Question: Answer ALL the questions in this section.
All working must be clearly shown.
1. (a) Calculate the exact value of
(i) (5.8² + 1.02) × 2.5 (2 marks)
(ii) 2 4/5 ÷ 3/4. (3 marks)
(b) A company pays its employees a basic wage of $9.50 per hour for a 40-hour week.
(i) Calculate the basic weekly wage for ONE employee. (1 mark)
Overtime is paid at a rate of time and a half.
(ii) Calculate the overtime wage for an employee who works 6 hours overtime in a certain week. (2 marks)
In a certain week, the company paid its 30 employees a total of $12 084.00 in basic and overtime wages. Calculate for that week:
(iii) The TOTAL paid in overtime wages (2 marks)
(iv) The TOTAL number of overtime hours worked by employees (1 mark)
Total 11 marks
2. (a) Simplify
2x/5 - x/3
expressing your answer as a single fraction. (2 marks)
(b) Factorise completely
a²b + 2ab. (1 mark)
(c) Express p as the subject of the formula
q = (p² - r)/t. (3 marks)
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(d) The students in a class sell donuts to raise money for their school project. The donuts are sold in small and large boxes. The number of donuts in EACH type of box is given in the table below:
Type of Box Number of Donuts per Box
Small box x
Large box 2x + 3
The students sold 8 small boxes and 5 large boxes in all.
(i) Write an expression in terms of x to represent the TOTAL number of donuts sold. (2 marks)
(ii) The total number of donuts sold was 195. Calculate the number of donuts in a
a) small box
b) large box. (4 marks)
Total 12 marks
3. (a) Simplify the expression
7p²q³ × 2p²q. (2 marks)
(b) Fresh Farms Dairy sells milk in cartons in the shape of a cuboid with internal dimensions 6 cm by 4 cm by 10 cm.
Graph/shape: a cuboid carton of milk is shown near the top-right/center area of the page, labeled “Milk”, with dimensions 6 cm along the front bottom edge, 4 cm along the depth edge, and 10 cm vertically.
(i) Calculate, in cm³, the volume of milk in EACH carton. (2 marks)
(ii) A recipe for making ice-cream requires 3 litres of milk. How many cartons of milk should be bought to make the ice-cream? (3 marks)
(iii) One carton of milk is poured into a cylindrical cup of internal diameter 5 cm. What is the height of milk in the cup? Give your answer to 3 significant figures. (4 marks)
[Use π = 3.14]
Total 11 marks
1. (a) (i) Calculate the exact value of $$(5.8^2 + 1.02) \times 2.5$$.
Step 1: Calculate $5.8^2$.
$$5.8^2 = 33.64$$
Step 2: Add $1.02$.
$$33.64 + 1.02 = 34.66$$
Step 3: Multiply by $2.5$.
$$34.66 \times 2.5 = 86.65$$
Answer: $$86.65$$
(ii) Calculate $$2 \frac{4}{5} \div \frac{3}{4}$$.
Step 1: Convert mixed number to improper fraction.
$$2 \frac{4}{5} = \frac{14}{5}$$
Step 2: Division of fractions is multiplication by reciprocal.
$$\frac{14}{5} \div \frac{3}{4} = \frac{14}{5} \times \frac{4}{3}$$
Step 3: Multiply numerators and denominators.
$$\frac{14 \times 4}{5 \times 3} = \frac{56}{15}$$
Step 4: Simplify if possible (not reducible).
Answer: $$\frac{56}{15}$$
(b) A company pays $9.50$ per hour for a 40-hour week.
(i) Calculate basic weekly wage.
Step 1: Multiply hourly wage by hours.
$$9.50 \times 40 = 380$$
Answer: $$380$$
(ii) Overtime wage rate is time and a half.
Step 1: Calculate overtime hourly rate.
$$9.50 \times 1.5 = 14.25$$
Step 2: Calculate overtime pay for 6 hours.
$$14.25 \times 6 = 85.5$$
Answer: $$85.5$$
(iii) Total paid to 30 employees is $12,084$.
Step 1: Calculate total basic wages for 30 employees.
$$380 \times 30 = 11400$$
Step 2: Calculate total overtime wages.
$$12084 - 11400 = 684$$
Answer: $$684$$
(iv) Calculate total overtime hours worked.
Step 1: Use overtime wage rate to find hours.
$$\frac{684}{14.25} = 48$$
Answer: $$48$$
2. (a) Simplify $$\frac{2x}{5} - \frac{x}{3}$$ as a single fraction.
Step 1: Find common denominator: $15$.
Step 2: Rewrite fractions.
$$\frac{2x}{5} = \frac{6x}{15}, \quad \frac{x}{3} = \frac{5x}{15}$$
Step 3: Subtract numerators.
$$\frac{6x - 5x}{15} = \frac{x}{15}$$
Answer: $$\frac{x}{15}$$
(b) Factorise completely $$a^2b + 2ab$$.
Step 1: Factor out common term $ab$.
$$ab(a + 2)$$
Answer: $$ab(a + 2)$$
(c) Express $p$ as the subject of $$q = \frac{p^2 - r}{t}$$.
Step 1: Multiply both sides by $t$.
$$qt = p^2 - r$$
Step 2: Add $r$ to both sides.
$$p^2 = qt + r$$
Step 3: Take square root.
$$p = \pm \sqrt{qt + r}$$
Answer: $$p = \pm \sqrt{qt + r}$$
(d) Donuts sold in boxes.
(i) Expression for total donuts sold.
Step 1: Small boxes: $8$ boxes with $x$ donuts each.
Step 2: Large boxes: $5$ boxes with $2x + 3$ donuts each.
Step 3: Total donuts:
$$8x + 5(2x + 3) = 8x + 10x + 15 = 18x + 15$$
Answer: $$18x + 15$$
(ii) Total donuts sold is $195$.
Step 1: Set equation.
$$18x + 15 = 195$$
Step 2: Subtract $15$.
$$18x = 180$$
Step 3: Divide by $18$.
$$x = 10$$
Step 4: Calculate donuts in large box.
$$2x + 3 = 2(10) + 3 = 23$$
Answer:
- Small box: $$10$$ donuts
- Large box: $$23$$ donuts
3. (a) Simplify $$7p^2q^3 \times 2p^2q$$.
Step 1: Multiply coefficients.
$$7 \times 2 = 14$$
Step 2: Add exponents for like bases.
$$p^{2+2} = p^4, \quad q^{3+1} = q^4$$
Answer: $$14p^4q^4$$
(b) Milk carton volume.
(i) Volume of cuboid:
$$V = l \times w \times h = 6 \times 4 \times 10 = 240 \text{ cm}^3$$
Answer: $$240$$ cm³
(ii) Recipe requires $3$ litres = $3000$ cm³.
Step 1: Number of cartons needed.
$$\frac{3000}{240} = 12.5$$
Step 2: Must buy whole cartons, so round up.
Answer: $$13$$ cartons
(iii) Height of milk in cylindrical cup.
Step 1: Volume of milk = $240$ cm³.
Step 2: Radius of cup.
$$r = \frac{5}{2} = 2.5 \text{ cm}$$
Step 3: Volume of cylinder formula.
$$V = \pi r^2 h$$
Step 4: Solve for height $h$.
$$h = \frac{V}{\pi r^2} = \frac{240}{3.14 \times (2.5)^2} = \frac{240}{3.14 \times 6.25} = \frac{240}{19.625} \approx 12.23$$
Answer: $$12.2$$ cm (3 significant figures)