Visual representation of a fraction with 3 out of 8 parts shaded.

Let's solve each question step by step from the worksheet.

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(1) Add the following fractions and reduce them to the simplest form:

We'll solve each part:

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#### A) $\frac{11}{12} + \frac{4}{9}$

- Find LCM of 12 and 9 → LCM = 36
- Convert:
- $\frac{11}{12} = \frac{11 \times 3}{12 \times 3} = \frac{33}{36}$
- $\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}$
- Add: $\frac{33}{36} + \frac{16}{36} = \frac{49}{36}$
- Simplify: $\frac{49}{36}$ is already in simplest form (since 49 and 36 have no common factors other than 1)
- Answer: $\boxed{\frac{49}{36}}$ or $1\frac{13}{36}$

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#### B) $\frac{23}{24} + \frac{17}{37}$

- LCM of 24 and 37 → since 37 is prime, LCM = $24 \times 37 = 888$
- Convert:
- $\frac{23}{24} = \frac{23 \times 37}{888} = \frac{851}{888}$
- $\frac{17}{37} = \frac{17 \times 24}{888} = \frac{408}{888}$
- Add: $\frac{851 + 408}{888} = \frac{1259}{888}$
- Check if reducible: GCD of 1259 and 888?
- Try dividing: 1259 ÷ 888 ≈ 1.42 → not divisible.
- Test small primes: 1259 is odd, not divisible by 2; sum digits = 1+2+5+9=17 → not divisible by 3.
- Likely irreducible.
- Answer: $\boxed{\frac{1259}{888}}$

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#### C) $\frac{23}{27} + \frac{19}{26}$

- LCM of 27 and 26 → 27 = 3³, 26 = 2×13 → LCM = $2 \times 3^3 \times 13 = 2 \times 27 \times 13 = 702$
- Convert:
- $\frac{23}{27} = \frac{23 \times 26}{702} = \frac{598}{702}$
- $\frac{19}{26} = \frac{19 \times 27}{702} = \frac{513}{702}$
- Add: $\frac{598 + 513}{702} = \frac{1111}{702}$
- Simplify: Check GCD of 1111 and 702.
- 1111 ÷ 11 = 101 → 11×101 = 1111
- 702 ÷ 11 = 63.8 → not divisible
- So, check if 101 divides 702? 101×6 = 606, 101×7 = 707 > 702 → no
- No common factors → irreducible
- Answer: $\boxed{\frac{1111}{702}}$

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#### D) $\frac{29}{32} + \frac{15}{37}$

- LCM of 32 and 37 → 32 = 2⁵, 37 is prime → LCM = $32 \times 37 = 1184$
- Convert:
- $\frac{29}{32} = \frac{29 \times 37}{1184} = \frac{1073}{1184}$
- $\frac{15}{37} = \frac{15 \times 32}{1184} = \frac{480}{1184}$
- Add: $\frac{1073 + 480}{1184} = \frac{1553}{1184}$
- Check simplification: 1553 and 1184
- 1553 ÷ 1184 ≈ 1.31 → not divisible
- Try GCD: seems irreducible
- Answer: $\boxed{\frac{1553}{1184}}$

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#### E) $\frac{23}{27} + \frac{2}{25}$

- LCM of 27 and 25 → 27 = 3³, 25 = 5² → LCM = $3^3 \times 5^2 = 27 \times 25 = 675$
- Convert:
- $\frac{23}{27} = \frac{23 \times 25}{675} = \frac{575}{675}$
- $\frac{2}{25} = \frac{2 \times 27}{675} = \frac{54}{675}$
- Add: $\frac{575 + 54}{675} = \frac{629}{675}$
- Check GCD: 629 and 675
- 629 ÷ 17 = 37 → 17×37 = 629
- 675 ÷ 17 = 39.7 → not divisible
- So no common factors
- Answer: $\boxed{\frac{629}{675}}$

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#### F) $\frac{29}{30} + \frac{15}{16}$

- LCM of 30 and 16 → 30 = 2×3×5, 16 = 2⁴ → LCM = $2^4 \times 3 \times 5 = 16 \times 3 \times 5 = 240$
- Convert:
- $\frac{29}{30} = \frac{29 \times 8}{240} = \frac{232}{240}$
- $\frac{15}{16} = \frac{15 \times 15}{240} = \frac{225}{240}$
- Add: $\frac{232 + 225}{240} = \frac{457}{240}$
- Simplify: 457 and 240 → check if divisible
- 457 is a prime number? Let's test: √457 ≈ 21.4 → check divisibility up to 19
- Not divisible by 2,3,5,7,11,13,17,19 → yes, likely prime
- So no common factor with 240
- Answer: $\boxed{\frac{457}{240}}$ or $1\frac{217}{240}$

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(2) Meenakshi is reading a book titled 'The mysteries of Mathematics', and this book has 336 pages. Meenakshi manages to read $\frac{2}{48}$ of the book every day. After 2 days, how many pages has she read?

- First simplify: $\frac{2}{48} = \frac{1}{24}$
- So, she reads $\frac{1}{24}$ of the book per day
- In 2 days: $2 \times \frac{1}{24} = \frac{2}{24} = \frac{1}{12}$ of the book
- Total pages: 336
- Pages read: $\frac{1}{12} \times 336 = \frac{336}{12} = 28$

✔ Answer: $\boxed{28}$ pages

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(3) Solve for the value of $z$:

$$
\frac{6}{100} + \frac{5}{10} + \frac{6}{z} + 3 = 3.62
$$

- Simplify known values:
- $\frac{6}{100} = 0.06$
- $\frac{5}{10} = 0.5$
- $3 = 3$
- Sum: $0.06 + 0.5 + 3 = 3.56$
- Equation becomes:
$$
3.56 + \frac{6}{z} = 3.62
$$
- Subtract:
$$
\frac{6}{z} = 3.62 - 3.56 = 0.06
$$
- So:
$$
\frac{6}{z} = 0.06 = \frac{6}{100}
$$
- Therefore: $z = 100$

✔ Answer: $\boxed{100}$

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(4) What is the LCM of $\frac{11}{42}, \frac{11}{28},$ and $\frac{8}{35}$?

Wait — LCM of fractions?

There’s a formula:

> LCM of fractions = $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$

But actually, LCM is defined for integers, not fractions directly.

However, in some contexts, especially when dealing with fractions with same numerator, we can interpret it differently.

But here, the question is ambiguous. But let's think: What does LCM of fractions mean?

Actually, LCM of fractions is typically used when you want the least common multiple of the values, but that's not standard.

Alternatively, perhaps the question wants us to find the LCM of the denominators?

But the wording says: "What is the LCM of $\frac{11}{42}, \frac{11}{28}, \frac{8}{35}$?"

That doesn't make sense unless we interpret it as:

> The least common denominator (LCD) of these fractions.

But LCD is usually for adding fractions.

Alternatively, maybe they mean: Find the LCM of the denominators?

Let’s see:

Denominators: 42, 28, 35

Find LCM of 42, 28, 35

Factorize:
- 42 = 2 × 3 × 7
- 28 = 2² × 7
- 35 = 5 × 7

So LCM = $2^2 \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7 = 420$

So LCD = 420

But the question says “LCM of the fractions” — which is not standard.

But sometimes, people use “LCM of fractions” meaning LCM of the denominators when finding a common denominator.

Alternatively, there's a formula:

> LCM of fractions = $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$ — but this is not correct.

Actually, the correct way is:

> To compare or add fractions, we need the LCM of denominators.

But the LCM of fractions is not a standard concept.

Alternatively, perhaps the question is asking: What is the least common multiple of the denominators?

Given that, and since the fractions are $\frac{11}{42}, \frac{11}{28}, \frac{8}{35}$, the denominators are 42, 28, 35.

As above: LCM = 420

✔ Answer: $\boxed{420}$

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(5) Which kind of fraction can never be represented using the following type of figure?

The figure is a circle divided into 8 equal parts, with some shaded.

This represents proper fractions where the numerator ≤ denominator, and the whole is one unit.

Let’s analyze options:

- a. Unlike fractions: These are fractions with different denominators (e.g., 1/2, 1/3). They can be represented using such figures — just draw separate circles.
- b. Like fractions: Same denominator (e.g., 1/8, 3/8). Easily shown on one circle.
- c. Improper fractions: Numerator ≥ denominator (e.g., 9/8, 10/8). This would require more than one whole. But the figure shows only one circle, so cannot represent improper fractions beyond 1 whole.
- d. Unit fractions: Fractions like 1/2, 1/3, etc. Can be shown easily.

So, improper fractions (like 9/8) cannot be shown on a single circle divided into 8 parts unless you have more than one circle.

But the figure is only one circle, so improper fractions cannot be represented using this single figure.

✔ Answer: $\boxed{\text{c. Improper Fractions}}$

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✔ Final Answers:

1.
A) $\frac{49}{36}$
B) $\frac{1259}{888}$
C) $\frac{1111}{702}$
D) $\frac{1553}{1184}$
E) $\frac{629}{675}$
F) $\frac{457}{240}$

2. $\boxed{28}$ pages

3. $\boxed{100}$

4. $\boxed{420}$

5. $\boxed{\text{c. Improper Fractions}}$

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Let me know if you'd like explanations in simpler terms!