33. Simplify: (a) $$ -\frac{1}{9} \cdot\left(-\frac{7}{3}\right) $$ (b) $$ \frac{16}{13}+\left(-\frac{4}{65}\right) $$ 34. How many times a wheel of radius $$ ? \mathrm{~cm} $$ must ratate to po 352 m ? 35. Simplify and then find the value if $$ \mathrm{a}=-1 $$ and $$ \mathrm{b}=2 $$ $$ 3 a b-2 b-5 a-2(5 a-7 a b)-30 $$ 36. The number of illiterate persons in a country decreased from 150 lakhs to 100 lakhs in 10 years. What is the percentage of decrease? 38. Simplify: $$ \frac{\left(2^{4}\right)^{2} \times 9^{3} \times 4 \times 1^{9}}{2^{3} \times 8^{2} \times 27 \times 1^{5}} $$ 39. What cross-section do you get when you give a (i) vertical cut (ii) herizontal cut to the following solids ? (a) A brick (b) A round apple (c) A die. 40. Kamal buys a book far Rx 275 and sells it at a loss of $$ 15 \% $$. How much does he sell if for ?

Question

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$275-\left(0.15\times 275\right)$$
- step1: Multiply the numbers:
  $$275-41.25$$
- step2: Subtract the numbers:
  $$233.75$$
Calculate or simplify the expression $$ -\frac{1}{9} \cdot \left(-\frac{7}{3}\right) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$-\frac{1}{9}\left(-\frac{7}{3}\right)$$
- step1: Use the rules for multiplication and division:
  $$\frac{1}{9}\times \frac{7}{3}$$
- step2: Multiply the fractions:
  $$\frac{7}{9\times 3}$$
- step3: Multiply:
  $$\frac{7}{27}$$
Calculate or simplify the expression $$ \frac{352}{2\pi} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{352}{2\pi }$$
- step1: Reduce the fraction:
  $$\frac{176}{\pi }$$
Calculate or simplify the expression $$ \frac{16}{13}+\left(-\frac{4}{65}\right) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{16}{13}+\left(-\frac{4}{65}\right)$$
- step1: Remove the parentheses:
  $$\frac{16}{13}-\frac{4}{65}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{16\times 5}{13\times 5}-\frac{4}{65}$$
- step3: Multiply the numbers:
  $$\frac{16\times 5}{65}-\frac{4}{65}$$
- step4: Transform the expression:
  $$\frac{16\times 5-4}{65}$$
- step5: Multiply the numbers:
  $$\frac{80-4}{65}$$
- step6: Subtract the numbers:
  $$\frac{76}{65}$$
Calculate or simplify the expression $$ \frac{150-100}{150} \times 100 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{150-100}{150}\times 100$$
- step1: Subtract the numbers:
  $$\frac{50}{150}\times 100$$
- step2: Reduce the fraction:
  $$\frac{1}{3}\times 100$$
- step3: Multiply:
  $$\frac{100}{3}$$
Calculate or simplify the expression $$ 3(-1)(2)-2(2)-5(-1)-2(5(-1)-7(-1)(2))-30 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3\left(-1\right)\times 2-2\times 2-5\left(-1\right)-2\left(5\left(-1\right)-7\left(-1\right)\times 2\right)-30$$
- step1: Simplify:
  $$3\left(-1\right)\times 2-2\times 2-5\left(-1\right)-2\left(-5-7\left(-1\right)\times 2\right)-30$$
- step2: Multiply:
  $$3\left(-1\right)\times 2-2\times 2-5\left(-1\right)-2\left(-5+14\right)-30$$
- step3: Add the numbers:
  $$3\left(-1\right)\times 2-2\times 2-5\left(-1\right)-2\times 9-30$$
- step4: Multiply:
  $$-6-2\times 2-5\left(-1\right)-2\times 9-30$$
- step5: Multiply the numbers:
  $$-6-4-5\left(-1\right)-2\times 9-30$$
- step6: Simplify:
  $$-6-4+5-2\times 9-30$$
- step7: Multiply the numbers:
  $$-6-4+5-18-30$$
- step8: Calculate:
  $$-53$$
Calculate or simplify the expression $$ \frac{\left(2^{4}\right)^{2} \times 9^{3} \times 4 \times 1^{9}}{2^{3} \times 8^{2} \times 27 \times 1^{5}} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(2^{4}\right)^{2}\times 9^{3}\times 4\times 1^{9}}{2^{3}\times 8^{2}\times 27\times 1^{5}}$$
- step1: Multiply by $$a^{-n}:$$
  $$\frac{\left(2^{4}\right)^{2}\times 9^{3}\times 4}{2^{3}\times 8^{2}\times 27}$$
- step2: Multiply the exponents:
  $$\frac{2^{4\times 2}\times 9^{3}\times 4}{2^{3}\times 8^{2}\times 27}$$
- step3: Multiply by $$a^{-n}:$$
  $$\frac{2^{4\times 2}\times 9^{3}\times 4\times 2^{-3}}{8^{2}\times 27}$$
- step4: Multiply the numbers:
  $$\frac{2^{8}\times 9^{3}\times 4\times 2^{-3}}{8^{2}\times 27}$$
- step5: Multiply:
  $$\frac{2^{7}\times 9^{3}}{8^{2}\times 27}$$
- step6: Factor the expression:
  $$\frac{2^{7}\times 9^{3}}{2^{6}\times 27}$$
- step7: Reduce the fraction:
  $$\frac{2\times 9^{3}}{27}$$
- step8: Factor the expression:
  $$\frac{2\times 3^{6}}{3^{3}}$$
- step9: Reduce the fraction:
  $$2\times 3^{3}$$
- step10: Evaluate the power:
  $$2\times 27$$
- step11: Multiply:
  $$54$$
Let's solve each part step by step.

### 33. Simplify:
(a) $$ -\frac{1}{9} \cdot\left(-\frac{7}{3}\right) $$

To simplify this expression:
1. Multiply the fractions: 
   $$
   -\frac{1}{9} \cdot -\frac{7}{3} = \frac{1 \cdot 7}{9 \cdot 3} = \frac{7}{27}
   $$

So, the simplified result is $$ \frac{7}{27} $$.

(b) $$ \frac{16}{13}+\left(-\frac{4}{65}\right) $$

To simplify this expression:
1. Convert $$ -\frac{4}{65} $$ to a common denominator with $$ \frac{16}{13} $$:
   $$
   \frac{16}{13} = \frac{16 \cdot 5}{13 \cdot 5} = \frac{80}{65}
   $$
2. Now add:
   $$
   \frac{80}{65} - \frac{4}{65} = \frac{80 - 4}{65} = \frac{76}{65}
   $$

So, the simplified result is $$ \frac{76}{65} $$.

### 34. How many times must a wheel of radius $$ r $$ cm rotate to cover 352 m?

1. First, convert 352 m to cm:
   $$
   352 \text{ m} = 352 \times 100 = 35200 \text{ cm}
   $$
2. The circumference $$ C $$ of the wheel is given by:
   $$
   C = 2\pi r
   $$
3. The number of rotations $$ N $$ is:
   $$
   N = \frac{35200}{2\pi r}
   $$

So, the number of rotations is $$ \frac{17600}{\pi r} $$.

### 35. Simplify and find the value if $$ a=-1 $$ and $$ b=2 $$:
$$
3 a b - 2 b - 5 a - 2(5 a - 7 a b) - 30
$$

1. Substitute $$ a $$ and $$ b $$:
   $$
   3(-1)(2) - 2(2) - 5(-1) - 2(5(-1) - 7(-1)(2)) - 30
   $$
2. Calculate each term:
   $$
   = -6 - 4 + 5 - 2(-5 + 14) - 30
   $$
   $$
   = -6 - 4 + 5 - 2(9) - 30
   $$
   $$
   = -6 - 4 + 5 - 18 - 30
   $$
   $$
   = -53
   $$

So, the final value is $$ -53 $$.

### 36. Percentage of decrease in illiterate persons:
1. Initial number: 150 lakhs, Final number: 100 lakhs.
2. Percentage decrease:
   $$
   \text{Percentage Decrease} = \frac{150 - 100}{150} \times 100 = \frac{50}{150} \times 100 = \frac{100}{3} \approx 33.33\%
   $$

So, the percentage of decrease is $$ \frac{100}{3} $$ or approximately $$ 33.33\% $$.

### 38. Simplify:
$$
\frac{\left(2^{4}\right)^{2} \times 9^{3} \times 4 \times 1^{9}}{2^{3} \times 8^{2} \times 27 \times 1^{5}}
$$

1. Simplify the numerator:
   $$
   (2^4)^2 = 2^8, \quad 9^3 = (3^2)^3 = 3^6, \quad 4 = 2^2, \quad 1^9 = 1
   $$
   So, the numerator becomes:
   $$
   2^8 \times 3^6 \times 2^2 = 2^{10} \times 3^6
   $$
2. Simplify the denominator:
   $$
   2^3 \times 8^2 = 2^3 \times (2^3)^2 = 2^3 \times 2^6 = 2^9, \quad 27 = 3^3, \quad 1^5 = 1
   $$
   So, the denominator becomes:
   $$
   2^9 \times 3^3
   $$
3. Now, simplify the entire fraction:
   $$
   \frac{2^{10} \times 3^6}{2^9 \times 3^3} = 2^{10-9} \times 3^{6-3} = 2^1 \times 3^3 = 2 \times 27 = 54
   $$

So, the simplified result is $$ 54 $$.

### 40. Kamal's selling price after a loss of $$ 15\% $$:
1. Calculate the loss:
   $$
   \text{Loss} = 0.15 \times 275 = 41.25
   $$
2. Selling price:
   $$
   \text{Selling Price} = 275 - 41.25 = 233.75
   $$

So, Kamal sells the book for $$ 233.75 $$.
(a) $$ \frac{7}{27} $$ (b) $$ \frac{76}{65} $$ Number of rotations: $$ \frac{17600}{\pi r} $$ Value when $$ a=-1 $$ and $$ b=2 $$: $$ -53 $$ Percentage decrease: $$ \frac{100}{3} $$ or approximately $$ 33.33\% $$ Simplified expression: $$ 54 $$ Selling price: $$ 233.75 $$

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