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Given:

The number 420.

Concept used:

Prime numbers are number greater than 1 they have two factors, 1 and the number itself.

Calculation:

Here,

Factors of 420 = 2 × 2 × 3 × 5 × 7

∴ The correct prime factors of 420 is  2 × 2 × 3 × 5 × 7.

Given:

LCM of any two numbers = 60

HCF = 5

Concept used:

Product of two numbers = LCM × HCF

Calculation:

Here,

LCM = 60 

HCF = 5

Let the two numbers are 5x, 5y

∴ 5x × 5y = 60 × 5

⇒ 25xy = 300

xy = 12

Possible co primes values of x, y are (3, 4) 

∴ the numbers are:-

5x = 5 × 3 = 15, 5y = 5 × 4 = 20

Product of two numbers = 20 × 15 = LCM × HCF = 300

∴ The numbers are:- 15, 20.

If  \(\frac{a}{b} = \frac{c}{d} \) ⇒ ad = bc 

Let the antecedents be x

⇒ 5 ∶ 7 ∶∶ x : 35 ⇒ 7 × x = 5 × 35 ⇒ x = 25

∴ antecedents is 25.

Given:

Men(1) = 4

Days(1) = 20

Men (2) = 10

Formula Used:

\(\frac{M1 \times D1}{W1} = \frac{M2 \times D2}{W2}\)

Calculation:

W₁  = W₂

Now,

\(⇒ \frac{4 \times 20}{W} = \frac{10 \times D2}{W}\)

⇒ D₂ = 8 days

∴ 10 men can complete the work in 8 days.

Given:

selling price of an article = ​₹329

Loss% = 6%

Concept used:

Loss% = loss × 100/C.P

C.P = Cost price

S.P = Selling price

Calculation:

S. P = (100 - loss%) × C.P/100 

⇒ 329 = (100 - 6) × C.P/100 

C.P = 329 × 100/94

C.P = ​₹350

∴ The cost price of article is  ​₹350.

Given:

Average of three numbers = 40

Average of two of them = 38

Concept used:

Average = Sum of all observations/ Total number of observations

Calculation:

Here, 

⇒ 40 = Sum of all observation/3

Sum of three numbers = 40 × 3 = 120

⇒ 38 = Sum of two observation/2

Sum of two observation = 38 × 2 = 76

Third number = 120 - 76 = 44

∴ The third number is 44.

Given:

Principal = ₹3,000

Time = 2 years

Rate = 11%

Formula used:

S.I = P × R × T/100

Amount = P + S.I

Where, 

P = principal

T = time 

R = rate 

Calculation:

Principal = ₹3,000

Time = 2 years

Rate = 11%

S.I = 3000 × 11 × 2/100 = ₹660

∴ Amount = 3000 + 660 = ₹3660

∴ Amount received is ₹3660.

Given:

Principal = ​₹2,100

Rate = 8%

Formula used:

S.I = (P × R × T) / 100

Calculation:

Principal = ​₹2,100

Rate = 8%

S.I for three year = 2100 × 3 × 8/100 = ​₹504

S.I for two years = 2100 × 2 × 8/100 = ​₹336

∴ Difference of S.I = ₹504 - ₹336 = ₹168

Shortcut Trick S.I = (P × R × T) / 100

⇒ {P× R× (T₂ - T₁)} / 100

⇒ {2100× 8× (3-2)} / 100 = 168.

Given:

Cost price of bicycle = ₹1,800

Profit = 12%

Concept used:

Selling price = Cost price × (100 + profit%)/100

Calculation:

Cost price of bicycle =₹1,800

Profit = 12%

Selling price = 1800 × 112/100 = ₹2016

∴ The selling price of bicycle is ₹2016.

Concept used:

Sum of linear pair angles = 180º

Calculation:

Let O be the intersection point of these lines.

⇒ ∠ AOC = ∠ FOD = 90º (Vertically opposite angle)

⇒ ∠ BOF + ∠ FOD + ∠ DOE = 180º (Linear pair)

⇒ 40º + 90º + 5a = 180º

⇒ 130º + 5a = 180º

⇒ 5a = 50º

∴ a = 10º