A school has 720 students. 1/8 of the students are seniors and 1/3 are juniors. a) i) Find the number of young learners. ii) Find the fraction of young learners. b) i) Find the distance between points

Here are the solutions to the questions.

QUESTION 1

a)

Step 1: Calculate the number of seniors.

$$\text{Number of seniors} = \frac{1}{8} \times 720 = 90$$

Step 2: Calculate the number of juniors.

$$\text{Number of juniors} = \frac{1}{3} \times 720 = 240$$

i) Step 3: Calculate the number of young learners.

$$\text{Number of young learners} = \text{Total students} - \text{Seniors} - \text{Juniors}$$ $$\text{Number of young learners} = 720 - 90 - 240 = 390$$

The number of young learners in the school is $\boxed{\text{390}}$.

ii) Step 4: Calculate the fraction of young learners.

$$\text{Fraction of young learners} = \frac{\text{Number of young learners}}{\text{Total students}} = \frac{390}{720}$$

Step 5: Simplify the fraction.

$$\frac{390}{720} = \frac{39}{72} = \frac{13}{24}$$

The fraction of young learners in the school is $\boxed{\frac{13}{24}}$.

b)

i) Step 1: Use the distance formula for points $A(1,1)$ and $B(4,5)$.
The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

$$d = \sqrt{(4 - 1)^2 + (5 - 1)^2}$$ $$d = \sqrt{(3)^2 + (4)^2}$$ $$d = \sqrt{9 + 16}$$ $$d = \sqrt{25}$$ $$d = 5$$

The length of $\overline{AB}$ is $\boxed{\text{5 units}}$.

ii) Step 1: Rearrange the formula $w + 2r = 2f - r$ to make $r$ the subject.

$$w + 2r = 2f - r$$

Step 2: Add $r$ to both sides.

$$w + 2r + r = 2f$$ $$w + 3r = 2f$$

Step 3: Subtract $w$ from both sides.

$$3r = 2f - w$$

Step 4: Divide by 3.

$$r = \frac{2f - w}{3}$$

The formula with $r$ as the subject is $\boxed{r = \frac{2f - w}{3}}$.

c)

Step 1: Determine the sum of interior angles of a pentagon.
The sum of interior angles for an $n$-sided polygon is $(n-2) \times 180^\circ$. For a pentagon, $n=5$.

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