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Given:

Total time taken = 10 hours

Speed in first half = 30 km/hr

Speed in second half = 20 km/hr

Formula used:

Time = Distance/Speed

Calculation:

Let the total distance covered be 2x km

As, he covered the first half (i.e. x km) with 30 km/hr

Time taken to cover first-half = Distance/Speed = x/30 hours

He covered the second half (i.e. x km) with 20 km/hr

Time taken to cover second half = x/20 hours

Total time taken = 10 hours

⇒ x/30 + x/20 = 10

⇒ 5x/60 = 10

⇒ x = 120

Total distance covered by him = 2x = 2 × 120 = 240 km

∴ Total distance covered by him is 240 km

Given:

A dealer sells an article at Rs. 403.20, after offering a discount of 16% on its marked price.

Concept used:

Selling price = Cost price + Cost price × Profit%

Selling price = Marked price - Markes price × Discount%

Calculation:

Marked price = 403.20 × 100/84

⇒ 480

Now if he sell the article at Rs. 480 there would be a profit of 28%

So, Cost price = 480 × 100/128

⇒ Rs. 375

∴ The cost price of the article is Rs. 375

Given:

Total length of rod = 50 m, Length of the cut piece = 5.25 m

Calculation:

No. of pieces =Total length/length of the cut piece

⇒ 50/5.25 = 9.52

Means 9 full such pieces can be cut from the rod.

Length of cut piece × no of pieces = length of the rod used

⇒ 5.25 × 9 = 47.25 m

⇒ Remaining length = 50 - 47.25 = 2.75

Remaining length = (Remaining length/total length) = (2.75/50) = 0.055

Given:

A  and B can complete a work in 20 days.

B can complete a work in 25 days.

Formula used:

Work done = time × efficiency

Calculation:

LCM of (A + B) and B = 100

Efficiency of A + B = 5

Efficiency of B = 4

Efficiency of A = 1

Time taken by A = 100/1 = 100 days

∴ The correct answer is 100 days.

Given,

2³ × 3⁴ × 1080 ÷ 15 = 6^x

⇒ 2³ × 3⁴ × 72 = 6^x

⇒ 2³ × 3⁴ × (2 × 6²) = 6^x

⇒ 2⁴ × 3⁴ × 6² = 6^x

⇒ (2 × 3)⁴ × 6² = 6^x           [∵ x^m × y^m = (xy)^m]

⇒ 6⁴ × 6² = 6^x

⇒ 6^{(4 + 2)} = 6^x

⇒ x = 6

For a number to be divisible by 6, The last digit must be even and sum of the digits of the number must be divisible by 3

Sum of the digits of 35432 = 3 + 5 + 4 + 3 + 2 = 17 it is not divisible by 3

Sum of the digits of 53322 = 5 + 3 + 3+ 2 + 2 =15 it is divisible by 3 and its last digit is even

Sum of the digits of 93242 = 9 + 3 + 2 + 4 + 2 = 20 it is not divisible by 3

Sum of the digits of 85232 = 8 + 5 + 2 + 3 + 2 = 20 it is not divisible by 3

∴ 53322 is divisible by 6

Given values:

A and B together finish work in 30 days

Worked together for 20 days, then B left

A worked for another 20 days

Concept:

If A and B together finish work in 'a' days, A finishes alone in 'b' days and B finishes alone in 'c' days, then 1/a = 1/b + 1/c.

Calculation:

⇒ Total work done by A and B in 20 days = 20/30 = 2/3 of the work

⇒ Work done by A in the next 20 days = 1 - 2/3 = 1/3 of the work

⇒ Therefore, A alone can finish the work in 20 × 3 = 60 days.

Hence, A alone can finish the whole work in 60 days.

Concept used:

Follow BODMAS rule to solve this question, as per the order given below:

Step-1: Parts of an equation enclosed in 'Brackets' must be solved first, and in the bracket,

Step-2: Any mathematical 'Of' or 'Exponent' must be solved next,

Step-3: Next, the parts of the equation that contain 'Division' and 'Multiplication' are calculated,

Step-4: Last but not least, the parts of the equation that contain 'Addition' and 'Subtraction' should be calculated.

Calculations:

⇒ 71% of 2200 – 122 = x + (22 × 25)

⇒ 71/100 × 2200 – 122 = x + (22 × 25)

⇒ 1562 – 122 = x + 550

⇒ 1440 = x + 550

⇒ x = 890.

Concept used:

a^-n = 1/aⁿ

Calculations:

\(\left(4^{-1}+8^{-1}\right) ÷\left(\frac{2}{3}\right)^{-1}=\left(\frac{1}{2}\right)^{2 x-1} \)

⇒  (1/4 + 1/8) ÷ (3/2) = (1/2)^{2x - 1}

⇒  (3/8) ÷ (3/2) = (1/2)2x - 1

⇒  (3/8) × (2/3) = (1/2)^2x-1

⇒ 1/4 = (1/2)^2x-1

⇒  (1/2)² = (1/2)^{2x - 1}

⇒  2 = 2x - 1

⇒ x = 3/2

so

x² + x - 2 = (3/2)² + 3/2 - 2 

⇒ 9/4 - 1/2

⇒ 7/4

\(\begin{array}{l} \frac{{0.7; \times 0.7 \times 0.7; + ;0.3 \times 0.3 \times 7.3}}{{0.7 \times 0.7; + ;0.3 \times 0.3; + ;0.42}}\ = ;\frac{{0.343; + ;0.09 \times 7.3}}{{{{0.7}^2}; + ;{{0.3}^2}; + ;2 \times 0.3 \times 0.7}};\ = ;\frac{{0.343; + ;0.657}}{{{{\left( {0.7; + ;0.3} \right)}^2}}} \end{array}\)

= 1/1 = 1