Population Density

Correct answer: B. 300 persons per square kilometre

Explanation: When area remains unchanged, density changes in the same proportion as population because density is population divided by area. A 20 percent population increase therefore produces a 20 percent density increase. Starting with 250 persons per square kilometre, calculate \(250\times1.20=300\). Thus, the correct choice is B.

The additional density is 20 percent of 250, which is \(250\times0.20=50\). Adding this to the original density gives \(250+50=300\). The unchanged area is important: if area had also changed, the result would require comparing both percentage changes. Here the denominator stays fixed, so the population increase passes directly to the density. The other choices do not represent a 20 percent rise from 250.

Correct answer: B. 25 percent

Explanation: Direct answer: Option B, 25 percent. Population density falls from 480 to 360, so the absolute decrease is 480 − 360 = 120 persons per square kilometre. For a percentage decrease, always compare the decrease with the original value, not the final value. Therefore, percentage decrease = (120 ÷ 480) × 100 = 25%. Option A is incorrect because 20% does not result from this calculation. Option B is correct because it matches the exact result. Option C is incorrect because 30% would mean a decrease of 144 from the original 480. Option D is incorrect because 33.3% is obtained by an unsuitable comparison with the new value or by rough estimation. Memory cue: percentage decrease uses original value as the denominator.

Correct answer: C. 40 percent

Explanation: Direct answer: Option C, 40 percent. First find the absolute increase: 210 − 150 = 60 persons per square kilometre. Percentage increase is calculated relative to the original density, so use 150 as the denominator: (60 ÷ 150) × 100 = 40%. Option A is wrong because 30% would give an increase of only 45 over 150. Option B is wrong because 35% would give 52.5, not 60. Option C is correct because 60 is 40% of 150. Option D is wrong because 60 is the absolute increase, not the percentage increase. Common confusion: the number 60 must not be written as 60% unless the base value is exactly 100.

Correct answer: A. Area will increase by 10 percent

Explanation: Density is calculated as population divided by area, \(D=P/A\). If density must remain unchanged, population and area must increase in the same proportion. Therefore, a population increase alone would raise density, so the area must also expand to accommodate the additional people at the original density.

Suppose the original population and area are P and A. After a 10 percent increase, the population is \(1.1P\). Keeping density equal gives \(1.1P/A' = P/A\), so \(A'=1.1A\). The area therefore increases by 10 percent. Hence option A is correct; it does not remain unchanged because the population has changed.

Correct answer: B. Area will decrease by 20 percent

Explanation: Population density is population divided by area, so when population stays fixed, density and area change in opposite directions. A higher density means the same number of people are being counted within a smaller area. The area therefore cannot increase or remain unchanged when density rises.

Let the original density and area be \(D\) and \(A\). A 25 percent rise makes density \(1.25D\). Since population is constant, \(P=DA=(1.25D)A_{new}\). Thus \(A_{new}=A/1.25=0.8A\). The new area is 80 percent of the old area, so it decreases by 20 percent. Therefore, option B is correct.

Correct answer: C. 800 square kilometres

Explanation: Population density tells us how many people live in one square kilometre. To find the total area, divide the population by the density. This works because population equals density multiplied by area. The given population is 360,000 and the density is 450 persons per square kilometre, so the required area must be measured in square kilometres.

Using the relation, area = population ÷ density = \(360000 \div 450 = 800\) square kilometres. The result can be checked by multiplying density by area: \(450 \times 800 = 360000\). Therefore, option C, 800 square kilometres, is correct. The other values would not reproduce the stated population at a density of 450.

Correct answer: C. 231,000

Explanation: Direct answer: Option C, 231,000 people. Density tells us how many persons live in one square kilometre. Therefore, total population equals density × area. Use the formula P = D × A. Here, P = 275 × 840. Break the multiplication into an easier step: 275 × 84 = 23,100, and multiplying by the extra 10 in 840 gives 231,000. Option A is incorrect because it does not equal the exact product. Option B is also too low and results from inaccurate multiplication. Option C is correct because 275 persons in each of 840 square kilometres gives 231,000 persons. Option D is too high and is not supported by the formula. Always check that the units cancel correctly: persons per square kilometre × square kilometres = persons.

Correct answer: C. 352 persons per square kilometre

Explanation: Direct answer: Option C, 352 persons per square kilometre. Density equals population divided by area. Population doubling gives a multiplier of 2, while a 25% area increase gives a new-area multiplier of 1.25. Thus the density multiplier is 2 ÷ 1.25 = 1.6. New density = 220 × 1.6 = 352. Option A is incorrect because it does not reflect the full effect of doubling population. Option B is incorrect because it is below the calculated value. Option C is correct. Option D is incorrect because it simply doubles the old density and ignores the increase in area. Memory cue: when both population and area change, use new density = old density × population multiplier ÷ area multiplier.

Correct answer: B. 600 persons per square kilometre

Explanation: Let the original population and area be P and A. An original density of 600 means \(P/A=600\). A 20 percent fall in population leaves \(0.8P\), and a 20 percent fall in area leaves \(0.8A\). The new density is therefore \(\frac{0.8P}{0.8A}=\frac{P}{A}=600\) persons per square kilometre.

Thus, option B is correct. Both parts of the ratio have been multiplied by the same factor, so their effect cancels. The population becomes smaller, but the area available for each remaining person also becomes smaller in exactly the same proportion. Answers such as 480 would result from reducing the population without reducing the area, which is not what the question states.

Correct answer: B. Fewer people live per unit area there than the national average

Explanation: Density measures how many people live per unit of area. Saying that a region's density is below the national average therefore means that, on average, fewer people live in each square kilometre of that region than in the country as a whole. This statement concerns a ratio, not the region's total population or its total area.

Option B is the only safe conclusion. A region may have a large total population but an even larger area, or a small population with a small area; density alone cannot decide its total population. It may contain cities, villages, or both, so C is unsupported. The region need not be larger than the whole country, making D impossible as a general inference. Thus only the per-unit-area comparison follows directly.

Correct answer: A. If its area is very small

Explanation: High density means that many people live in relation to each unit of area. Total population, however, depends on both density and the size of the region. A small region may have a high number of people per square kilometre but still contain fewer people in total than a much larger, less densely settled region.

For example, a small area of 10 square kilometres with 100 people per square kilometre has 1,000 people altogether. A larger area may have a lower density but a much larger total population. Therefore, a very small area can allow a region to be classified as high-density while its total population remains low. Option A is correct. A large area would generally increase total population for a given density.

Correct answer: A. If its area is extremely large

Explanation: Low density describes the relationship between population and area; it does not directly tell us that the total population is small. A region may have only a few people per square kilometre but still contain many people if the region covers a very large area.

For example, a large region can have a low ratio of people to land while its total population remains substantial. A small area would generally need a high density to contain a large population. Zero density would mean no population, and living in cities does not by itself determine the total population. Therefore, option A is the appropriate answer.

Correct answer: B. Its area will be three times

Explanation: Population density is the number of people living in one unit of area. It is calculated as population divided by area: \(D=P/A\). If two regions have equal density, their population-to-area ratios must be equal. Therefore, a region with more people must also have proportionally more land if its density is unchanged. This is the central meaning of equal density.

Let the first region have population \(P\) and area \(A\), so its density is \(P/A\). The second has population \(3P\). To keep the same density, its area must satisfy \(3P/A_2=P/A\), giving \(A_2=3A\). Thus its area is three times as large, so option B follows. One-third would make the density nine times higher, not equal.

Correct answer: C. 4:3

Explanation: Density is population divided by area. Therefore, when comparing two regions, population ratios and area ratios must be combined carefully. The first region has population represented by 2 and area by 1, while the second has population 3 and area 2.

Their density ratio is \((2/1):(3/2)\). Dividing the first density by the second gives \((2/1)\div(3/2)=2\times2/3=4/3\). Hence the density ratio is 4:3, so option C is correct. Simply using the population ratio would ignore the fact that the regions have different areas.

Correct answer: A. 5:4

Explanation: Density is population divided by area, so area is population divided by density. For two regions, the area ratio must therefore be found by dividing each population ratio term by its corresponding density ratio term. This prevents the common mistake of multiplying the ratios in the wrong direction.

Using the given ratios, the area ratio is \(3/6 : 2/5\), which is \(1/2 : 2/5\). Multiplying both parts by 10 gives 5:4. Therefore, option A is correct. The result also makes sense: the first region has a relatively larger population compared with its density, so its area is larger in this ratio.

Correct answer: B. 360 persons per square kilometre

Explanation: Population density means the number of people living in one unit of area. If the population stays fixed but the area becomes smaller, the same people are concentrated in less space, so density increases. The required relationship is density = population ÷ area.

The original population is \(300 \times 900 = 270000\) people. After the area changes, the population is still 270000, while the new area is 750 square kilometres. Thus, new density = \(270000 \div 750 = 360\) persons per square kilometre. Therefore, option B is correct. The result is higher than 300 because the area decreased without a population decrease.

Correct answer: B. 442 persons per square kilometre

Explanation: Direct answer: Option B, approximately 442 persons per square kilometre. A 10% population decrease leaves 90% of the original population, so its multiplier is 0.9. A 10% area increase makes the new area 110% of the original, so its multiplier is 1.1. Since density is population divided by area, new density = 540 × 0.9 ÷ 1.1 = 441.8, which rounds to about 442. Option A is too low and does not follow the formula. Option B is correct after rounding. Option C would result from subtracting only 10% from density and ignoring the area increase. Option D incorrectly increases density even though population falls and area expands. Warning: percentage changes in numerator and denominator should be applied as multipliers, not simply added or subtracted.

Correct answer: B. 150 persons per square kilometre

Explanation: Density is calculated by dividing population by area, so a change in both quantities must be considered together. If population rises by 50 percent, it becomes \(1.5P\). If area doubles, it becomes \(2A\). The new density is therefore the new population divided by the new area, rather than simply adding or subtracting the percentage changes.

The new density is \(\frac{1.5P}{2A}=0.75\frac{P}{A}\). Starting from 200 persons per square kilometre, the result is \(200\times0.75=150\) persons per square kilometre. Although the population increases, the area increases more strongly, so density falls. Therefore, option B is correct.

Correct answer: B. The first region has lower density

Explanation: Direct answer: Option B, the first region has lower density. Calculate its density using density = population ÷ area. Thus, 500,000 ÷ 2,500 = 200 persons per square kilometre. The other region has density 300 persons per square kilometre. Since 200 is less than 300, the first region is less densely populated. Option A is incorrect because 200 is not greater than 300. Option B is correct because the calculated density is lower. Option C is incorrect because the two values are 200 and 300, not equal. Option D is incorrect because both densities are expressed in the same unit, so comparison is possible. Core idea: before comparing two densities, calculate the unknown one and make sure both use identical units.

Correct answer: A. It may dilute the apparent residential concentration

Explanation: Arithmetic density uses the entire stated area, whether or not every part is equally inhabited. In this case, the population is divided by 1,000 square kilometres. That denominator includes the 600 square kilometres where few people may live. As a result, the calculated average is pulled downward compared with the much greater concentration inside the 400 square kilometres where most residents actually live.

Option A is correct because an overall average can dilute the apparent density of the inhabited zone. It remains useful for comparing whole cities or regions, but it does not reveal the density of a particular neighbourhood or the built-up area. Option B is incorrect because exact local density requires smaller-area data. Option C confuses density with population change, and option D is false because area is essential in the formula. A map or sub-area calculation would show the concentration more clearly.

Correct answer: B. It may understate internal concentration

Explanation: Arithmetic density is calculated by dividing a country's total population by its total area. If a large part of the country is desert and contains few people, that empty land is still included in the denominator. The resulting national average is therefore pulled downward, even though people may be tightly concentrated in a fertile or well-watered river valley.

Thus, option B is correct: national arithmetic density may understate the internal concentration of population. It does not show where people actually live within the country. Option A reverses the effect, because the national average does not overstate the valley's exact density. Option C is false because desert land does not automatically have a high population, and option D is false because the national density can still be calculated.

Correct answer: B. Density will decrease

Explanation: Arithmetic density is calculated as total population divided by the total area included in the calculation. If a large lake is added to the municipal boundary, the counted area becomes larger. The question says that population does not change, so the numerator stays fixed while the denominator increases.

Consequently, the density becomes lower, because the same number of people is now spread over a larger measured area. Option B is correct. The result is not necessarily half or any other fixed amount; the exact decrease depends on the lake’s area. This is a change in calculated arithmetic density, not necessarily in actual settlement conditions.

Correct answer: C. 60 percent

Explanation: Direct answer: Option C, 60 percent. The difference between the densities is 200 − 125 = 75 persons per square kilometre. Because the question asks how much higher the second value is than the first, the first density, 125, is the reference value. Percentage increase = (75 ÷ 125) × 100 = 60%. Option A is incorrect because 40% of 125 is 50. Option B is incorrect because 50% of 125 is 62.5. Option C is correct because 75 is 60% of 125. Option D is incorrect because 75 is the absolute difference, not 75%. Common confusion: the denominator changes if the comparison direction changes; “second is higher than first” uses the first value as the base.

Correct answer: B. 200 persons per square kilometre

Explanation: A percentage decrease is taken from the original quantity. Saying that Region A is 20 percent less dense than Region B means A retains 80 percent of B's density. It does not mean that 20 persons are subtracted, because the stated percentage must first be calculated from 250.

Twenty percent of 250 is \(0.20\times250=50\) persons per square kilometre. Subtracting this decrease from Region B gives \(250-50=200\), or directly \(250\times(1-0.20)=250\times0.8=200\). Hence Region A has a density of 200 persons per square kilometre, so option B is correct.

Correct answer: C. 80 percent

Explanation: The question asks what percentage the first density represents of the second density. It is not asking how much lower or higher the first city is, so the correct base is the second city’s density, 400. Divide the first value by the reference value and multiply by 100.

The calculation is \(\frac{320}{400}\times100\). Since 320 is four-fifths of 400, the result is 80 percent. Thus option C is correct. A value of 70 or 75 percent would use an incorrect ratio, while 85 percent would overstate the first city’s density. If the question had asked for percentage decrease, the answer would be 20 percent, but that is not what is asked here.