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Given:

374 × 374 – 374 × 174 = ?

Concept used:

Distributive property

(a × b) + (a × c) = a × (b + c)

Calculation:

374 × 374 – 374 × 174 

⇒ 374 × (374 – 174)

⇒ 374 × 200

⇒ 74800

∴ The value of 374 × 374 – 374 × 174 is 74800.

Formula used

Area of rectangle = length ×  breadth

Calculation

Let length of rectangle = L and Breadth of rectangle = B

Initial area= L × B

Final length =L + (40/100) L = 1.4 L

Final breadth = B - (10/100) B = 0.9 B

Final area = 1.4 L × 0.9 B

⇒ 1.26 L × B

Change in area = 1.26 (L × B) - (L × B)

⇒ 0.26 L × B

∴ Percentage Change in area = [(0.26 L × B)/(L × B)] × 100

⇒ 26% increase

Alternate method-

Length - increased 40% = +40%

Breathe - decrease 10% = -10%

% change in area = L + B +  L × B/100

= +40 + (-10) + (+40) × (-10) / 100

= +30 - 4 = +26 = Increased by 26%

Given:

The rate of onion increased from Rs. 70 per kilogram to Rs. 120 per kilogram

Concept Used:

Concept of percentage. It is calculated on the basis of 100.

For example, x% means x out of 100

Calculation:

Initial price Rs. 70 per kg

Final price Rs. 120 per kg

Price increased by (120 - 70) = Rs. 50

Price increased in percentage (50/70) × 100

⇒ 71.42 ≈ 71.5%

∴ The rate of onion was increased by 71.5%

Given:

Initial salary = Rs. 27,000

Percentage increase in first year = 11.11% = 1/9

Percentage increase in second year = 37.5% = 3/8

Calculations:

New salary = 27,000 × (10/9) × (11/8)

⇒ Rs. 41,250

∴ His approximate salary at the end of second year is Rs. 41,250.

Given :

703 ÷ 37 = 19 

Calculations :

7.03 ÷ 0.0037 = 7030000 ÷ 3700 

⇒ (703 ÷ 37) × 100 

⇒ 19 × 100 

⇒ 1900 

∴ The value of 7.03 ÷ 0.0037 will be 1900 

Given:

\(\frac{1}{{a\: + \;\frac{1}{{b\: + \;\frac{7}{c}}}}} = \frac{{22}}{{37}}\)

Calculation:

\(\frac{{22}}{{37}} = \;\frac{1}{{37/22}}\) (on dividing by 22)

\(\Rightarrow \frac{1}{{37/22}} = \;\frac{1}{{1\: +\: \frac{{15}}{{22}}}}\)

\( \Rightarrow \frac{1}{{1\: + \:\frac{{15}}{{22}}}} = \;\frac{1}{{1\: + \;\frac{1}{{22/15}}}} = \;\frac{1}{{1\: + \;\frac{1}{{1\: + \;\frac{7}{{15}}}}}}\)

\( \Rightarrow \frac{1}{{a\: + \;\frac{1}{{b \:+ \;\frac{7}{c}}}}} = \;\frac{1}{{1\: + \;\frac{1}{{1\: + \;\frac{7}{{15}}}}}}\)

On comparing both side we get,

a = 1, b = 1 and c = 15

∴ a + b + c is 17.

Given∶ x^a = y^b = z^c, y² = xz

Formula Used∶

^m√a = (a)^1/m, (a^m × aⁿ) = a^{m + n}

Calculation∶

Let x^a = y^b = z^c = k, then

x = k^1/a, y = k^1/b, z = k^1/c

Also, y² = xz

(k^1/b)² = k^1/a × k^1/c = k1^/a+1/c k^(c+a)/ac

K^2/b = k^(c+a)/ac

Or 2/b = (c + a)/ac

2 × ac = b × (c + a)

∴ The value of b is (2ac)/(c + a).

Given:

The square root of number = -64

Calculation:

Let the number be x.

ATQ

\( { \sqrt{x} }=\) -64

so, x = 642 

So, cube root of x = (8 × 8 × 8 × 8)1/3 = 8 × 2 = 16

Given:

(0.08)² + (0.74)² + (0.084)² / (0.008)² + (0.074)² + (.0084)²

Calculation:

Taking 1/(10)² common from denominator

1/{1/(10)²} {(0.08)² + (0.74)² + (0.084)² /(0.08)² + (0.74)² + (0.084)²}

⇒ 100

∴ The value of (0.08)² + (0.74)² + (0.084)² / (0.008)² + (0.074)² + (.0084)² is 100.

Given:

[(0.3)3 + (0.6)3] ÷ [(0.6)3 + (0.2)3]

Calculation:

[(0.3)3 + (0.6)3] ÷ [(0.6)3 + (0.2)3]  

⇒ [0.027 + 0.216] ÷ [0.216 + 0.008]

⇒ 0.243/0.224 = 243/224

∴ 243/224

Given∶

7^{(a + b)} = 343 and 7^{(a – b)} = 16807

Formula Used∶

Basic knowledge of cube and cube root, solution of linear equations in two variables

Calculation∶

7^{(a + b)} = 343

⇒ 7^{(a + b)} = 7³

⇒ a + b = 3……..…(1)

⇒ 7^{(a – b)} = 16807

⇒ 7^{(a – b)} = 7⁵

⇒ a – b = 5................(2)

Add equation (1) and (2), we get

a + b + a – b =  3 + 5

⇒ 2a = 8

⇒ a = 4

Put the value of 8 in equation (1)

a + b = 3

⇒ 4 + b = 3

⇒ b = 3 – 4

⇒ b = -1

∴ The value of a is 4 and b is -1.

Given∶

256^2.5 × 16^4.5 ÷ 4^4.8 = 16^x

Formula Used∶

a^m × aⁿ = a^{m + n}, (a^m)ⁿ = a^{m × n}, a^m÷ aⁿ = a^{m - n}

Calculation∶

256^2.5 × 16^4.5 ÷ 4^4.8 = 16^x

⇒ (16²)^2.5 × 16^4.5 ÷ 4^{2× 2.4} = 16^x

⇒ 16^{2 × 2.5} × 16^4.5 ÷ 16^2.4 = 16^x

⇒ 16⁵ × 16^4.5 ÷ 16^2.4 = 16^x

⇒ 16^{5 + 4.5 - 2.4} = 16^x

⇒ 16^{9.5 - 2.4} = 16^x

⇒ x = 9.5 – 2.4 = 7.1

∴ The value of x is 7.1.