(a) The area of Cuba, in square kilometres, is one hundred and five thousand eight hundred and six.
Write this number in figures.
(b) The population of an island is 103000.
Write this number in standard form.
(c) The table shows some populations in 2014.
(i) Write the population of St Maarten as an ordinary number.
(ii) Complete the statement.
The population of Haiti is approximately …………………. times the population of the US Virgin Islands.
(iii) Find the difference between the population of Haiti and the population of Puerto Rico.
Give your answer in standard form.
(d) In 2013 the population of a town was 30405.
In 2014 the population was 30851.
Calculate the percentage increase in the population.
▶️ Answer/Explanation
(a) Ans: 105,806
One hundred five thousand = 105,000
Eight hundred six = 806
Combined: 105,806
(b) Ans: \(1.03 \times 10^5\)
Standard form requires \(1 \leq a < 10\): 103,000 = 1.03 × 105
(c)(i) Ans: 46,100
From table: St Maarten population = 4.61 × 104 = 46,100
(c)(ii) Ans: 100
Haiti: 1.04 × 107 = 10,400,000
US Virgin Islands: 1.04 × 105 = 104,000
Ratio: 10,400,000 ÷ 104,000 ≈ 100
(c)(iii) Ans: \(6.82 \times 10^6\)
Haiti: 10,400,000
Puerto Rico: 3,580,000
Difference: 10,400,000 – 3,580,000 = 6,820,000 = 6.82 × 106
(d) Ans: 1.47%
Increase = 30,851 – 30,405 = 446
Percentage increase = (446 ÷ 30,405) × 100 ≈ 1.47%
(a) (i) Calculate the cube root of 68 921.
(ii) Write 68 921 in standard form.
(iii) Write 68 921 correct to 2 significant figures.
(b) 96 550 kg is reduced to 88 826 kg. Calculate the percentage reduction.
(c) (i) Work out 5–2.
(ii) Simplify \((\sqrt{5})^{2}\).
(iii) Explain why 6 is not a prime number.
(iv) Explain the term ‘irrational number’.
▶️ Answer/Explanation
(a) (i) Ans: 41
Check \(41^3 = 41 \times 41 \times 41 = 68,\!921\). Thus, \(\sqrt[3]{68,\!921} = 41\).
(ii) Ans: \(6.8921 \times 10^4\)
Move the decimal point 4 places left: \(68,\!921 = 6.8921 \times 10^4\).
(iii) Ans: 69 000
The third digit (8) rounds the second digit (9) up: \(68,\!921 \approx 69,\!000\) (2 s.f.).
(b) Ans: 8%
Reduction: \(96,\!550 – 88,\!826 = 7,\!724\) kg. Percentage: \(\frac{7,\!724}{96,\!550} \times 100 \approx 8\%\).
(c) (i) Ans: \(\frac{1}{25}\) or 0.04
Negative exponents give reciprocals: \(5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04\).
(ii) Ans: 5
Squaring a square root cancels both: \((\sqrt{5})^2 = 5\).
(iii) Ans: 6 has more than two factors
Prime numbers have exactly two distinct factors (1 and themselves). 6 has four factors: 1, 2, 3, 6.
(iv) Ans: Non-terminating, non-repeating decimal
An irrational number cannot be written as a fraction \(\frac{a}{b}\) (where \(a, b\) are integers). Its decimal form never terminates nor repeats (e.g., \(\pi\), \(\sqrt{2}\)).