a. State the number of roads (ARCS) on the diagram by identifying their endpoints (nodes) b. State the number…

Question

a. State the number of roads (ARCS) on the diagram by identifying their endpoints (nodes) b. State the number of regions in the network

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Question 6:
A woman gave $\frac{1}{3}$ of her farm to her daughter and sold $\frac{1}{4}$ of the remainder.

a) Calculate the fraction sold.

Determine the fraction of the farm remaining after giving to her daughter.
$\text{Fraction remaining} = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}$
Calculate the fraction of the farm sold, which is $\frac{1}{4}$ of the remainder.
$\text{Fraction sold} = \frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}$
The fraction sold is $\boxed{\frac{1}{6}}$ .

b) Calculate the fraction of the farm remaining.
The fraction remaining after giving to her daughter was $\frac{2}{3}$ .
She sold $\frac{1}{4}$ of this remainder, meaning $\frac{3}{4}$ of this remainder is still left.
$\text{Fraction remaining} = \frac{3}{4} \times \frac{2}{3}$
Multiply the fractions.
$\text{Fraction remaining} = \frac{6}{12} = \frac{1}{2}$
The fraction of the farm remaining is $\boxed{\frac{1}{2}}$ .

Question 7:
Solve for $x$ in the equation $4^{x+1} = 32^{x-1}$ .
Express both bases as powers of the same prime number, which is 2.
$4 = 2^2$ $32 = 2^5$
Substitute these into the equation.
$(2^2)^{x+1} = (2^5)^{x-1}$
Apply the exponent rule $(a^m)^n = a^{mn}$ .
$2^{2(x+1)} = 2^{5(x-1)}$ $2^{2x+2} = 2^{5x-5}$
Since the bases are equal, equate the exponents.
$2x+2 = 5x-5$
Solve for $x$ .
$2+5 = 5x-2x$ $7 = 3x$ $x = \frac{7}{3}$
The value of $x$ is $\boxed{\frac{7}{3}}$ .

Question 8:
A trader took a loan of 3,000,000 FCFA from a bank at the rate of 2% simple interest per annum.

a) Calculate how much interest he pays per year.
Use the simple interest formula for one year
$I = P \times R$ .
$I = 3,000,000 \text{ FCFA} \times 0.02$ $I = 60,000 \text{ FCFA}$
The interest paid per year is $\boxed{60,000 \text{ FCFA}}$ .

b) Calculate the total interest paid in three years.
Use the simple interest formula
$I = P \times R \times T$ .
$I = 3,000,000 \text{ FCFA} \times 0.02 \times 3$ $I = 60,000 \text{ FCFA} \times 3$ $I = 180,000 \text{ FCFA}$
The total interest paid in three years is $\boxed{180,000 \text{ FCFA}}$ .

c) Calculate the total amount paid at the end of three years.
The total amount paid is the principal plus the total interest.
$\text{Total Amount} = \text{Principal} + \text{Total Interest}$ $\text{Total Amount} = 3,000,000 \text{ FCFA} + 180,000 \text{ FCFA}$ $\text{Total Amount} = 3,180,000 \text{ FCFA}$
The total amount paid at the end of three years is $\boxed{3,180,000 \text{ FCFA}}$ .

Question 9:
The diagram below represents roads linking the offices in an establishment.

a) State the number of roads (ARCS) on the diagram by identifying their endpoints (nodes).
Identify the nodes (offices)
Principal, V Principal, BC Workshop, Bursary.
Count the lines connecting these nodes, which represent the roads (arcs).
- Principal to V Principal
- Principal to Bursary
- Principal to BC Workshop
- V Principal to Bursary
- BC Workshop to Bursary There are $\boxed{5}$ roads (arcs) on the diagram.
b) State the number of regions in the network.
Identify the number of vertices (nodes, V) and edges (arcs, E) in the planar graph.
$V = 4 \quad (\text{Principal, V Principal, BC Workshop, Bursary})$ $E = 5 \quad (\text{The 5 roads identified in part a})$
Apply Euler's formula for planar graphs, which states $V - E + F = 2$ , where $F$ is the number of faces (regions).
$4 - 5 + F = 2$ $-1 + F = 2$ $F = 2 + 1$ $F = 3$
The number of regions in the network is $\boxed{3}$ .

A. State the number of roads (ARCS) on the diagram by identifying their endpoints (nodes) b. State the number of regions in the network

Step-by-step solution

Determine the fraction of the farm remaining after giving to her daughter.

Calculate the fraction of the farm sold, which is $\frac{1}{4}$ of the remainder.

The fraction remaining after giving to her daughter was $\frac{2}{3}$ .

Multiply the fractions.

Express both bases as powers of the same prime number, which is 2.

Substitute these into the equation.

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Since the bases are equal, equate the exponents.

Solve for $x$ .

Use the simple interest formula for one year

Use the simple interest formula

The total amount paid is the principal plus the total interest.

Identify the nodes (offices)

Count the lines connecting these nodes, which represent the roads (arcs).

Identify the number of vertices (nodes, V) and edges (arcs, E) in the planar graph.

Apply Euler's formula for planar graphs, which states $V - E + F = 2$ , where $F$ is the number of faces (regions).

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