A) State the total value of the digit 8 in 84,265,739. b) Round off 84,265,739 to the nearest million. 7. a) Ben’s age is t years and is not less than 14 years. Write an inequality for Ben’s age. b) I

a) To find the total value of the digit 8 in 84,265,739, we identify its place value. The digit 8 is in the ten millions place.

$$\text{Value of 8} = 8 \times 10,000,000 = 80,000,000$$

The total value of the digit 8 is Sh. 80,000,000.

b) To round off 84,265,739 to the nearest million shillings, we look at the digit in the millions place (4) and the digit to its right (2, in the hundred thousands place). Since 2 is less than 5, we round down, keeping the millions digit as 4 and changing all digits to its right to zero.

$$\text{Rounded amount} = 84,000,000$$

The rounded amount is Sh. 84,000,000.

a) Ben's age is $t$ years and is "not less than 14 years". This means her age is greater than or equal to 14.

$$t \ge 14$$

The inequality is $\boxed{t \ge 14}$.

b) To illustrate the inequality $t \ge 14$ on a number line, we draw a closed circle at 14 (indicating that 14 is included) and an arrow extending to the right (indicating all values greater than 14).

$$\text{Number line illustration:}$$ $$\begin{array}{cccccccccccccccc} \bullet & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ \\ \hline 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \end{array}$$

(The line should extend from 14 to the right, with a solid circle at 14).

Let $j$ be the cost of one jacket and $s$ be the cost of one shirt.
From the problem statement, we can form two linear equations:

1. Gideon: $3j + 2s = 4600$
2. John: $2j + 3s = 4400$

To solve this system of equations, we can use the elimination method. Multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of $j$ equal:

$$2 \times (3j + 2s) = 2 \times 4600 \implies 6j + 4s = 9200 \quad (\text{Equation 3})$$ $$3 \times (2j + 3s) = 3 \times 4400 \implies 6j + 9s = 13200 \quad (\text{Equation 4})$$

Subtract Equation 3 from Equation 4:

$$(6j + 9s) - (6j + 4s) = 13200 - 9200$$ $$5s = 4000$$ $$s = \frac{4000}{5}$$ $$s = 800$$

Now substitute the value of $s$ into Equation 1:

$$3j + 2(800) = 4600$$ $$3j + 1600 = 4600$$ $$3j = 4600 - 1600$$ $$3j = 3000$$ $$j = \frac{3000}{3}$$ $$j = 1000$$

The cost of each jacket is Sh. 1000 and the cost of each shirt is Sh. 800.

Total attendees = 5024.
Learners were $\frac{5}{8}$ of the total attendees:

$$\text{Number of learners} = \frac{5}{8} \times 5024 = 5 \times 628 = 3140$$

Remaining attendees after learners:

$$\text{Remaining attendees} = 5024 - 3140 = 1884$$

Parents were $\frac{2}{3}$ of the remaining attendees:

$$\text{Number of parents} = \frac{2}{3} \times 1884 = 2 \times 628 = 1256$$

The rest were community members.

$$\text{Number of community members} = \text{Remaining attendees} - \text{Number of parents}$$ $$\text{Number of community members} = 1884 - 1256 = 628$$

There were 628 community members.

Given the algebraic equation:

$$(6t-2)+3(7-2t)=18$$

First, distribute the 3 into the parenthesis:

$$6t - 2 + (3 \times 7) - (3 \times 2t) = 18$$ $$6t - 2 + 21 - 6t = 18$$

Combine like terms ($6t$ and $-6t$, and $-2$ and 21):

$$(6t - 6t) + (-2 + 21) = 18$$ $$0t + 19 = 18$$ $$19 = 18$$

This statement is a contradiction, meaning there is no value of $t$ that can satisfy the given equation.
The equation has no solution.

An employee earns Sh. 3600 for working 5 days.
First, calculate the daily earning rate:

$$\text{Daily earning rate} = \frac{\text{Total earnings}}{\text{Number of days}} = \frac{3600 \text{ Sh.}}{5 \text{ days}} = 720 \text{ Sh./day}$$

Now, calculate the earnings for 12 days at the same rate:

$$\text{Earnings for 12 days} = \text{Daily earning rate} \times 12 \text{ days}$$ $$\text{Earnings for 12 days} = 720 \text{ Sh./day} \times 12 \text{ days}$$ $$\text{Earnings for 12 days} = 8640 \text{ Sh.}$$

The employee will earn Sh. 8640 after working for 12 days.