Population Density

Correct answer: C. 540 persons per square kilometre

Explanation: Direct answer: Option C, 540 persons per square kilometre. First calculate the original population: density × area = 450 × 800 = 360,000 people. Add the increase of 72,000: new population = 360,000 + 72,000 = 432,000. The area remains 800 square kilometres, so new density = 432,000 ÷ 800 = 540 persons per square kilometre. Option A is too low and does not reflect the complete increase. Option B is also below the exact result. Option C is correct. Option D is too high because it would require a larger population increase than 72,000. The safe method is to find the original population first, add or subtract the population change, and then divide by the unchanged area.

Correct answer: B. 300 persons per square kilometre

Explanation: Direct answer: Option B, 300 persons per square kilometre. First find the original population: 360 × 500 = 180,000 people. Out-migration removes 30,000 people, so the remaining population is 180,000 − 30,000 = 150,000. The area is unchanged at 500 square kilometres. Therefore, new density = 150,000 ÷ 500 = 300 persons per square kilometre. Option A is incorrect because it subtracts too much density. Option B is correct because it follows the complete calculation. Option C is too high and does not remove the full migration amount. Option D is also higher than the correct result. Remember: when area remains fixed, a fall in population causes density to fall in the same proportion as population.

Correct answer: A. From 300 to 240 persons per square kilometre

Explanation: Density depends on both population and area, using \(D=\frac{P}{A}\). At first, the population is 360,000 and the area is 1,200 square kilometres, so the original density is \(360000\div1200=300\) persons per square kilometre. After the boundary change, the population is treated as unchanged, but the area becomes 1,500 square kilometres.

The new density is therefore \(360000\div1500=240\) persons per square kilometre. The density falls from 300 to 240, because the same population is now spread over a larger area. Thus option A is correct. The change in boundary changes the denominator in the formula; it does not automatically change the stated population. This explains why options B, C, and D are unsuitable.

Correct answer: B. Area growth rate is greater than population growth rate

Explanation: Population density is a ratio: \(\text{Density}=\frac{\text{Population}}{\text{Area}}\). When both population and area increase, the result depends on which one grows faster in relative terms. For density to decrease, the denominator, area, must increase proportionally more than the numerator, population. Thus the area growth rate must be greater than the population growth rate. This makes the amount of area available per person larger.

For example, if population rises by 10% but area rises by 20%, the new density multiplier is \(\frac{1.10}{1.20}\), which is less than 1. Therefore density falls. If population grew faster, density would rise instead. Equal growth rates would leave density unchanged, not reduce it. Population being zero is not the required condition here. Hence option B is correct.

Correct answer: B. Area decreases proportionally faster

Explanation: Density is a ratio: population is the numerator and area is the denominator. If both values decrease, the result does not depend only on whether they decrease; it depends on their proportional changes. For density to rise, area must shrink by a greater percentage than population. Then the denominator becomes relatively smaller, making the population per square kilometre larger. Therefore, choice B is correct.

For example, suppose population falls by 10 percent, so it becomes 90 percent of its original value, while area falls by 20 percent, becoming 80 percent. The new ratio is \(0.90P/0.80A=1.125(P/A)\), so density rises by 12.5 percent. Equal percentage decreases would leave density unchanged. A faster population decrease would lower density, and density does not automatically become zero.

Correct answer: B. Different boundary definitions or reference years may be used

Explanation: Population density uses the same basic relationship: population divided by land area. Differences between reported values usually arise because sources do not always measure the same geographic unit or the same time period. One source may count only the municipal area, while another may include the wider urban agglomeration or metropolitan region.

A census year and a later estimate can also contain different population totals. These changes affect either the numerator, the denominator, or both, so the final density differs even though the method is unchanged. Option B is correct. Language does not alter a population count, area is essential to density, and reputable sources do not need different density formulas.

Correct answer: B. No because their areas may differ

Explanation: A density class on a map represents a range of people per unit area, not an exact total population. Two regions may both belong to the same class because their densities are similar or fall within the same mapped interval. However, their areas can be different, and total population depends on both density and area: \(P=D\times A\).

For example, if two regions have roughly the same density but one is twice as large, the larger region can contain approximately twice as many people. Therefore, equal map classes do not require equal total populations. Option B is correct because their areas may differ. Option A confuses density with total population, option C incorrectly says density ignores area, and option D is false because density is calculated from population and area.

Correct answer: B. When the boundary changed between years but this was ignored

Explanation: Direct answer: Option B. A density trend is meaningful only when the areas being compared are spatially comparable. If the administrative or regional boundary changes between two years and the change is ignored, the area may include different places in each year. The calculated density may then appear to rise or fall because of the boundary change rather than a real change in population concentration. Option A is not misleading because the same boundary and unit make comparison consistent. Option B is correct because an ignored boundary change can distort the trend. Option C is not a problem; knowing population and area is necessary for calculating density. Option D is also safe because the same unit makes values comparable. Memory cue: compare like with like—same boundary, same area basis, and same unit.

Correct answer: B. Analyzing percentage changes, migration, and boundary effects along with the population-to-area ratio

Explanation: The basic measure of population density is the ratio of population to area, written as \(Population\ Density=\frac{Population}{Area}\). Medium-level analysis should use this ratio but should not stop at memorizing a total or labelling every large population as dense. It should examine how density changes over time and why.

Percentage changes can show whether density is rising or falling. Migration explains movement into or out of a region, while administrative boundary changes can alter the measured ratio without an equivalent change in residents. Thus option B is most appropriate because it combines calculation with interpretation. Options A, C and D ignore area or the factors needed to understand the result.

Correct answer: B. 10%

Explanation: Answer: B, 10%. Population density means population divided by area: D = P/A. Take the original population and area as 1 each, so the original density is 1. After the changes, population becomes 1.54 times the original and area becomes 1.40 times the original. Therefore, the new density is 1.54/1.40 = 1.10 times the old density. This means density is 110% of its original value, so its increase is 10%. A is incorrect because the ratio does not produce 5%. C is the common error of subtracting 40% from 54%; percentage changes in a ratio cannot normally be found by simple subtraction. D is not supported by the calculation. Memory cue: for density, multiply the population factor by the reciprocal of the area factor.

Correct answer: B. 20%

Explanation: Answer: B, 20%. Density is population divided by area. After a 36% population decrease, the population factor is 64% or 0.64. After a 20% area decrease, the area factor is 80% or 0.80. The new density factor is therefore 0.64/0.80 = 0.80. Thus, the new density is 80% of the old density, which means it has decreased by 20%. A is not obtained from the ratio. C may result from an incorrect comparison of the two decreases. D incorrectly treats the population decrease as the density decrease. The key point is that both the numerator and denominator change. Memory cue: convert decreases into remaining fractions before dividing.

Correct answer: B. 66.7%

Explanation: Answer: B, approximately 66.7%. The phrase ‘what percentage of the higher density’ means that the lower value must be divided by the higher value. Therefore, percentage = (360/540) x 100 = 2/3 x 100 = 66.666..., which rounds to 66.7%. A would correspond to 360 being 60% of 600, not 540. C and D are also incorrect because they use the wrong ratio. Do not calculate the percentage difference unless the question asks how much one value is greater or smaller than the other. For comparison, 540 is 50% greater than 360, but 360 is 66.7% of 540. Memory cue: ‘percentage of’ uses the reference value in the denominator.

Correct answer: C. 490 persons per square kilometre

Explanation: Answer: C, 490 persons per square kilometre. ‘75% higher than 280’ means the original 280 plus an additional 75% of 280. The increase is 0.75 x 280 = 210. Adding this to the original gives 280 + 210 = 490. Equivalently, use the multiplier 1 + 0.75 = 1.75: 280 x 1.75 = 490. A is 280 x 1.5 and represents only a 50% increase. B does not follow from 75%. D would be 280 x 2 and would represent a 100% increase. A percentage is not an absolute number, so 75 cannot simply be added to 280. Memory cue: ‘x% higher’ means multiply by 1 + x/100.

Correct answer: C. 120%

Explanation: Answer: C, 120%. The basic relationship is population = density x area, because density = population/area. If density becomes 1.25 times its original value and area becomes 0.96 times its original value, multiply these factors: new population factor = 1.25 x 0.96 = 1.20. Therefore, the population will be 120% of the original population, which is a 20% increase. A ignores the density change. B would result from an incorrect combination of the percentages. D uses only the density factor and ignores the reduction in area. The wording asks for the percentage remaining, so answer 120% is correct; it also means the population increased by 20%. Memory cue: when P = D x A, multiply the density and area multipliers.

Correct answer: B. 66.7%

Explanation: Answer: B, approximately 66.7%. To find what percentage one quantity is of another, divide the first quantity by the reference quantity and multiply by 100. Here, resident density is the first quantity and daytime density is the reference: (1200/1800) x 100 = 66.666..., or about 66.7%. A, C and D use incorrect ratios or approximations. This question does not ask the percentage by which daytime density exceeds resident density. That different question would use (1800 - 1200)/1200 x 100 = 50%. Thus, the denominator must be 1800 because the wording says ‘of daytime density’. Memory cue: identify the words after ‘of’; that value is usually the base or denominator.