Population Density

Gross density counts all residents over the entire city area. Net residential density counts those same residents only over the part of the city where they live. Since the residential area is only 60 percent of the total area, the same population is concentrated into a smaller area, so the net density must be higher than the gross density.

Let total area be \(A\). Population is \(300A\), while residential area is \(0.6A\). Thus net density is \(\frac{300A}{0.6A}=\frac{300}{0.6}=500\) persons per square kilometre. The correct choice is C; 300 would incorrectly use the whole area.

When the question asks for density over habitable land, only the habitable area should be used as the denominator. The relevant population is 320,000 and the relevant area is 800 square kilometres. Therefore, habitable-area density is \(320000\div800=400\) persons per square kilometre.

Option D is correct. If total area were used instead, the result would be \(320000\div2000=160\), which is the ordinary density over the whole region, not the density of the land where people can live. The distinction between total-area density and habitable-area density is important: changing the denominator changes the meaning of the answer, even though the population remains the same.

Area and population density are connected through the relation population = density × area. Therefore, when population and density are known, area is found by dividing population by density. Here the population is 750,000 and the density is 625 persons per square kilometre, so the units of the answer are square kilometres.

Area = \(750000 \div 625 = 1200\) square kilometres. A useful check is to multiply the answer by the given density: \(625 \times 1200 = 750000\). Since this exactly gives the stated population, option B, 1,200 square kilometres, is correct. Choosing 1,000 or 1,250 would produce different populations at the same density.

Population is found by multiplying density by area. Density tells us how many people occupy each square kilometre, so multiplying it by the number of square kilometres gives the total number of people. The units also confirm this: persons per square kilometre multiplied by square kilometres leaves persons.

Here, population = \(420 \times 1750\). We can calculate \(1750 \times 42 = 73500\), and then multiply by 10 because 420 is 42 times 10. This gives 735000 people. Equivalently, the product can be written directly as \(420 \times 1750 = 735000\). Therefore, option B, 735000, is correct. The other choices do not match this product.

When population and area both change, the new density is found by dividing the new population by the new area. A 40 percent population increase changes the population to 140 percent, or 1.40 times, its original value. A 12 percent area increase changes the area to 112 percent, or 1.12 times, its original value.

The density multiplier is \(1.40\div1.12=1.25\). Thus, the new density is \(350\times1.25=437.5\) persons per square kilometre. Therefore, option C is correct. The result is not 490, because that would multiply the original density by the population factor alone and ignore the increase in area, which reduces the effect on density.

Correct answer: B, the area remains unchanged. The formula connecting these quantities is density = population ÷ area, or area = population ÷ density. A 20% decrease changes both population and density by the factor 0.80. Thus the new area factor is 0.80 ÷ 0.80 = 1. The area is therefore exactly the same as before. A is wrong because the area does not automatically follow the population decrease; density also decreases by the same proportion. C and D are wrong because there is no excess population change relative to density that would require a larger area. For example, if population is 800 and density is 40, area is 20. After both fall by 20%, they become 640 and 32, and 640 ÷ 32 is still 20. The memory rule is: if population and density change by equal proportions, their ratio—and therefore area—does not change.

The relationship among population, density, and area is \(\text{Density}=\frac{\text{Population}}{\text{Area}}\), so \(\text{Area}=\frac{\text{Population}}{\text{Density}}\). A 50% population increase multiplies population by 1.5. A 25% density decrease means the new density is 75% of the old density, so it is multiplied by 0.75. The area multiplier is therefore \(\frac{1.5}{0.75}=2\).

Thus the new area is twice the original area, making option C correct. It is important not to subtract 25 from 150 or simply add the percentage changes, because percentage changes apply multiplicatively to different quantities. The population increases while density decreases, so a substantially larger area is required to accommodate the population at the lower density. The supplied answer and explanation are correct.

Let the country’s total population be represented by 70 units and its total land area by 30 units for the selected comparison. The selected portion contains 70 percent of the population but only 30 percent of the area. Its density relative to the national average is therefore the population share divided by the area share: \(\frac{0.70}{0.30}=2.333\ldots\). The national density of 180 is not needed to identify the multiplier.

For a numerical check, the portion’s average density is approximately \(180\times\frac{0.70}{0.30}=420\) persons per square kilometre. This is about 2.33 times the national average, so option C is correct. The comparison is not 70 divided by 30 as ordinary percentages without interpreting them as shares; the ratio of shares is what matters.

Average density uses the total area, including land where people cannot live. If 40 percent is uninhabitable, only 60 percent of the total area is habitable. The total population remains the same, but it is concentrated in this smaller area. Consequently, density on habitable land is higher than the overall average. The required adjustment is to divide the average density by 0.60.

Thus the effective density is \(200/0.60=333.33\) persons per square kilometre, approximately 333.3. Option C is therefore correct. The answer is not 240 because removing unusable land does not reduce the population; it concentrates the same residents in the remaining 60 percent. This calculation assumes all residents live in the habitable portion.

Gross density uses the whole city area, while net residential density uses only the land where residents live. If residential land occupies just 40 percent of the total area and the entire population lives there, the same population is concentrated on a smaller area. Therefore, net residential density must be higher than gross density.

Let total area be A and population be P. Gross density gives \(P/A=240\). Residential area is \(0.4A\), so net density is \(P/(0.4A)=240/0.4=600\) persons per square kilometre. Thus, option C is correct. The value 480 would correspond to a different proportion and does not follow from the stated 40 percent.

Gross density is calculated using the whole city area, including homes, parks, factories, roads, and other land. Residential density uses the area actually available for housing. If two cities have the same population per total square kilometre, but one devotes more land to parks and industry, less of its total area remains residential. The same residents can therefore be concentrated in a smaller residential area, producing a higher residential density.

Option A is the best answer because it says residential density may be higher, not that it must be higher in every possible measurement. Option B reverses the likely relationship, C assumes equality without evidence, and D is incorrect because gross density does include area. The distinction between gross and residential density is the key idea.

Density compares population with the area occupied by that population. It is found by dividing population by area, so a change in both quantities must be considered as a ratio rather than separately. If population and area increase by the same percentage, density does not change. This is because the numerator and denominator are multiplied by the same factor.

Initially, density is \(100000/500=200\) persons per square kilometre. Later it is \(140000/700=200\) persons per square kilometre. Both population and area become 1.4 times their original values, so the ratio remains the same. Therefore, density is unchanged and option B is correct. A 20 percent increase or decrease would be possible only if population and area changed by different proportions.

Density must be recalculated before finding its percentage change, because both population and area change. Initially, density is population divided by area: \(100000/400=250\) persons per square kilometre. After the changes, it is \(150000/500=300\). The increase is therefore 50 persons per square kilometre.

Percentage increase is measured against the original density, not the new one. Thus it is \(50/250 \times 100=20\%\). Option B is correct. A 50 percent rise in population does not produce a 50 percent density rise here because the area also increases, spreading the population over more land.

Density is a ratio between population and area, so both quantities must be compared together. A fall in population does not necessarily reduce density; the result depends on whether area falls by the same proportion, a smaller proportion, or a larger proportion. Here both figures fall by 20 percent.

Initially, density is \(250000\div1000=250\) persons per square kilometre. Afterwards, it is \(200000\div800=250\) persons per square kilometre. Equivalently, both numerator and denominator become 80 percent of their original values, so their ratio stays unchanged. Therefore, option C is correct; it does not become zero merely because both quantities decline.

To compare the density of a selected portion with the density of the whole region, compare the fraction of population in that portion with the fraction of area it occupies. The portion contains 60% of the total population but occupies only 20% of the total area. Because it contains population three times as large relative to its area, its density is three times the regional average.

The ratio is \(0.60 \div 0.20 = 3\). The total population value of 450,000 is not needed, because both densities use the same total population and total area, which cancel in the comparison. Thus the selected portion has three times the total regional density, so option C is correct. This is not a comparison of percentage-point differences.

Average regional density is total population divided by total area. If a portion contains 45 percent of the population but only 15 percent of the area, compare its population share with its area share. The regional average can be represented by 100 percent population spread over 100 percent area. The portion’s relative density is therefore its population fraction divided by its area fraction.

Compute \(0.45\div0.15=3\). This means the selected portion has three times the regional average density, so option B is correct. It is not 45 minus 15, because density depends on a ratio. A threefold result means that each unit of area in this portion supports, on average, three times as many people as the regional average.

Density is population divided by area. The daytime and resident populations occupy the same 500 square kilometres, so calculate each density separately and then subtract. Daytime density is \(600000\div500=1200\) persons per square kilometre. Resident density is \(400000\div500=800\) persons per square kilometre.

The difference is \(1200-800=400\) persons per square kilometre. Therefore, option C is correct. The same result can be found more quickly by subtracting the populations first: 600,000 minus 400,000 equals 200,000, and \(200000\div500=400\). The answer is a density difference, not a difference in total population.

A percentage increase must be measured from the original or reference value. Here the nighttime resident density is the reference because the question asks how much higher the daytime density is than resident density. The absolute difference alone is 280 persons per square kilometre, but that difference must be converted into a percentage using 700, not 980.

The difference is \(980-700=280\). The percentage increase is \((280/700)\times100=40\%\). Therefore, daytime density is 40 percent higher than nighttime resident density, making option C correct. Dividing by 980 would answer a different question: what percentage of daytime density the difference represents. The higher daytime value may reflect people present during the day who do not reside there permanently.

Population density is a ratio: population divided by area. If a region's boundaries change, its measured area may change even when its population does not. When the area becomes smaller while the population stays constant, the quotient becomes larger. Therefore, a rise in calculated density after boundary contraction does not by itself prove population growth. Option B is correct.

For example, a fixed population of 100 people has density 10 in an area of 10 units, but density 20 if the measured area becomes 5 units. The increase comes only from the denominator becoming smaller. Boundary changes can therefore create an apparent statistical increase. Population growth could also raise density, but the given information does not establish that it happened.

Population density is calculated by dividing the population included in a boundary by the area inside that same boundary. A municipal area usually refers to the official limits of a city, while a metropolitan area may include the city and surrounding suburbs or connected settlements. These are different spatial units, so their population and area values are not directly equivalent.

Because the denominator, and often the numerator too, changes with the boundary, two density figures may describe different realities even when the same formula is used. Therefore, option A is correct: different spatial definitions change the denominator. The formula itself does not change, and metropolitan density can certainly be calculated when consistent data are available.

Density is population divided by area. If density must remain unchanged, population and area have to change by the same multiplier. An 80 percent increase in population means that the new population is 180 percent of the old population, or 1.8 times the original. To keep the ratio unchanged, area must also become 1.8 times as large.

Thus the area increases by 80 percent, so option B is correct. If area stayed unchanged, density would also rise by 80 percent because there would be more people in the same space. A 40 percent increase would not be enough to preserve the original ratio. The conclusion follows directly from keeping population divided by area constant.

Answer: B, 75 percent. Population density is calculated as population divided by area: D = P/A. Let the original population and area be P and A, so the original density is P/A. A 50 percent population increase makes the new population 1.5P. Doubling density makes it 2P/A. Therefore the new area is (1.5P)/(2P/A) = 0.75A, or 75 percent of the original area. A is wrong because it ignores the population increase. B is correct because it gives the calculated area factor. C would mean area stayed unchanged, which would make density rise only 50 percent. D would mean area increased, although density increased faster than population. Memory cue: area factor = population factor ÷ density factor.

Population density equals population divided by area. When population does not change, any reduction in area makes the same number of people occupy a smaller space, so density rises. The percentage increase in density is not equal to the percentage decrease in area, because the new, smaller area becomes the denominator.

Let the original area be \(A\) and population be \(P\). A 40 percent decrease leaves \(0.6A\). New density is \(P/(0.6A)=1.6667(P/A)\). The increase is therefore \(1.6667-1=0.6667\), or about 66.7 percent. Hence option C is correct; 40 percent would incorrectly use the original denominator.

Answer: C, approximately 33.3 percent. Since D = P/A, when population is fixed, area varies inversely with density. A 25 percent decrease changes density from D to 0.75D. The new area is P/(0.75D) = (1/0.75)(P/D) = 1.333A. Thus the area becomes 133.3 percent of its former value, so its increase is 33.3 percent. A is wrong because inverse changes are not equal in percentage terms. B is the approximate result of the correct reciprocal calculation, not exactly 30 percent. C is correct. D is too large. Common confusion: a 25 percent fall in one quantity does not produce a 25 percent rise in its inverse.

Population density tells us how many people live in one square kilometre. The relationship among population, density, and area is population = density × area. Therefore, when population and density are known, area must be found by dividing population by density. The units also confirm this: people divided by people per square kilometre leaves square kilometres.

Use the calculation area = 640,000 ÷ 800. Dividing by 8 gives 80,000, and then accounting for the two zeros in 800 gives 800 square kilometres. A reverse check is useful: 800 people per square kilometre × 800 square kilometres = 640,000 people. Therefore, option C is correct. The other choices would produce populations different from the stated total when multiplied by the density.