Triangle Angles

Here are the solutions to the problems:

5. Triangle Angles
A triangle has angles in the ratio 2:3:4. What is the measure of the largest angle?

Step 1: Find the sum of the ratio parts.

$$\text{Sum of ratio parts} = 2 + 3 + 4 = 9$$

Step 2: The sum of angles in a triangle is $180^\circ$. Divide this by the sum of the ratio parts to find the value of one part.

$$\text{Value of one part} = \frac{180^\circ}{9} = 20^\circ$$

Step 3: The largest ratio part is 4. Multiply this by the value of one part to find the largest angle.

$$\text{Largest angle} = 4 \times 20^\circ = 80^\circ$$

The measure of the largest angle is $\boxed{80^\circ}$.

6. LCM and GCD Product
Find the product of the L.C.M and G.C.D of 10, 20 and 40.

Step 1: Find the prime factorization of each number.

$$10 = 2 \times 5$$ $$20 = 2^2 \times 5$$ $$40 = 2^3 \times 5$$

Step 2: Determine the G.C.D (Greatest Common Divisor). This is the product of the lowest powers of common prime factors.

$$\text{G.C.D}(10, 20, 40) = 2^1 \times 5^1 = 10$$

Step 3: Determine the L.C.M (Least Common Multiple). This is the product of the highest powers of all prime factors present.

$$\text{L.C.M}(10, 20, 40) = 2^3 \times 5^1 = 8 \times 5 = 40$$

Step 4: Find the product of the L.C.M and G.C.D.

$$\text{Product} = \text{L.C.M} \times \text{G.C.D} = 40 \times 10 = 400$$

The product of the L.C.M and G.C.D is $\boxed{400}$.

7. Prime Factorization
Write 2700 and 750 as product of their prime factors.

For 2700:

Step 1: Divide 2700 by the smallest prime factors.

$$2700 = 100 \times 27 = (10^2) \times (3^3)$$ $$2700 = (2 \times 5)^2 \times 3^3$$ $$2700 = 2^2 \times 5^2 \times 3^3$$

The prime factorization of 2700 is $\boxed{2^2 \times 3^3 \times 5^2}$.

For 750:

Step 1: Divide 750 by the smallest prime factors.

$$750 = 10 \times 75 = (2 \times 5) \times (3 \times 25)$$ $$750 = 2 \times 5 \times 3 \times 5^2$$ $$750 = 2 \times 3 \times 5^3$$

The prime factorization of 750 is $\boxed{2 \times 3 \times 5^3}$.

8. Hire Purchase Calculation
Samson was offered to purchase a mobile phone whose cash price is 50000 at hire purchase which included a deposit of 20% of the cash price followed by 12 equal monthly installments of 4000 each. Calculate how much more did Samson pay for the phone.

Step 1: Calculate the deposit amount.

$$\text{Deposit} = 20\% \text{ of } 50000 = 0.20 \times 50000 = 10000$$

Step 2: Calculate the total amount paid in installments.

$$\text{Total installments} = 12 \text{ months} \times 4000/\text{month} = 48000$$

Step 3: Calculate the total hire purchase price.

$$\text{Hire purchase price} = \text{Deposit} + \text{Total installments} = 10000 + 48000 = 58000$$

Step 4: Calculate how much more Samson paid compared to the cash price.

$$\text{Extra paid} = \text{Hire purchase price} - \text{Cash price} = 58000 - 50000 = 8000$$

Samson paid $\boxed{8000}$ more for the phone.

9. Equation of a Line
The equation of a line is $4y - 16x - 20 = 0$.