}$, so Group 1 provides a larger share per child.

Key Concept: Equivalent Fractions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: Check if $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{3}{6}$ represent equal shares.
- $\frac{1}{2}$ means 1 roti divided among 2 children, each gets $\frac{1}{2}$.
- $\frac{2}{4}$ means 2 rotis divided among 4 children, each gets $\frac{1}{2}$.
- $\frac{3}{6}$ means 3 rotis divided among 6 children, each gets $\frac{1}{2}$.
Step 2: Confirm that equivalent fractions represent the same value despite having different numerators and denominators.
Thus, the Assertion is true, and the Reason correctly explains it.

Key Concept: Equivalent Fractions, Writing fractions in lowest terms

b) 36 and 84
[Solution Description]
First, reduce $\frac{36}{84}$ to its lowest terms:
Find the GCD of 36 and 84. The prime factors of 36 are $2^2 \times 3^2$, and the prime factors of 84 are $2^2 \times 3 \times 7$. The GCD is $2^2 \times 3 = 12$. Now divide numerator and denominator by 12:
$\frac{36 \div 12}{84 \div 12} = \frac{3}{7}$.
Since $\frac{a}{b}$ is equivalent to $\frac{3}{7}$, we can write $a = 3k$ and $b = 7k$ for some integer $k$. Given that the GCD of $a$ and $b$ is 12, the GCD of $3k$ and $7k$ must be $k$ (since 3 and 7 are co-prime). Thus, $k = 12$. Therefore, $a = 3 \times 12 = 36$ and $b = 7 \times 12 = 84$.

Key Concept: Equivalent Fractions

d) $\frac{6}{7}$
[Solution Description]
The HCF of 126 and 147 is 21. Divide both numerator and denominator by 21:
$\frac{126 \div 21}{147 \div 21} = \frac{6}{7}$
The simplest form is $\frac{6}{7}$.

Key Concept: Equivalent Fractions

b) $\frac{2}{3}$
[Solution Description]
To simplify $\frac{24}{36}$, we find the highest common factor (HCF) of 24 and 36. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The HCF is 12. Now, divide both the numerator and denominator by 12: $\frac{24 \div 12}{36 \div 12} = \frac{2}{3}$. Therefore, the simplest form is $\frac{2}{3}$.

Key Concept: Ordering Fractions

a) $\frac{7}{12} < \frac{3}{4} < \frac{5}{6}$
[Solution Description]
Find a common denominator for $\frac{3}{4}, \frac{5}{6}, \frac{7}{12}$. The least common multiple of 4, 6, and 12 is 12. Convert them: $\frac{3}{4} = \frac{9}{12}$, $\frac{5}{6} = \frac{10}{12}$, and $\frac{7}{12}$ remains the same. Now compare numerators: 7 < 9 < 10. Thus, the ascending order is $\frac{7}{12} < \frac{3}{4} < \frac{5}{6}$.

Key Concept: Comparing Fractions

c) $\frac{5}{6} > \frac{7}{9}$
[Solution Description]
To compare $\frac{5}{6}$ and $\frac{7}{9}$, first find a common denominator. The least common multiple of 6 and 9 is 18. Convert both fractions to equivalent fractions with denominator 18: $\frac{5}{6} = \frac{15}{18}$ and $\frac{7}{9} = \frac{14}{18}$. Comparing numerators, 15 > 14, so $\frac{5}{6} > \frac{7}{9}$.

Key Concept: Comparing Fractions

b) $\frac{5}{2} < \frac{8}{3}$
[Solution Description]
To compare $\frac{8}{3}$ and $\frac{5}{2}$, we find equivalent fractions with a common denominator. The least common multiple of 3 and 2 is 6.
$\frac{8}{3} = \frac{8 \times 2}{3 \times 2} = \frac{16}{6}$
$\frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6}$
Comparing the numerators, $15 < 16$, so $\frac{5}{2} < \frac{8}{3}$.

Key Concept: Comparing fractions, Ascending order

a) $\frac{7}{12}, \frac{5}{8}, \frac{2}{3}, \frac{3}{4}$
[Solution Description]
To arrange the fractions in ascending order, we first find a common denominator. The LCM of 12, 8, 4, and 3 is 24. Convert each fraction:
$\frac{7}{12} = \frac{14}{24}$
$\frac{5}{8} = \frac{15}{24}$
$\frac{3}{4} = \frac{18}{24}$
$\frac{2}{3} = \frac{16}{24}$
Now, comparing the numerators: $\frac{14}{24} < \frac{15}{24} < \frac{16}{24} < \frac{18}{24}$.
Thus, the ascending order is: $\frac{7}{12}, \frac{5}{8}, \frac{2}{3}, \frac{3}{4}$.

Key Concept: Comparing Fractions with Same Denominator

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare two fractions with the same denominator, we only need to compare their numerators. For example, $\frac{5}{8}$ and $\frac{3}{8}$ have the same denominator 8. Since 5 > 3, it follows that $\frac{5}{8} > \frac{3}{8}$. The reason correctly explains the assertion because when denominators are identical, the size of the numerator determines the value of the fraction. Thus, the assertion and reason are both true, and the reason correctly explains the assertion.

Key Concept: Comparing Fractions with Same Denominator

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare $\frac{5}{12}$ and $\frac{3}{12}$, both fractions share the same denominator, 12. According to the rule of comparing fractions with the same denominator, the fraction with the larger numerator is greater. Here, 5 > 3, so $\frac{5}{12} > \frac{3}{12}$. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Comparing fractions, Equivalent fractions with LCM

c) $\frac{15}{18} > \frac{8}{18}$
[Solution Description]
To compare $\frac{5}{6}$ and $\frac{4}{9}$, we need to find equivalent fractions with a common denominator. The LCM of 6 and 9 is 18. Now, convert both fractions:
$\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$,
$\frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18}$.
Clearly, $\frac{15}{18} > \frac{8}{18}$. So, $\frac{5}{6} > \frac{4}{9}$.

Key Concept: Comparing Fractions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare $\frac{5}{6}$ and $\frac{7}{9}$, we find their equivalent fractions with a common denominator. The LCM of 6 and 9 is 18.
Step 1: Convert $\frac{5}{6}$ to an equivalent fraction with denominator 18:
$\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$
Step 2: Convert $\frac{7}{9}$ to an equivalent fraction with denominator 18:
$\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}$
Step 3: Compare the numerators of the equivalent fractions:
$15 > 14 \implies \frac{15}{18} > \frac{14}{18} \implies \frac{5}{6} > \frac{7}{9}$
Both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Comparing fractions, Equivalent fractions with LCM

c) $\frac{6}{14} > \frac{5}{14}$
[Solution Description]
To compare $\frac{3}{7}$ and $\frac{5}{14}$, the LCM of 7 and 14 is 14. Convert $\frac{3}{7}$ to a fraction with denominator 14:
$\frac{3}{7} = \frac{3 \times 2}{7 \times 2} = \frac{6}{14}$.
Now, compare $\frac{6}{14}$ and $\frac{5}{14}$: clearly $\frac{6}{14} > \frac{5}{14}$.

Key Concept: Direct Comparison of Fractions

b) $\frac{4}{7}$
[Solution Description]
To compare $\frac{3}{7}$ and $\frac{4}{7}$, we look at the numerators since the denominators are the same. The fraction with the larger numerator is greater. Here, $4 > 3$, so $\frac{4}{7} > \frac{3}{7}$.

Key Concept: Comparing Fractions Using Reasoning

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, we need to verify if both the Assertion and Reason are true.
- **Assertion**: $\frac{5}{8} > \frac{1}{2}$ is correct because converting $\frac{1}{2}$ to eighths gives $\frac{4}{8}$, and $\frac{5}{8} > \frac{4}{8}$.
- **Reason**: The statement "When comparing fractions with the same denominator, the fraction with the larger numerator is greater" is indeed a valid rule for comparing fractions.
Now, we check if the Reason correctly explains the Assertion.
The Assertion compares $\frac{5}{8}$ and $\frac{1}{2}$, but they do not have the same denominator initially. The comparison is only valid after converting them to equivalent fractions with the same denominator ($\frac{5}{8}$ and $\frac{4}{8}$). Therefore, the Reason does not directly explain the Assertion because it assumes the denominators are already the same, which is not the case here without conversion.
Hence, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Cross-Multiplication for Comparing Fractions

c) $\frac{4}{9}$ is greater
[Solution Description]
To compare $\frac{4}{9}$ and $\frac{3}{7}$:
1. Cross-multiply: $4 \times 7 = 28$ and $3 \times 9 = 27$
2. Compare the products: $28 > 27$
3. Therefore, $\frac{4}{9} > \frac{3}{7}$
The fraction with the larger cross-product is greater.

Key Concept: Subtracting fractions with different denominators

b) $\frac{5}{12}$
[Solution Description]
To subtract $\frac{1}{3}$ from $\frac{3}{4}$, first find a common denominator. The LCM of 4 and 3 is 12.
Convert both fractions to equivalent fractions with denominator 12:
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}, \quad \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$
Now subtract them:
$\frac{9}{12} - \frac{4}{12} = \frac{5}{12}$
So, the result is $\frac{5}{12}$.

Key Concept: Addition of fractions with different denominators

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $\frac{1}{3} + \frac{1}{6}$, we need a common denominator. The least common multiple (LCM) of 3 and 6 is 6. Convert $\frac{1}{3}$ to an equivalent fraction with denominator 6: $\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$. Now add $\frac{2}{6} + \frac{1}{6} = \frac{3}{6}$. Simplify $\frac{3}{6}$ by dividing numerator and denominator by 3: $\frac{3 \div 3}{6 \div 3} = \frac{1}{2}$. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Subtraction of Fractions with Different Denominators

b) $\frac{1}{24}$
[Solution Description]
Total eaten initially: $\frac{7}{8}$, so remaining: $\frac{1}{8}$. Given away fraction from remaining: $\frac{2}{3} \times \frac{1}{8} = \frac{2}{24} = \frac{1}{12}$. Subtract from remaining: $\frac{1}{8} - \frac{1}{12}$. Common denominator is 24: $\frac{3}{24} - \frac{2}{24} = \frac{1}{24}$.

Key Concept: Adding fractions with the same denominator

c) $\frac{4}{10}$
[Solution Description]
The denominators are the same (10), so we add the numerators: $1 + 3 = 4$. The result is $\frac{4}{10}$. This can be simplified to $\frac{2}{5}$ by dividing numerator and denominator by 2. However, since the options are in terms of denominator 10, the correct answer is $\frac{4}{10}$.

Key Concept: Addition of Fractions with Same Denominator

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To add two fractions with the same denominator, we simply add their numerators and keep the denominator unchanged. Here, both fractions $\frac{3}{8}$ and $\frac{2}{8}$ have the same denominator 8. Adding the numerators gives $3 + 2 = 5$, so the sum is $\frac{5}{8}$. The reason correctly explains why this operation works, as it describes the rule for adding fractions with identical denominators.

Key Concept: Adding fractions with same denominator, Converting improper fractions to mixed numbers

b) $4$
[Solution Description]
Add the fractions: $\frac{9}{4} + \frac{7}{4} = \frac{16}{4}$. Simplify $\frac{16}{4}$ by dividing numerator and denominator by 4: $4$. However, if we express it as a mixed number, it would still be $4$ since $\frac{16}{4} = 4$ exactly.

Key Concept: Addition and subtraction of fractions with same denominator, Mixed operations

b) $\frac{3}{5}$
[Solution Description]
First, subtract the juice given away from the initial amount:
$\frac{7}{10} - \frac{3}{10} = \frac{4}{10}$
Then, add the newly bought juice:
$\frac{4}{10} + \frac{2}{10} = \frac{6}{10}$
Finally, simplify the fraction:
$\frac{6 \div 2}{10 \div 2} = \frac{3}{5}$

Key Concept: Addition of Fractions with Same Denominator

a) 1
[Solution Description]
Both fractions have the same denominator, so we add their numerators directly.
Step 1: Add the numerators: $5 + 1 = 6$.
Step 2: The denominator remains 6.
Hence, $\frac{5}{6} + \frac{1}{6} = \frac{6}{6} = 1$.

Key Concept: Adding fractions with same denominator

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $\frac{3}{7} + \frac{2}{7} = \frac{5}{7}$. To verify this, follow the rule mentioned in the reason: when two fractions have the same denominator, we simply add their numerators and keep the denominator unchanged. Here, 3 (numerator of first fraction) + 2 (numerator of second fraction) = 5, and the denominator remains 7. Thus, $\frac{3}{7} + \frac{2}{7} = \frac{5}{7}$, which matches the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Adding fractions with different denominators

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$. Let us verify this using Brahmagupta's method:
1. Find the LCM of 2 and 3, which is 6.
2. Convert the fractions to have denominator 6:
$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$
$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$
3. Add the converted fractions:
$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$
The assertion is correct. The reason explains the method used to add fractions with different denominators, and it is indeed the correct explanation for the assertion.

Key Concept: Concept Check: Simplifying fraction result

c) $\frac{1}{2}$
[Solution Description]
First, find a common denominator for $\frac{1}{6}$ and $\frac{1}{3}$. The least common multiple of 6 and 3 is 6.
Convert $\frac{1}{3}$:
$\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$.
Keep $\frac{1}{6}$ as it is since the denominator is already 6.
Add them: $\frac{1}{6} + \frac{2}{6} = \frac{3}{6}$.
Simplify $\frac{3}{6}$ by dividing numerator and denominator by 3:
$\frac{3 \div 3}{6 \div 3} = \frac{1}{2}$.
Final answer is $\frac{1}{2}$.

Key Concept: Adding three fractions with different denominators

b) $\frac{19}{20}$
[Solution Description]
To add $\frac{1}{4} + \frac{2}{5} + \frac{3}{10}$, first find a common denominator. The denominators are 4, 5, and 10. The least common multiple (LCM) of 4, 5, and 10 is 20.
Step 1: Convert $\frac{1}{4}$ to an equivalent fraction with denominator 20.
$\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$
Step 2: Convert $\frac{2}{5}$ to an equivalent fraction with denominator 20.
$\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$
Step 3: Convert $\frac{3}{10}$ to an equivalent fraction with denominator 20.
$\frac{3}{10} = \frac{3 \times 2}{10 \times 2} = \frac{6}{20}$
Step 4: Add the three fractions.
$\frac{5}{20} + \frac{8}{20} + \frac{6}{20} = \frac{19}{20}$

Key Concept: Subtracting Fractions with Different Denominators

c) $\frac{19}{40}$
[Solution Description]
Find the least common multiple (LCM) of the denominators 8 and 5. The LCM of 8 and 5 is 40.
Convert each fraction to an equivalent fraction with denominator 40:
$\frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40}$,
$\frac{2}{5} = \frac{2 \times 8}{5 \times 8} = \frac{16}{40}$.
Now subtract these fractions:
$\frac{35}{40} - \frac{16}{40} = \frac{19}{40}$.

Key Concept: Addition, Subtraction, Brahmagupta’s method

c) $\frac{1}{6}$
[Solution Description]
To find the remaining sugar, subtract the used amount from the original purchase. The denominators are 6 and 3. The LCM of 6 and 3 is 6. Convert both fractions to have denominator 6:
$\frac{5}{6} \text{ remains as } \frac{5}{6}, \quad \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}.$
Subtract the numerators: $\frac{5}{6} - \frac{4}{6} = \frac{1}{6}$.

Key Concept: Subtraction of fractions with same denominator, multi-step solution

c) $\frac{1}{3}$
[Solution Description]
First subtract $\frac{5}{15}$ from $\frac{12}{15}$: $\frac{12}{15} - \frac{5}{15} = \frac{7}{15}$. Then subtract $\frac{2}{15}$ from this result: $\frac{7}{15} - \frac{2}{15} = \frac{5}{15}$. Finally, simplify $\frac{5}{15}$ by dividing numerator and denominator by 5 to get $\frac{1}{3}$.

Key Concept: Subtraction of fractions with same denominator

b) $\frac{3}{11}$
[Solution Description]
To solve $\frac{12}{11} - \frac{9}{11}$, subtract the numerators since denominators are the same. So, $12 - 9 = 3$. Thus, $\frac{12}{11} - \frac{9}{11} = \frac{3}{11}$.

Key Concept: Fraction Operations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that adding $\frac{1}{2}$ and $\frac{1}{4}$ gives $\frac{3}{4}$, which is mathematically correct ($\frac{1}{2} = \frac{2}{4}$; $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$). The reason explains why visual methods work—they rely on equal divisions. Both are true, and the reason correctly justifies the assertion.

Key Concept: Multi-step fraction operations

d) $1 \frac{4}{5}$ km
[Solution Description]
Add all distances: $\frac{3}{5} + \frac{1}{2} + \frac{7}{10}$.
Find LCM of 5, 2, and 10, which is 10.
Rewrite fractions: $\frac{3}{5} = \frac{6}{10}$, $\frac{1}{2} = \frac{5}{10}$, $\frac{7}{10}$ remains the same.
Add numerators: $\frac{6}{10} + \frac{5}{10} + \frac{7}{10} = \frac{18}{10}$.
Simplify: $\frac{18}{10} = 1 \frac{8}{10} = 1 \frac{4}{5}$ km.

Key Concept: Subtraction of Fractions with Different Denominators

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $\frac{5}{6} - \frac{1}{4}$, we follow Brahmagupta's method:
Step 1: Find LCM of 6 and 4, which is 12.
Step 2: Convert $\frac{5}{6}$ to an equivalent fraction with denominator 12: $\frac{5 \times 2}{6 \times 2} = \frac{10}{12}$.
Step 3: Convert $\frac{1}{4}$ to an equivalent fraction with denominator 12: $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$.
Step 4: Subtract the numerators: $\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$.
Thus, both Assertion (A) and Reason (R) are true, and R correctly explains A.

Key Concept: Subtraction of fractions with different denominators

c) $\frac{41}{84}$
[Solution Description]
For subtracting $\frac{11}{12}$ and $\frac{3}{7}$, the least common denominator (LCD) of 12 and 7 is 84.
Convert $\frac{11}{12}$ to an equivalent fraction with denominator 84:
$\frac{11}{12} = \frac{11 \times 7}{12 \times 7} = \frac{77}{84}$
Convert $\frac{3}{7}$ to an equivalent fraction with denominator 84:
$\frac{3}{7} = \frac{3 \times 12}{7 \times 12} = \frac{36}{84}$
Perform the subtraction:
$\frac{77}{84} - \frac{36}{84} = \frac{41}{84}$
Thus, the answer is $\frac{41}{84}$.

Key Concept: Subtraction of Fractions

b) $\frac{11}{40}$ liters
[Solution Description]
Subtract $\frac{3}{5}$ from $\frac{7}{8}$ to find remaining milk. Convert to equivalent fractions with denominator 40:
$\frac{7}{8} = \frac{35}{40}$, $\frac{3}{5} = \frac{24}{40}$.
Milk left = $\frac{35}{40} - \frac{24}{40} = \frac{11}{40}$ liters.

Key Concept: Addition and Subtraction of Fractions

d) Assertion is false, but Reason is true.
[Solution Description]
To solve this problem, we need to subtract $\frac{5}{6}$ from $\frac{3}{4}$.
Step 1: Find a common denominator for $\frac{3}{4}$ and $\frac{5}{6}$. The least common denominator (LCD) for 4 and 6 is 12.
Step 2: Convert both fractions to have the denominator 12:
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
Step 3: Subtract the second fraction from the first:
$\frac{9}{12} - \frac{10}{12} = \frac{-1}{12}$
The Assertion states that the remaining ribbon is $\frac{-1}{12}$ meters, which is mathematically correct based on the calculation above. However, negative lengths are not meaningful in practical contexts, indicating an error in the original subtraction setup or assumptions.
The Reason correctly explains the process of converting fractions to the same denominator before subtraction, which is fundamental when performing such operations.

Key Concept: Fraction Notation

d) Bakshali manuscript
[Solution Description]
The question asks about the ancient Indian manuscript that introduced fraction notation similar to today's format. As per the syllabus, it was the Bakshali manuscript around 300 CE.

Key Concept: Egyptian Fractions

c) As sums of fractional units
[Solution Description]
Ancient Egyptians used sums of fractional units (fractions with numerator 1) to represent general fractions, known as Egyptian fractions. For example, $\frac{19}{24}$ was written as $\frac{1}{2} + \frac{1}{6} + \frac{1}{8}$.

Key Concept: Ancient Indian Mathematical Texts

c) Bakshali manuscript
[Solution Description]
The Bakshali manuscript, dating back to around 300 CE, is one of the earliest known Indian mathematical texts to use notations for fractions similar to modern-day representations. For instance, $\frac{1}{2}$ was written as it appears today.

Key Concept: Transmission of Indian Fractions

d) Via the Arabs
[Solution Description]
Indian mathematical knowledge, including the concept of fractions, was transmitted to Europe via Arab scholars over several centuries. This exchange occurred during the medieval period, leading to widespread adoption in Europe by the 17th century.

Key Concept: Ancient notation, Egyptian fractions

b) $\frac{1}{2} + \frac{1}{12}$
[Solution Description]
To express $\frac{7}{12}$ as an Egyptian fraction, we need to break it into unit fractions (fractions with numerator 1).
First, find the largest unit fraction less than $\frac{7}{12}$, which is $\frac{1}{2}$.
Subtract this from $\frac{7}{12}$: $\frac{7}{12} - \frac{1}{2} = \frac{7}{12} - \frac{6}{12} = \frac{1}{12}$.
Thus, $\frac{7}{12} = \frac{1}{2} + \frac{1}{12}$.

Key Concept: Historical context, Fraction addition

c) $\frac{19}{12}$
[Solution Description]
First, find a common denominator for $\frac{3}{4}$ and $\frac{5}{6}$. The least common multiple of 4 and 6 is 12. Now convert both fractions to have denominator 12:
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$,
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$.
Add the numerators: $9 + 10 = 19$.
So the sum is $\frac{19}{12}$.

Key Concept: Fractions, Brahmagupta's Contributions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that Brahmagupta codified the method of adding and subtracting fractions by reducing them to a common denominator in the 7th century CE. This is true as per historical records, specifically mentioned in his work *Brahmasphuṭasiddhānta*. The reason states that these rules were transmitted to Europe via Arab mathematicians, which is also historically accurate, as Arabic scholars like Al-Hassar played a key role in transmitting Indian mathematical knowledge to Europe. Moreover, the reason correctly explains how Brahmagupta's contributions became globally influential.

Key Concept: Fraction representation, Egyptian fractions

c) $\frac{1}{2} + \frac{1}{4}$
[Solution Description]
The Egyptian fraction system represents general fractions as sums of unit fractions (fractions with numerator 1). To represent $\frac{3}{4}$, we look for unit fractions that sum up to it. The correct representation is $\frac{1}{2} + \frac{1}{4}$ because: $\frac{1}{2} = \frac{2}{4}$, so $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$.