Elize, Lily and Marco start a business.
(a) Elize invests \$5000.
Lily invests \$8000.
Marco invests \$3000.
After one year they make a profit of \$40 000.
They share this profit in the ratio of their investments.
Work out how much they each receive.
(b)(i) Lily buys 20 rolls of ribbon.
8 are red, 6 are blue, 4 are yellow and 2 are pink.
A roll of ribbon is chosen at random.
On the probability scale, draw an arrow (↓) to show the probability that this roll is
(a) yellow
(b) not red
(c) green.
(ii) The length, l m, of a roll of ribbon is 120 m, correct to the nearest metre.
Complete this statement about the value of l: $…… < l < ……$
(c) Elize buys some picture frames.
The frames cost \$5.80 each in New York and 4.50 euros each in Paris.
The exchange rate is 1 euro = \$1.37.
Calculate the difference in the cost in euros. Give your answer correct to 2 decimal places.
(d) Elize buys a framed picture.
(i) 
The picture is a circle with diameter 18 cm.
The frame is a square of side length 18 cm.
Calculate the shaded area.
(ii) Elize buys the framed picture for \$12.50.
She sells the framed picture for \$20.25.
Calculate the percentage profit.
▶️ Answer/Explanation
(a) Elize: \$12,500, Lily: \$20,000, Marco: \$7,500
Total investment ratio is 5:8:3. Profit share is calculated by dividing \$40,000 in this ratio.
(b)(i) (a) Arrow at 0.2 (4/20), (b) Arrow at 0.6 (12/20), (c) Arrow at 0 (no green ribbons)
Probabilities are calculated by dividing favorable outcomes by total outcomes.
(ii) 119.5 < l < 120.5
For measurements correct to nearest meter, the range is ±0.5m.
(c) 0.27 euros
Convert \$5.80 to euros (5.80/1.37 ≈ 4.2336), then subtract Paris price (4.50) to find difference.
(d)(i) 69.5 cm²
Shaded area = square area (18×18) minus circle area (π×9²). Calculation gives 324 – 254.469 ≈ 69.5.
(ii) 62%
Profit = (20.25-12.50)/12.50 × 100 = 7.75/12.50 × 100 = 62%.
(a) Ancel has 80 tea bags, $\frac{1}{2}$ kg of sugar and 1 litre of milk.
To make a cup of tea he uses:
- 1 tea bag
- 8 grams of sugar
- 40 millilitres of milk.
(i) In the morning, Ancel makes 15 cups of tea.
Work out
(a) the fraction of the tea bags he uses, in its simplest form
(b) the mass of sugar, in grams, he has left.
(ii) During the day, Ancel uses all of the milk to make cups of tea.
Work out the total number of cups of tea Ancel makes.
(b) Bobby, Carl and Davood share $6875 in the ratio
Bobby : Carl : Davood = 6 : 8 : 11
Calculate the amount of money they each receive.
(c) (i) Write $\frac{3^2 \times 3^4}{3^6}$ as a power of 3.
(ii) Write the value of $2^{-4}$ as a decimal.
(d) Simplify.
(i) $(b^5)^3$
(ii) $\left(\frac{4}{m}\right)^{-2}$
(e) $30 = 2 \times 3 \times 5$ $84 = 2^2 \times 3 \times 7$
Use this information to find the lowest common multiple (LCM) of 30 and 84.
(f) $\frac{2}{9}$ $\sqrt{7}$ $\frac{5}{4}$ $\sqrt{16}$ $2^3$
Put a ring around the irrational number in this list.
▶️ Answer/Explanation
3(a)(i)(a): $\frac{3}{16}$
15 tea bags used out of 80 gives $\frac{15}{80}$ which simplifies to $\frac{3}{16}$.
3(a)(i)(b): 380 g
$\frac{1}{2}$ kg = 500 g. Used 15 × 8 g = 120 g. Left with 500 – 120 = 380 g.
3(a)(ii): 25
1000 ml milk available. Each cup uses 40 ml, so $\frac{1000}{40} = 25$ cups.
3(b): Bobby \($1650\), Carl \($2200\), Davood \($3025\)
Total parts = 6 + 8 + 11 = 25. Each part = $\frac{6875}{25} = 275$. Multiply by ratio parts.
3(c)(i): $3^0$
Numerator: $3^{2+4} = 3^6$. Then $\frac{3^6}{3^6} = 3^0 = 1$.
3(c)(ii): 0.0625
$2^{-4} = \frac{1}{2^4} = \frac{1}{16} = 0.0625$.
3(d)(i): $b^{15}$
Multiply exponents: $(b^5)^3 = b^{5×3} = b^{15}$.
3(d)(ii): $\frac{m^2}{16}$
Reciprocal and square: $\left(\frac{4}{m}\right)^{-2} = \left(\frac{m}{4}\right)^2 = \frac{m^2}{16}$.
3(e): 420
LCM = product of highest powers: $2^2 × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420$.
3(f): $\sqrt{7}$
$\sqrt{7}$ is irrational as it cannot be expressed as a fraction and has non-terminating decimal.