Population Density

Correct answer: B. 667 persons per square kilometre

Explanation: Density is calculated by dividing population by area. If the population stays unchanged while the area becomes larger, the same number of people is spread across more square kilometres, so the density must decrease. A 20 percent increase means multiplying the original area by 1.20, not simply adding 20 square kilometres.

First find the original area: \(960000\div800=1200\) square kilometres. After a 20 percent increase, the area is \(1200\times1.20=1440\) square kilometres. The new density is therefore \(960000\div1440=666.67\), which rounds to about 667 persons per square kilometre. Hence option B is correct; 800 would be the old density, not the new one.

Correct answer: B. 3 times

Explanation: Let the region’s total population be \(P\) and its total area be \(A\). The selected part contains 30 percent of the population, or \(0.30P\), and 10 percent of the area, or \(0.10A\). Its density is therefore \(\frac{0.30P}{0.10A}=3\frac{P}{A}\). Since \(\frac{P}{A}\) is the region’s average density, the selected area has three times the average density.

Option B is correct. The key idea is that density rises when a population share is larger than the corresponding area share. Here 30 percent of people occupy only 10 percent of the land, so that part is relatively concentrated. The answer is not obtained by subtracting the percentages; it is obtained by dividing the population share by the area share. Thus \(0.30\div0.10=3\), not 2 or 4.

Correct answer: C. 440 persons per square kilometre

Explanation: National arithmetic density equals total population divided by total area. If 70% of the population lives in 35% of the area, the selected area contains population twice as large, relative to its area, as the national average. The concentration factor is therefore \(\frac{0.70}{0.35}=2\).

Multiplying the national density by this factor gives \(220\times2=440\) persons per square kilometre. Option C is correct. The calculation does not require the country’s actual population or total area because the percentages provide the relative concentration directly. Options A, B, and D do not follow from the given ratio.

Correct answer: A. Density alone does not fully represent resources technology and internal distribution

Explanation: Population density is an average: it tells us how many people live per unit of area. It is useful for comparing the spatial concentration of people, but it does not show the complete level of pressure on land and resources. Pressure also depends on the amount and quality of resources, technology, infrastructure, consumption, income, and how people are distributed within the region. A single average can hide crowded cities and nearly empty areas.

Therefore, density alone cannot prove that a highly populated area has a severe resource crisis. Technology and good infrastructure may help a dense region support many people, while a sparsely populated region may still face shortages or poor access. Total population and density are related but not identical: total population ignores area, whereas density includes it. Hence option A is the most careful and correct statement.

Correct answer: C. 400 persons per square kilometre

Explanation: Population density is calculated by dividing population by land area: \(\text{Density}=\frac{\text{Population}}{\text{Area}}\). The population remains 360,000, but the relevant new area is 900 square kilometres, not the former area of 1,200 square kilometres. Thus, \(\text{Density}=\frac{360000}{900}=400\) persons per square kilometre. The decrease in area makes the same population more concentrated.

Therefore, option C is correct. Option A would result from incorrectly using the old area, because \(360000\div1200=300\). Option B does not follow from the given division, and option D is also an incorrect calculation. The unit must be persons per square kilometre, since population is divided by area measured in square kilometres.

Correct answer: C. 33.3 percent

Explanation: Density increases when population rises and area becomes smaller, because the same or greater number of people is being compared with less land. Let the original population and area be \(P\) and \(A\), so the original density is \(P/A\). A 20% population increase gives \(1.2P\), and a 10% area decrease gives \(0.9A\). The new density is therefore \(\frac{1.2P}{0.9A}=1.3333\frac{P}{A}\).

The new value is about 133.33% of the original, so the increase is approximately 33.3%. Thus option C follows. The answer is not 20%, because the area decrease also raises density. This example shows why percentage changes in a ratio must be calculated by dividing the new numerator and denominator, rather than by adding or subtracting percentages casually.

Correct answer: B. 312.5 persons per square kilometre

Explanation: Use the density formula and apply the two percentage changes separately. A 25 percent population decrease leaves (0.75) of the original population. A 20 percent area increase makes the area (1.20) times its original size. Starting from 500 persons per square kilometre, the new density is (500 times 0.75 divided by 1.20 = 312.5).

Therefore option B is correct. Density falls by more than population alone would suggest because the denominator, area, also becomes larger. It would be 375 if only population changed, but expanding the area reduces it further to 312.5. The unit remains persons per square kilometre, and no rounding is needed because one option gives the exact decimal result.

Correct answer: A. 15:8

Explanation: Population density is population divided by area. For a ratio between two regions, divide the population ratio by the area ratio: \(D_1:D_2=(P_1/P_2)\div(A_1/A_2)\). This works because the first region's population and area must both be compared with those of the second region.

Here the population ratio is \(5:4\), and the area ratio is \(2:3\). Therefore, the density ratio is \((5/4)\div(2/3)=(5/4)(3/2)=15/8\). Equivalently, using proportional values, the densities are \(5/2\) and \(4/3\), whose ratio is also \(15:8\). Thus option A is correct.

Correct answer: B. 3:2

Explanation: Density equals population divided by area, so area equals population divided by density. Let the population ratio be 9:4 and density ratio be 3:2. The area ratio is therefore \(\frac{9}{3}:\frac{4}{2}=3:2\). Hence option B is correct.

The calculation must preserve the order of the two regions. For the first region, population units divided by density units give 9/3 = 3. For the second, they give 4/2 = 2. Comparing these results gives 3:2. Using the population ratio alone or the density ratio alone would not determine the area ratio. The relationship among all three quantities is necessary.

Correct answer: B. 576 persons per square kilometre

Explanation: Density changes when population and area change, because density is population divided by area. A population increase of 12.5% gives a population multiplier of 1.125. An area increase of 25% gives an area multiplier of 1.25. The new density is therefore the old density multiplied by the population multiplier and divided by the area multiplier.

The calculation is \(640 \times \frac{1.125}{1.25}=640 \times 0.9=576\) persons per square kilometre. Thus option B is correct. Although population rises, area rises by a larger percentage, so the average number of people per square kilometre falls. Option C would be possible only if population and area increased by the same proportion. The other numerical choices do not follow from the two percentage changes.

Correct answer: B. 25 percent

Explanation: When area remains unchanged, population density changes in exactly the same proportion as population. The initial density is 360 persons per square kilometre and the final density is 450. The increase is \(450-360=90\) persons per square kilometre. To express this increase relative to the starting value, divide 90 by 360 and multiply by 100.

Thus the percentage increase is \(\frac{90}{360}\times100=25\%\), so option B is correct. Another quick method is to compare the values directly: \(450/360=1.25\), meaning the final density is 125% of the initial density, or 25% higher. The unchanged area is essential; if area had also changed, density alone could not be used to infer the same percentage change in population.

Correct answer: B. 62.5 percent

Explanation: Density is population divided by area. If population does not change, a rise in density must result from a fall in area. Because population is constant, area and density are inversely related: multiplying density by a factor requires area to be divided by the same factor. The original density is 250 and the new density is 400 persons per square kilometre.

The area ratio is \(A_{new}/A_{old}=D_{old}/D_{new}=250/400=0.625\). Therefore the new area is 0.625, or 62.5%, of the original area. Option B is correct. The result does not mean that 62.5% area was added; it means that 37.5% of the original area was reduced.

Correct answer: C. 1500 square kilometres

Explanation: Arithmetic density connects population, area, and density through \(Density = Population \div Area\). To find area, rearrange the relationship as \(Area = Population \div Density\). The units also confirm the operation: persons divided by persons per square kilometre leaves square kilometres. This is a useful check against multiplying by mistake.

Substituting the values gives \(Area = 480,000 \div 320 = 1,500\) square kilometres. Therefore, option C is correct. A quick verification is \(1,500 \times 320 = 480,000\), which reproduces the stated population. The other choices do not satisfy the density equation. The answer describes total area, not only cultivated land or built-up land.

Correct answer: B. 396000

Explanation: The total population of a region is obtained by multiplying its population density by its area. Density gives the number of persons in each square kilometre, while area gives the number of such units. Thus the formula is \(\text{Population}=\text{Density}\times\text{Area}\). Keeping the units in mind helps prevent confusing multiplication with division.

Substitute the given values: \(275\times1440\). One simple method is \(275\times144=39600\), then multiply by 10 because 1440 is ten times 144. This gives 396000 people. Therefore, option B is correct. The other choices do not result from the stated density and area, so they cannot represent the total population.

Correct answer: B. Large sparsely populated areas can lower the national average

Explanation: Arithmetic density is a national average. It divides the total population by the total area, so it combines crowded and empty places into one number. A country may contain large mountains, deserts, forests, or other sparsely settled areas, while its agricultural valleys contain many people and have very high local density.

Those lightly populated areas add substantial land to the denominator without adding many residents. Consequently, they lower the national average even though some valleys are crowded. Option B gives this interpretation correctly. Option A wrongly treats an average as the exact density everywhere, while C and D ignore the clear relationship between local concentration and the national average.

Correct answer: A. Internal distribution can change even when average density stays the same

Explanation: Average density is calculated for the whole metropolitan region as total population divided by total area. It does not show how that population is arranged inside the region. If some residents move from the central city to suburbs, the internal pattern changes, but the total population and total area may remain unchanged.

In that case, the metropolitan average can stay exactly the same while central density falls and suburban density rises. Option A is therefore correct. Option B confuses an unchanged average with an unchanged distribution. Options C and D are also incorrect because suburban movement is possible and an average cannot reveal every local pattern.

Correct answer: B. Density will decrease because the denominator area increases

Explanation: Correct answer: B, density will decrease because the denominator area increases. Arithmetic density is calculated as total population divided by official area. When a large forest reserve with no residents is added to the city boundary, the population remains approximately the same, but the official area becomes larger. With the numerator unchanged and the denominator increased, the ratio becomes smaller. A is wrong because a larger area does not by itself increase density; it normally lowers density when population does not rise with it. C is wrong because an uninhabited area still counts in the official geographical area used for arithmetic density. D is wrong because density can still be calculated using the revised population and revised area. This example also shows why boundary definitions matter when comparing densities. Memory cue: adding empty land increases the denominator, so the average density falls.

Correct answer: A. Increased by about 18.2%

Explanation: Correct answer: A, the area increased by about 18.2%. Since density D = population P ÷ area A, rearranging gives area A = P ÷ D. A 30% population increase gives a population factor of 1.30, and a 10% density increase gives a density factor of 1.10. Therefore the area factor is 1.30 ÷ 1.10 = 1.1818..., meaning the new area is about 118.18% of the old area. The increase is therefore about 18.2%. B wrongly assumes area changes by the same percentage as density. C has the wrong direction: because population grows faster than density, area must expand. D ignores the density increase and treats the area as if it followed population exactly. The key relationship is that population grows faster than density only when the area also increases. Use factors rather than subtracting percentages directly.

Correct answer: A. Population remained unchanged

Explanation: Population can be found by multiplying density by area: \(P=D\times A\). A 20 percent decrease in density leaves 80 percent of the original density, so its multiplier is 0.8. A 25 percent increase in area makes the new area 125 percent of the original, giving a multiplier of 1.25.

The population multiplier is therefore \(0.8\times1.25=1.00\). A multiplier of 1 means the final population is exactly equal to the initial population. Thus option A is correct. The decrease in density is exactly balanced by the increase in area: fewer people per unit area are spread across proportionally more area. Options B, C, and D do not match this product calculation.

Correct answer: C. 1.25

Explanation: Area is related to population and density by \(Area = Population \div Density\). A percentage increase is first changed into a multiplier: a 60 percent population increase gives 1.60, and a 28 percent density increase gives 1.28. The new area compared with the old area is therefore the population multiplier divided by the density multiplier.

The area multiplier is \(1.60 \div 1.28 = 1.25\). This means the new area is 1.25 times the original area, so it has increased by 25 percent. Hence option C is correct. Simply subtracting 28 from 60 would not be the correct method because the two percentage changes act through a ratio, not through direct subtraction.

Correct answer: B. Population is concentrated in a few small urban clusters while the rest is nearly empty

Explanation: Arithmetic density is an average for an entire region, whereas settlement concentration describes where people are actually clustered. The greatest difference occurs when the average hides a highly uneven pattern. If most residents live in a few small urban clusters and the remaining land is almost empty, the whole-region average may appear moderate or low even though the clusters are extremely dense.

Option B describes this contrast directly. An evenly distributed population would make the average more representative, so A would produce less difference. A small area or stable population does not by itself create a contrast between average density and concentration. Thus B is the most accurate choice.

Correct answer: C. Density will double

Explanation: Density depends on how many people occupy each unit of the area being considered. If the total population remains P and the inhabited area changes from A to half of A, the same people are now concentrated in a smaller space. Consequently, the number of people per unit of inhabited area increases.

The original density is \(P/A\). The new density is \(P/(A/2)=2P/A\), which is twice the original value. Therefore, option C is correct. It does not become half because the population has not decreased; it becomes double because the denominator, the inhabited area, has been reduced by half.

Correct answer: C. 500 persons per square kilometre

Explanation: 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.

Correct answer: D. 400 persons per square kilometre

Explanation: 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.

Correct answer: B. 1200 square kilometres

Explanation: 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.