Population Density

Population density is calculated by dividing population by area. The initial density is \(\frac{720000}{1800}=400\) persons per square kilometre. An 18% population increase multiplies the population by \(1.18\), while a 10% area increase multiplies the area by \(1.10\). Hence the new density is \(400\times\frac{1.18}{1.10}=429.09\), approximately 429 persons per square kilometre.

Therefore option B is correct. It is important to increase both quantities before dividing, because density depends on population and area together. The answer is not 472, which would overlook the effect of the enlarged area, and it is not 440 because the percentage changes do not transfer directly to density.

Density means population divided by area. Let the original population be P and area be A, so the original density is \(P/A=560\). A 16 percent fall makes the new population \(0.84P\). A 12 percent increase makes the new area \(1.12A\). Therefore, the new density is \(\frac{0.84P}{1.12A}=560\times\frac{0.84}{1.12}\). Since \(0.84/1.12=0.75\), the result is \(560\times0.75=420\) persons per square kilometre.

Thus option B is correct. A common mistake is to subtract 16 percent and 12 percent directly from 560, but density changes through both the numerator and denominator. The area increase lowers density further than population decline alone, while the population decline also lowers it. The calculated value, 420 persons per square kilometre, exactly matches option B.

First find the original population from density multiplied by area: \(480\times1250=600000\). When 90,000 people move into the region, the population becomes \(690000\). Because the area remains 1,250 square kilometres, the new density is found by dividing the new population by that unchanged area.

The calculation is \(\frac{690000}{1250}=552\) persons per square kilometre. Therefore option C is correct. A quick check also works: the incoming population adds \(\frac{90000}{1250}=72\) persons per square kilometre to the original density of 480, giving \(552\). The unchanged area is essential; if it changed, the answer would be different.

Population density means the number of people living in one square kilometre. First find the original area by dividing population by original density: \(960000\div800=1200\) square kilometres. The added land makes the total area \(1200+300=1500\) square kilometres.

The population does not change, so the new density is \(960000\div1500=640\) persons per square kilometre. Thus option B is correct. The density falls because the same number of people is now spread over a larger area. Option C is only a rough approximation, while 600 and 700 do not result from the required calculation.

Density is population divided by area. If the original values are P and A, the new population is \(1.25P\) and the new area is \(0.75A\). Therefore the new density compared with the old one is multiplied by \(1.25/0.75=1.6667\). This means the new density is about 166.67 percent of the original density. The increase is not 166.67 percent; it is the amount above 100 percent, namely about 66.67 percent. Thus option C is correct.

The area reduction makes each square kilometre represent an even larger share of the population, while the population itself also rises. Algebraically, \(D'=(1.25P)/(0.75A)=1.6667D\), so \(D'-D=0.6667D\). Option A incorrectly adds 25 percent and 25 percent without considering the ratio, while option D overstates the increase. Hence 66.67 percent is the correct change.

Population density is defined as the ratio of total population to total area, written as \(D=P/A\). Both P and A may change, but if their ratio after the changes is exactly the same as before, then the value of D is unchanged. For example, if both population and area double, the quotient remains the same. Therefore option C follows directly from the definition.

The question does not ask whether population or area individually rises or falls. It asks about density, which depends on their relationship. A larger population alone would increase density, and a smaller area alone would also increase it, but simultaneous changes can offset one another. Since the population-to-area ratio is stated to remain constant, options A and B cannot be selected. Density does not become zero unless population is zero, so D is also incorrect.

Let the whole country have area \(A\). Its total population is therefore \(300A\), because national density is 300 persons per square kilometre. The selected part has 20% of the area, or \(0.20A\), and contains 60% of the population, or \(0.60\times300A\). Its density is therefore \(\frac{0.60\times300A}{0.20A}=900\) persons per square kilometre.

Thus, option C is correct. A quick way is to compare population share with area share: \(0.60\div0.20=3\). The selected part has three times the national average density, so \(300\times3=900\). The calculation assumes the percentages refer to the same total population and total area.

Let the total area be represented by \(A\) square kilometres. An arithmetic density of 420 means that the population is \(420A\). Only 70 percent of the area is suitable for settlement, so the inhabited area is \(0.70A\). Effective density is therefore \(\frac{420A}{0.70A}=600\) persons per square kilometre. The total area cancels, so no particular value of \(A\) is needed.

Thus option C, 600 persons per square kilometre, is correct. The effective density is higher than the arithmetic density because the same population is concentrated on a smaller usable area. Option B, 560, would not result from dividing by 0.70, and 500 or 650 do not match the calculation. The calculation assumes that the entire population lives within the suitable area.

Arithmetic density is calculated as total population divided by total land area, expressed as an average such as people per square kilometre. It tells us how many people are present on average across the whole region, but it does not show where those people are located within that region. Population may be concentrated in one part and scattered in another while the average remains identical.

Thus, two regions can have the same total population and area, or simply the same population-to-area ratio, even though one has clustered settlements and the other has dispersed settlements. Option A is correct. The other choices make unsupported claims: clustering does not necessarily raise average density, and dispersion does not prove that population is lower.

Population density means population divided by area. At the start, the density is \(\frac{1,500,000}{600}=2,500\) persons per square kilometre. A 6% population increase changes the population to \(1,500,000\times1.06=1,590,000\). A 4% area increase changes the area to \(600\times1.04=624\) square kilometres.

The new density is therefore \(\frac{1,590,000}{624}\), or equivalently \(2,500\times\frac{1.06}{1.04}\approx2,548.08\) persons per square kilometre. Rounded to the nearest whole number, this is about 2,548. Option B is correct. The density rises only slightly because both population and area increase, and the population percentage increase is larger than the area percentage increase.

A population increase does not necessarily raise density if area grows faster. The original density is 840. Population is multiplied by (1.12) after a 12 percent increase, while area is multiplied by (1.40) after a 40 percent increase. Hence the new density is (840 times 1.12 divided by 1.40 = 840 times 0.8 = 672) persons per square kilometre.

Option B is correct. Although the population becomes larger, the land area grows proportionally much more, so people are spread over more space. The factor 0.8 means the new density is 80 percent of the old density, or a 20 percent decrease. Options with values near the original density do not account correctly for the larger denominator.

When population remains constant, density changes inversely with area. A 16 percent reduction leaves 84 percent of the original area, so the new area is \(0.84A\). Because the same population is now contained in a smaller area, the numerical density must be higher than 375.

The unchanged population can be represented as \(375A\). Dividing it by the new area gives \(375A/(0.84A)=375/0.84=446.43\) persons per square kilometre. Rounded to the nearest whole number, this is approximately 446. Thus option B is correct. The answer 450 is close but not the appropriate result of the stated percentage change.

The new population is the original population plus the added people: \(680000+120000=800000\). The new area is the original area plus the additional area: \(1600+400=2000\) square kilometres. Density is population divided by area, so the new density is \(800000/2000=400\) persons per square kilometre.

Option B is correct. It is important to update both quantities before dividing; using the original area would give an incorrect result, and using only the added population would ignore the existing residents. The units are persons per square kilometre because people are divided by square kilometres. Therefore the region's new density is 400 persons per square kilometre.

A change in both population and area must be handled by applying both percentage factors. A 40 percent population increase multiplies population by 1.40. A 12 percent area decrease leaves 88 percent of the original area, so its factor is 0.88. Density rises more than population alone would suggest because the available area becomes smaller.

Starting with density 450, the new density is \(450\times\frac{1.40}{0.88}=715.909...\). Rounded to the nearest whole number, this is about 716 persons per square kilometre. Therefore option C is correct. Option 650 ignores the area decrease, while 682 and 750 do not result from applying the two factors correctly.

Density equals population divided by area. A population decrease of 22 percent leaves a factor of \(0.78\), while an area decrease of 18 percent leaves a factor of \(0.82\). The new density is therefore \(720\times(0.78/0.82)\). Since \(0.78/0.82\approx0.9512\), the result is about \(684.9\) persons per square kilometre. Rounding to the nearest listed value gives 685.

Option B is correct. The density does not fall by the full 22 percent because the area also becomes smaller; fewer people are spread over an even smaller area. Option A is too low, C is not the calculated result, and D incorrectly assumes that the ratio remains unchanged. Using percentage factors for both population and area prevents a common calculation error.

First find the original population from density multiplied by area. The calculation is \(312.5\times1920=600,000\) people. After an increase of 120,000, the population becomes 720,000. The area is unchanged at 1,920 square kilometres, so the new density is \(\frac{720000}{1920}=375\) persons per square kilometre.

Therefore, option C is correct. A useful check is that the population rises by 20 percent, from 600,000 to 720,000, while the area remains fixed; the density must consequently rise by 20 percent, from 312.5 to 375. The result should not be found by adding 120,000 directly to the density, because population and density have different units.

Answer: B, 500 persons per square kilometre. First find the original area using A = P/D. Thus A = 625,000 ÷ 625 = 1,000 square kilometres. The area increases by 250, so the new area is 1,250 square kilometres. Population remains 625,000. New density = 625,000 ÷ 1,250 = 500 persons per square kilometre. Option B is correct. Option A is too low, while options C and D do not result from dividing the unchanged population by the new area. A shortcut also works: area rises from 1,000 to 1,250, which is 1.25 times the original; with fixed population, density becomes 1/1.25 = 0.8 times 625 = 500. Memory cue: when population is fixed, density and area move in opposite directions.

The stated density uses the whole region’s area, including the part covered by water bodies. If 10% of the area cannot support settlement, only 90% remains available, but the same total population must be considered for effective density. Consequently, people are concentrated over a smaller usable area.

Let total area be A. The population is \(540A\). Usable area is \(0.90A\), so effective density is \(\frac{540A}{0.90A}=600\) persons per square kilometre. Therefore option B is correct. This assumes all residents are counted in the effective-density calculation.

Density is found by dividing population by area. For the first region, the calculation is \(840000/2100=400\) persons per square kilometre. For the second region, it is \(675000/1350=500\) persons per square kilometre. The second region therefore has more people for each square kilometre of land. The difference is \(500-400=100\) persons per square kilometre.

Option B is correct: the second region is denser by 100 persons per square kilometre. The comparison must use both population and area; a larger population alone does not guarantee greater density. Option A reverses the result and gives the wrong difference. Option C gives an excessive difference, while option D ignores that the two areas and populations produce different ratios. Checking the units also confirms that both answers are densities, not total populations.

Density changes according to both population and area. If the original density is 350 persons per square kilometre, a 15 percent population increase makes population 1.15 times its original value. A 38 percent area increase makes area 1.38 times its original value. Therefore the new density is \(350\times 1.15/1.38\), which is about \(291.67\), or approximately 292 persons per square kilometre.

Option B is correct after rounding to the nearest whole number. Although population grows, area grows much faster, so density falls from 350 to about 292. Option A is too low, while options C and D are too high because they do not sufficiently account for the larger denominator. The calculation also illustrates that density is not changed by population growth alone; the relative percentage change in area must be included. Thus the ratio of the two growth factors is essential.

Let the original population be P and the original area be A. The original density is 640 persons per square kilometre, so it equals P/A. A 35% population decrease leaves 65% of the population, or 0.65P. A 20% area decrease leaves 80% of the area, or 0.80A. The new density is therefore \(640\times\frac{0.65P}{0.80A}=640\times\frac{0.65}{0.80}=520\) persons per square kilometre.

Option C is correct. It is not enough simply to subtract both percentages from 640, because density changes according to the ratio of the new population to the new area. Although the population falls, the area also becomes smaller, which partly offsets the fall in density. The resulting density is 520, not 480, 500, or 540. The calculation assumes the original density and the stated percentage changes refer to the same region and comparable units.

The original area can be found from density = population divided by area, so area = population divided by density. Thus the original area is \(1280000\div1600=800\) square kilometres. Adding 128,000 people gives a new population of 1,408,000. Since the area remains 800 square kilometres, the new density is \(1408000\div800=1760\) persons per square kilometre. Therefore, choice C is correct.

A quicker check is possible because 128,000 is one-tenth of 1,280,000. With area unchanged, density also rises by one-tenth of 1,600, which is 160. Adding 160 to 1,600 gives 1,760. The area must first be inferred because it is not directly stated. The other options do not match either the full calculation or this proportional check.

With area fixed, a decrease in population produces a proportional decrease in density. The question gives an area of 3200 square kilometres and a population loss of 160000. The density fall can therefore be found directly by dividing the population decrease by the unchanged area: \(Change\ in\ density=Population\ change/Area\).

The calculation is \(160000/3200=50\) persons per square kilometre. The original density of 625 would become 575, but the question asks how much it falls, so the answer is 50 rather than 575. Therefore option B is correct. The unchanged area is essential for using this direct division.

Answer: C, 480 square kilometres. Let the original area be A square kilometres. Since the original density is 600, the unchanged population is 600A. After 120 square kilometres is added, the area is A + 120 and the density is 480. Therefore 600A = 480(A + 120). Expanding gives 600A = 480A + 57,600. Subtracting 480A gives 120A = 57,600, so A = 480 square kilometres. Option C is correct. Option A, B, and D do not satisfy the equation. Check the result: original population = 600 × 480 = 288,000; new area = 600; new density = 288,000 ÷ 600 = 480. The check confirms the answer. Memory cue: with fixed population, equate old density × old area to new density × new area.

The area can be found from the increase in population and the corresponding increase in density. Since the area does not change, the added population is spread over that same area. Therefore, added population equals increase in density multiplied by area.

The density increase is \(510 - 420 = 90\) persons per square kilometre. Hence area = added population ÷ density increase = \(36000 \div 90 = 400\) square kilometres. Option B is correct. Using 420 or 510 alone would be wrong because only the difference in density is caused by the arriving people.