Question
The park has an area of $4620$ m$^{2}$
(a) Work out the area of the gardens and the area of the playground
(b) The park area of $4620$ m$^{2}$ is made up of paths and grassland
$18\%$ of the park area is paths.
(i) Show that the grassland area is $3788.4$ m$^{2}$
(ii) Seed for the grassland is sold in bags.
The seed in one bag covers an area of $280\mathrm{m}^{2}$.
The bags cost \(\$72\) each for the first 5 bags and then \(\$58\) each for any extra bags
Calculate the cost of the seed needed to cover the grassland
(c) The owners of the land buy new equipment for the playground.
They borrow \(\$8500\) for 4 years at a rate of $6.5\%$ per year compound interest.
Calculate the amount they repay at the end of the $4$ years.
Give your answer correct to the nearest dollar.
(d) The café in the park sells water in bottles A, B and C
Work out which bottle is the best value
You must show all your working
▶️Answer/Explanation
(a): $840$, $1260$
(b)(i): $\frac{(100 – 18)}{100} \times 4620$
(b)(ii): $882$
(c): $10935$
(d): A With correct comparisons made of the 3 bottles with suitable accuracy shown.
Detailed Answer
(a)
The ratio of park : gardens : playground
$
11 : 2 : 3
$
The park area is given as \( 4620 \, \mathrm{m^2} \).
$
11 \text{ parts} = 4620 \implies 1 \text{ part} = \frac{4620}{11} = 420 \, \mathrm{m^2}
$
Calculate the area of the gardens
$
2 \times 420 = 840 \, \mathrm{m^2}
$
Calculate the area of the playground:
$
3 \times 420 = 1260 \, \mathrm{m^2}
$
(b)(i)
18% of the park area is paths
$
0.18 \times 4620 = 831.6 \, \mathrm{m^2}
$
Grassland area
$
4620 – 831.6 = 3788.4 \, \mathrm{m^2}
$
(b)(ii)
Number of bags required
$
\frac{3788.4}{280} \approx 13.53
$
Round up to 14 bags.
First 5 bags: \( 5 \times 72 = 360 \)
Remaining 9 bags: \( 9 \times 58 = 522 \)
Total cost
$
360 + 522 = 882
$
(c)
compound interest
$
A = P \left( 1 + \frac{r}{100} \right)^t
$
\( P = 8500 \)
\( r = 6.5\% \)
\( t = 4 \)
$
A = 8500 \left( 1 + \frac{6.5}{100} \right)^4
$
$
= 8500 \times (1.065)^4
$
$
= 8500 \times 1.2748
$
$
\approx 10936
$
(d)
Bottle A
$
\frac{1.98}{330} \approx 0.006
$
Cost: $\$0.006/mL$
Bottle B
$
\frac{3.20}{500} = 0.0064
$
Cost: $\$0.0064/mL$
Bottle C
$
\frac{5.10}{750} = 0.0068
$
Cost: $\$0.0068/mL$
Bottle A is the best value at $\$0.006/mL$.
Question
(a) Chen asks some people if they prefer a beach, cruise, lake or mountain holiday.
The pie chart shows the results.
(i) Find the fraction of people who prefer a mountain holiday.
Give your answer in its simplest form.
(ii) Find the percentage of people who prefer a beach holiday.
(iii) Find the ratio of people who prefer each type of holiday in the form
$\text{beach:cruise: lake: mountain.}$
Give your answer in its simplest form.
(iv) One person is chosen at random
Find the probability that this person prefers a cruise or a lake holiday
(v) $675$ people prefer a beach holiday.
Show that the total number of people Chen asks is $1800$
(vi) (a) Complete the table.
(b) Complete the bar chart, including the scale on the frequency axis.
(b) Mr Gibb pays $\$2208$ for a holiday. Mr Shah pays 2050 euros for the same holiday.The exchange rate is $1$ euro=$\$1.15$.
Work out how much more, in curos, Mr Shah pays for the holiday than Mr Gibb.
▶️Answer/Explanation
(a)(i): $\frac{1}{3}$
(a)(ii): $37.5$
(a)(iii): $9:2:5:8$
(a)(iv): $\frac{105}{360}$
(a)(v): $\frac{675 \times 360}{135} = 1800$
(a)(vi)(a): $375$, $600$
(a)(vi)(b): A correct bar chart drawn with linear scale labelled.
(b): $130$
Detailed Answer
Total angle in a circle: \( 360^\circ \)
Mountain: \( 120^\circ \)
Beach: \( 135^\circ \)
Lake: \( 75^\circ \)
Cruise: \( 30^\circ \)
(i)
Fraction = \(\frac{\text{Angle for mountain}}{360}\)
$
= \frac{120}{360} = \frac{1}{3}
$
(ii)
Percentage = \(\frac{\text{Angle for beach}}{360} \times 100\%\)
$
= \frac{135}{360} \times 100 = 37.5\%
$
(iii)
$
\text{Beach : Cruise : Lake : Mountain} = 135^\circ : 30^\circ : 75^\circ : 120^\circ
$
Divide by 15 (the GCD of all the angles):
$
= 9 : 2 : 5 : 8
$
(iv)
Add the angles for cruise and lake:
$
30^\circ + 75^\circ = 105^\circ
$
Probability = \(\frac{105}{360}\)
$
= \frac{7}{24}
$
(v)
675 people prefer a beach holiday, and the angle for the beach is 135°.
total number of people as x.
$
\frac{135}{360} \times x = 675
$
$
135x = 675 \times 360
$
$
x = 1800
$
(vi) (a)
$
\text{Frequency} = \frac{\text{Angle}}{360} \times 1800
$
Cruise: \( \frac{30}{360} \times 1800 = 150 \)
Lake: \( \frac{75}{360} \times 1800 = 375 \)
Mountain: \( \frac{120}{360} \times 1800 = 600 \)
(vi)(b)
(b)
Mr. Gibb pays: $\$2208$
Exchange rate: €\(1 = \$1.15\)
Mr. Shah pays: €\(2050\)
$
\text{Amount in euros} = \frac{2208}{1.15} = 1920 \text{ euros}
$
$
2050 – 1920 = 130 \text{ euros}
$
Question
(a) 6 144 63 11 288 72 8
From the list, write down
(i) the multiple of 7,
(ii) the cube of 2,
(iii) the prime number,
(iv) the lowest common multiple (LCM) of 16 and 18.
(b) Without using a calculator explain why the square of 4.86 must be between 16 and 25.
(c) Find the value of
(i) \(4^7\).
(ii) \(12^0\)
(iii) \(8.3^2 + \sqrt{27}\)
(d) Write 90 as the product of its prime factors.
Answer/Explanation
Answer:
(a) (i) 63
(ii) 8
(iii) 11
(iv) 144
(b) \(4^2[=]16\) \(5^2[=]25\)
(c) (i) 16384
(ii) 1
(iii) 74.1 or 74.08 to 74.09
(d) \(2 \times 3^2 \times 5\) or \(2 \times 3 \times 3 \times 5\)