Population Density

Density must be calculated separately for each region. For the first region, divide 600,000 people by 1,500 square kilometres: \(600000/1500=400\) persons per square kilometre. For the second, divide 450,000 people by 900 square kilometres: \(450000/900=500\) persons per square kilometre. Comparing the results shows that the second region is more densely populated.

The difference is \(500-400=100\) persons per square kilometre. Therefore option B, “the second by 100,” is correct. A larger population does not automatically mean a higher density, because area must also be considered. The first region has more people, but it occupies much more land; the second has fewer people concentrated in a smaller area, producing the higher density.

The original population is density multiplied by area: \(250\times1600=400000\). After a 20 percent population increase, population becomes \(400000\times1.20\). After a 25 percent area increase, area becomes \(1600\times1.25\). It is simpler to compare density factors: new density equals old density multiplied by \(1.20\div1.25=0.96\). Thus \(250\times0.96=240\) persons per square kilometre, so C is correct.

Although population rises, area rises by a larger percentage. The denominator therefore grows faster than the numerator, causing density to fall by 4 percent. The exact new values are 480,000 people and 2,000 square kilometres, and \(480000\div2000=240\). This also shows why simply adding 20 percent to density would be wrong: the area change must be included.

Density changes according to both population and area. The original density is 480 persons per square kilometre. A 25 percent population decrease leaves 75 percent of the original population, or a factor of 0.75. A 20 percent area decrease leaves 80 percent of the original area, or a factor of 0.80. The new density factor is therefore 0.75/0.80=0.9375.

The new density is 480×0.9375=450 persons per square kilometre. It falls because population decreases proportionally more than area. Thus option B is correct. Simply subtracting 25 percent from 480 would give 360, but that ignores the simultaneous reduction in area. Likewise, subtracting 20 percent from the density is not valid, because density is a ratio and both parts of that ratio have changed.

The given density and population can first be used to find the unchanged area. The relationship is \(D=P/A\), so \(A=P/D\). The original area is \(720000/1200=600\) square kilometres. When 60,000 people are added, the new population becomes \(720000+60000=780000\), while the area remains 600 square kilometres.

The new density is therefore \(780000/600=1300\) persons per square kilometre. The increase in population raises density because the same land must now accommodate more people. Hence option B, 1300 persons per square kilometre, is correct. The other choices do not result from dividing the new population by the unchanged area.

For a fixed area, a change in population produces a density change equal to population change divided by area. The region loses 70000 people, while its area remains 2000 square kilometres. Thus the decrease in density is \(70000\div2000=35\) persons per square kilometre. The question asks how much density decreases, so the answer is the positive amount 35, not the signed change \(-35\). Therefore, option B is correct.

The original population would be \(350\times2000=700000\), and after the loss it would be 630000. The new density is \(630000\div2000=315\), which is 35 less than 350. This second method confirms the result. Options 25, 50, and 70 would imply different population changes for the same area and are not supported by the data.

Correct answer: B, 450 square kilometres. Let the original area be A square kilometres. At the original density, population equals 400A. After 50 square kilometres are added, the new area is A + 50 and the new density is 360. Since population is unchanged, 400A = 360(A + 50). Expanding gives 400A = 360A + 18,000, so 40A = 18,000 and A = 450 square kilometres. A would give a population of 180,000 initially, but the new density over 500 square kilometres would be 360, so it does not satisfy the original condition. C and D similarly produce different initial populations and fail the equation. The reliable method is to equate the old and new population because population is fixed.

The area remains unchanged, so the extra population is related directly to the increase in density. The density rises from 250 to 300 persons per square kilometre, an increase of \(300-250=50\) persons per square kilometre. These 50 additional persons per square kilometre represent the arrival of 20000 people. Therefore, \(Area=20000\div50=400\) square kilometres. Option B is correct.

The unit check is useful: people divided by people per square kilometre gives square kilometres. We can also verify the result by multiplying the density increase by the area: \(50\times400=20000\). If the area were 500 square kilometres, the increase would require 25000 people, not 20000. Thus, the unchanged-area condition makes option B definite.

With a fixed area, a change in population produces a proportional change in density. Here density falls from 600 to 500 persons per square kilometre, so the decrease is 100 persons per square kilometre. This decrease represents the 40,000 people who left the region.

If the area is A square kilometres, the population loss is \(100A\), because every square kilometre contains 100 fewer people after the change. Setting this equal to 40,000 gives \(100A=40000\), so \(A=400\) square kilometres. Therefore option B is correct. The other values would imply population losses different from 40,000 when multiplied by the density decrease.

Correct answer: C, density remains unchanged. Population density is the ratio population ÷ area. Let the original population be P and the original area be A, so original density is P/A. After a 10% decrease, the population becomes 0.90P and the area becomes 0.90A. New density is therefore 0.90P ÷ 0.90A = P/A, exactly the original density. A is wrong because the population decrease is not acting alone; the area decreases by the same proportion. B is wrong because there is no relative increase in population compared with area. D is also wrong because the two equal factors cancel completely, not partially. For example, if population is 1,000 and area is 100, density is 10; after both become 900 and 90, density is still 10. Memory cue: equal percentage change in numerator and denominator leaves a ratio unchanged.

Density compares population with area. If both population and area are multiplied by exactly the same factor, their ratio does not change. A 40 percent increase means each quantity becomes 1.4 times its original amount; it does not mean that density automatically rises by 40 percent.

Let the original population and area be \(P\) and \(A\). The new density is \(\frac{1.4P}{1.4A}=\frac{P}{A}\), which is the original density. Therefore, density remains unchanged and option C is correct. The equal percentage increases cancel because the numerator and denominator are changed equally.

Density equals population divided by area. If population becomes 150 percent of its original value and area becomes 125 percent, the new density compared with the old density is \(\frac{1.50}{1.25}=1.20\). Thus the new density is 120 percent of the original, which means an increase of 20 percent.

Option B is correct. The increase in population is larger, in relative terms, than the increase in area, so density rises, but it does not rise by the full 50 percent because the area also expands. Choosing 25 percent would incorrectly treat the population and area changes separately rather than taking their ratio. The calculation assumes the stated percentage changes apply to the same region and that density is measured consistently. The supplied answer is accurate.

Density means population divided by area. When both values change, their percentage changes cannot simply be subtracted because area is in the denominator. The population becomes 80% of its original value, while the area becomes 75% of its original value. The density therefore changes by the ratio of these two factors.

New density factor is \(0.80/0.75=1.0667\). Thus the new density is about 106.67% of the original, meaning an increase of about 6.7%. Option A is correct. A direct subtraction of 20% and 25% would miss the effect of the smaller area.

When both population and area change, density must be adjusted by the ratio of their growth factors. The original density is 720. Population increases by 5 percent, giving a factor of 1.05, while area increases by 12.5 percent, giving a factor of 1.125. Hence new density is \(720\times\frac{1.05}{1.125}\). The ratio is about 0.9333, so the result is approximately \(720\times0.9333=672\) persons per square kilometre.

Option B is correct. Density decreases because the area grows faster than the population. Option A would represent a much larger decrease, C is not the calculated result, and D would ignore the area increase. The word “approximately” is suitable because the ratio is expressed as a rounded decimal, although the result is exactly 672 using the given percentages. The supplied answer is consistent.

With a fixed area, a percentage change in population causes the same percentage change in density. The original density is 900 persons per square kilometre. A 20 percent decrease is \(0.20\times900=180\). Subtracting this loss gives \(900-180=720\) persons per square kilometre.

Therefore option B is correct. The area of 400 square kilometres confirms the result if calculated fully: the original population is \(900\times400=360000\), and after a 20 percent decrease it becomes 288,000. Dividing by 400 gives \(720\). The area does not change, so it must not be reduced by 20 percent as well. Option A subtracts too much, while C and D do not represent the stated percentage decrease.

When people migrate into a region, population increases. If the area does not change, density changes in exactly the same percentage as population because density is population divided by a fixed area. Therefore, it is enough to compare the incoming people with the original population.

The increase is \(50000\) out of \(250000\). Its percentage is \((50000\div250000)\times100=20\%\). The original density is \(250000\div625=400\) persons per square kilometre, and the new density is \(300000\div625=480\); this also shows a 20 percent increase. Thus option C is correct.

First find the original area using the density formula. Since density equals population divided by area, area is \(\frac{480000}{600}=800\) square kilometres. The area increases by 200 square kilometres, so the new area is 1,000 square kilometres. The population increases by 60,000, giving a new population of 540,000.

Now calculate the new density: \(\frac{540000}{1000}=540\) persons per square kilometre. Therefore, option C is correct. The answer cannot be 600 because both population and area changed; the new values must be used together. The calculation assumes the stated increases are exact and that density is measured over the whole new area.

To find the new density, first recover the original area from the given population and density. Since density equals population divided by area, area equals population divided by density. Then apply the population increase and the area increase separately before dividing again.

The original area is \(180000\div300=600\) square kilometres. A 20% population increase gives \(180000\times1.2=216000\), and the new area is \(600+50=650\) square kilometres. Thus, new density is \(216000\div650\approx332.31\) persons per square kilometre, which rounds to 332. Option B is correct.

Let total population be P and total area be A. The national density is P/A = 180. The selected part contains 25% of the population but only 5% of the area. Its density is therefore \(\frac{0.25P}{0.05A}=5\frac{P}{A}\). It is five times the national average, or 900 persons per square kilometre. Thus option C is correct.

The important comparison is the population share divided by the area share. The selected area holds five times as large a share of population as of area: 25 divided by 5 equals 5. This makes its density five times the average. It is not merely equal to the national average, and it is not ten times the average. The concentration of people in a much smaller area explains the result.

The national average density compares the whole population with the whole land area. The selected part contains 40% of the population but only 20% of the area. Its density relative to the national average is found by dividing the population share by the area share: \(\frac{0.40}{0.20}=2\). Therefore, that part has twice the national average density, so option B is correct.

The result does not mean that its density is two people per square kilometre. It means its density is two times the countrywide average. For instance, if the national average were 100 persons per square kilometre, this part would have 200. The concentration is higher because its population share is proportionally twice its area share.

Density compares population share with area share. In the first part, 60 percent of the total population occupies 30 percent of the area, so its relative density is (0.60 \/ 0.30 = 2). In the remaining part, 40 percent of the population occupies 70 percent of the area, giving (0.40 \/ 0.70 approximately 0.571).

The first part is therefore much denser than the second. Option A is correct. The comparison does not require the actual population or actual area, because the total values cancel when shares are used. A common mistake is to compare only population shares and ignore area shares. Here, the first area has both a larger population share and a much smaller area share, so its population is concentrated more strongly.

Correct answer: C, 400,000 people. The basic formula is density = population ÷ area. Rearranging it gives population = density × area. Substitute the given values: population = 625 × 640. A quick calculation is 625 × 64 = 40,000, and multiplying by 10 gives 625 × 640 = 400,000. The units also confirm the operation: persons per square kilometre multiplied by square kilometres leaves persons. A, 360,000, and B, 380,000, are lower than the correct product and result from inaccurate multiplication. D, 420,000, is higher and also does not equal 625 × 640. A useful check is that 625 is close to 600 and 640 is large, so a result near 400,000 is reasonable. Do not divide by area here; division is used when finding density, whereas population requires multiplication.

Area can be calculated when total population and population density are known. Since density means population per square kilometre, dividing the population by the density removes the people unit and leaves square kilometres. Decimal densities should be handled carefully, but the same division rule remains valid.

Area = population ÷ density = 360000 ÷ 562.5. Multiplying both numbers by 2 gives 720000 ÷ 1125, which equals 640. Another check is 562.5 × 640 = 360000. Therefore, option B is correct. The other options would give populations that do not match the stated total at the given density.

Correct answer: C, 600 persons per square kilometre. Density is population divided by area. A 15% population increase gives the multiplier 1.15, while an 8% area decrease gives the multiplier 0.92. Therefore the density multiplier is 1.15 ÷ 0.92 = 1.25. New density = 480 × 1.25 = 600. A and B are too low because they do not fully account for the simultaneous population increase and area decrease. D is too high and does not follow the correct factor calculation. Notice that both changes raise density: there are more people, and they occupy a smaller area. The percentage effects should not be handled by simply adding 15% and 8% to the original density, because one change affects the numerator and the other affects the denominator. Use the ratio of the new population factor to the new area factor.

To compare the new density with the old density, use separate multipliers for population and area. An 8 percent population decrease leaves 92 percent of the population, or a factor of 0.92. A 15 percent area increase makes the area 115 percent of its old value, or a factor of 1.15.

Because density equals population divided by area, the density factor is \(\frac{0.92}{1.15}=0.80\). Thus the new density is 80 percent of the old density, meaning it has fallen by 20 percent. Option B is correct. The result is not 92 percent because the larger area also lowers the density.

Density is the ratio of population to area. If both quantities are multiplied by the same factor, their ratio does not change. A 10 percent increase changes each original quantity to 110 percent, or \(1.1\) times, its original value. The equal change in numerator and denominator therefore cancels when the new density is calculated.

Let the original population and area be \(P\) and \(A\). The new density is \(1.1P \div 1.1A=P\div A\). Since the original density was 360, the new density remains 360 persons per square kilometre. Thus, option B is correct; 396 would incorrectly apply the population increase without applying the area increase.