Population Density

Density allows a fair comparison between regions of different sizes. It is calculated as population divided by area, and the difference is found by subtracting the smaller density from the larger one.

The first region has density \(720000\div1800=400\) persons per square kilometre. The second has density \(560000\div1120=500\) persons per square kilometre. The second region is therefore denser, and the difference is \(500-400=100\) persons per square kilometre. Thus option B is correct. Equal total populations are not required for equal density.

The original population is obtained from density times area: \(300 \times 2000 = 600000\). A 10 percent population increase gives a multiplier of 1.10, while a 25 percent area increase gives a multiplier of 1.25. New density can therefore be found directly by multiplying the old density by the population multiplier and dividing by the area multiplier.

The new density is \(300 \times (1.10 \div 1.25) = 300 \times 0.88 = 264\) persons per square kilometre. Option C is correct. Although population rises, area rises more quickly, so the number of people per square kilometre falls from 300 to 264. This also explains why 275, a smaller decrease, is not correct.

A percentage change should be applied separately to population and area because density is a ratio. A 30 percent population decrease leaves 70 percent of the original population, or a factor of 0.70. A 20 percent area decrease leaves 80 percent of the original area, or a factor of 0.80.

The new density is \(560\times\frac{0.70}{0.80}=560\times0.875=490\) persons per square kilometre. Hence option C is correct. Although population falls, area falls by a larger relative amount, so the denominator decreases more strongly and the density becomes lower by only 70 persons per square kilometre.

The original density and population can be used to find the unchanged area. Since \(Density=Population\div Area\), the area is \(980000\div1400=700\) square kilometres. The population then increases by 70000, so the new population is \(980000+70000=1050000\). Dividing this by the same area gives \(1050000\div700=1500\) persons per square kilometre. Therefore, option B is correct.

Another quick method is to calculate the density increase directly: \(70000\div700=100\) persons per square kilometre. Adding this to the original 1400 gives 1500. Option A would represent an increase of only 50, while the fixed area and added population require an increase of 100. The unchanged-area condition is essential to both calculations.

When area remains fixed, a fall in population produces a directly proportional fall in density. The region covers 2,500 square kilometres and initially has a density of 480 persons per square kilometre. The reduction in population is 120,000. The density reduction is found by distributing this loss over the unchanged area: 120000/2500=48 persons per square kilometre.

The original population would be 480×2500=1,200,000, and after the loss it would be 1,080,000. Dividing the new population by 2,500 gives 432 persons per square kilometre, which is 48 less than 480. Therefore option C is correct. The answer is not 40 or 60 because those values do not result from dividing the stated population change by the stated fixed area.

Let the original area be \(A\) square kilometres. Since the original density is 500 persons per square kilometre, the population is \(500A\). After adding 80 square kilometres, the area becomes \(A+80\). The population does not change, but the new density is 400, so the same population is also \(400(A+80)\).

Equating the two expressions gives \(500A=400(A+80)\). Expanding, \(500A=400A+32000\), so \(100A=32000\) and \(A=320\) square kilometres. Therefore, the original area was 320 square kilometres, making option C correct. Substitution confirms that the population is 160,000 before and after the area increase.

When the area remains unchanged, an increase in population produces an increase in density. The amount by which density rises is caused by the additional people divided by the same area. Therefore, first subtract the old density from the new density, then divide the added population by that increase.

The density increase is \(425-350=75\) persons per square kilometre. Thus, area is \(30000\div75=400\) square kilometres. Option B is correct. Checking it confirms the result: 30,000 extra people spread over 400 square kilometres add 75 people per square kilometre. The unchanged-area condition is essential.

Answer: B, 400 square kilometres. Density is population per unit area. Because the area does not change, the fall in total population equals the fall in density multiplied by the area. The density decrease is 720 − 600 = 120 persons per square kilometre. Therefore area = population decrease ÷ density decrease = 48,000/120 = 400 square kilometres. A, C and D do not produce the stated population loss: multiplying 120 by them gives 36,000, 54,000 and 60,000 respectively. B gives 120 × 400 = 48,000, so it is correct. As a check, the original population was 720 × 400 = 288,000 and the final population was 600 × 400 = 240,000; their difference is 48,000.

Answer: C, density remains unchanged. Let the initial density be P/A. A decrease of 18 percent changes population to 0.82P and area to 0.82A. New density is (0.82P)/(0.82A), and the common factor 0.82 cancels, leaving P/A, exactly the original density. A is wrong because it considers only the population change. B is wrong because there is no unmatched increase in population relative to area. D is also wrong because equal proportional changes cancel completely, not partially. For example, if population is 100 and area is 10, density is 10; after both become 82 and 8.2, density is still 10. Memory cue: equal percentage change in numerator and denominator leaves a ratio unchanged.

Density is a ratio: population divided by area. If both population and area are multiplied by the same factor, that factor cancels from the ratio. A 60 percent increase changes each original quantity to 160 percent of its former value, or 1.6 times the original value.

Let the original population and area be \(P\) and \(A\). The new density is \(\frac{1.6P}{1.6A}=\frac{P}{A}\), exactly the original density. Therefore density neither increases nor decreases, and option C is correct. A 60 percent increase in both quantities does not mean a 120 percent density increase because density depends on their ratio, not their sum.

Answer: A, density increases by about 16.7 percent. A 30 percent population decrease leaves 0.70P, while a 40 percent area decrease leaves 0.60A. The new density factor is (0.70P)/(0.60A) = 0.70/0.60 = 1.1667. Thus density becomes about 116.67 percent of its original value, an increase of about 16.7 percent. B reverses the direction. C incorrectly copies the population percentage and ignores the area change. D incorrectly uses the area percentage as the density change. Although both population and area fall, area falls more sharply, so the remaining people occupy less space per person overall and density rises. Cue: compare the remaining factors, not just the declines.

Answer: B, 742.5 persons per square kilometre. The initial density is P/A = 810. Population growth multiplies P by 1.10, and area growth multiplies A by 1.20. Therefore new density = 810 × (1.10/1.20) = 810 × 11/12 = 742.5 persons per square kilometre. A is too low, while C and D do not apply both growth factors correctly. Since area grows faster than population, the result must be below 810; this rules out any value above the original and supports the calculation. Do not subtract 20 from 10 as if density were a simple difference; density is a quotient. Memory cue: new density = old density × population factor ÷ area factor.

If area remains fixed, population density changes in the same proportion as population. The original density is 960 persons per square kilometre. A decrease of 25 percent means 75 percent of the original population remains. Therefore, the new density is \(0.75\times960\), or equivalently, 25 percent of 960 is first subtracted.

Twenty-five percent of 960 is \(960\times0.25=240\). Subtracting gives \(960-240=720\) persons per square kilometre. The original area of 500 square kilometres is not needed for the final calculation, but it confirms that fixed area would carry the same proportional change from population to density. Thus option C is correct.

Answer: C, 25 percent. The area is unchanged, so density changes in the same proportion as population. The original population is 360,000 and 90,000 people enter. The proportional increase is 90,000/360,000 = 0.25 = 25 percent. For a second method, original density is 360,000/900 = 400 persons per square kilometre. New population is 450,000, so new density is 450,000/900 = 500. The increase is 100, and 100/400 × 100 = 25 percent. A, B and D give incorrect fractions of the original population. The key condition is fixed area; if area had also changed, the density percentage would need a ratio calculation.

Answer: B, approximately 643 persons per square kilometre. First find the original area: A = 600,000/750 = 800 square kilometres. After the increase, new area is 800 + 250 = 1,050 square kilometres, and new population is 600,000 + 75,000 = 675,000. New density = 675,000/1,050 = 642.857..., which rounds to 643 persons per square kilometre. A is too low, while C and D result from inaccurate area or population handling. Although population rises, area rises proportionally more: population factor is 1.125 but area factor is 1.3125, so density falls. Always calculate the original area before adding an absolute area increase.

The original area can be recovered from the density formula \(D=P/A\), so \(A=P/D\). The population grows by 25 percent, while the area receives an additional 100 square kilometres. The new density must be calculated using both the new population and the new area; comparing only population growth would be incorrect.

Original area is \(270000/450=600\) square kilometres. New population is \(270000\times1.25=337500\), and new area is \(600+100=700\) square kilometres. Thus new density is \(337500/700\approx482.14\) persons per square kilometre. Rounded approximately, this is 482, so option B is correct.

Answer: C, 720 persons per square kilometre. Population density means population divided by area. Let the total population be P and the total area be A. The country’s average density is P/A = 240. The selected part contains 30% of the population, or 0.30P, and 10% of the area, or 0.10A. Its density is therefore (0.30P)/(0.10A) = (0.30/0.10)(P/A) = 3 × 240 = 720 persons per square kilometre. A is wrong because it does not represent the threefold concentration. B is wrong because it uses an insufficient increase over the average. C correctly applies the population-share to area-share ratio. D is wrong because it gives four times the average, whereas the ratio is only three. Memory cue: sub-area density = average density × population-share ÷ area-share.

National average density is total population divided by total land area. To compare a particular part with that national average, compare its population share with its area share. The relative density factor is population share divided by area share. Here, the part contains 45% of the population but occupies only 15% of the land.

The factor is \(\frac{0.45}{0.15}=3\). Therefore, that part has three times the national average density, so option C is correct. This does not mean that every square kilometre inside the part has exactly the same number of people; it is a comparison of average densities. The result is high because the population share is three times as large as the area share. The other options do not match this ratio.

To compare density, compare the share of population with the share of area. The first part contains 50 percent of the population in only 20 percent of the area. Relative to the regional average, its density factor is \(0.50\div0.20=2.5\). The remaining part contains the other 50 percent in 80 percent of the area.

Its relative density factor is \(0.50\div0.80=0.625\). Thus the first 20 percent area has far more people per unit of area and is denser. The correct choice is A. The comparison is possible because both population and area shares are given.

Population density gives the number of people per unit area. To find the total population, multiply density by area because the area units cancel: persons per square kilometre multiplied by square kilometres gives persons. This is the reverse of the formula used to find density, which is population divided by area.

The required calculation is \(875\times640\). First, \(875\times64=56000\), and multiplying by ten gives \(560000\). Thus the total population is 560,000 people, so option C is correct. The other choices come from using an incorrect multiplication or place value and do not match the stated density and area.

Area is obtained by dividing population by density: \(Area=\frac{Population}{Density}\). Here the calculation is \(\frac{441000}{612.5}\). To remove the decimal, multiply numerator and denominator by 2: \(\frac{882000}{1225}\). Since \(1225\times720=882000\), the quotient is 720 square kilometres.

Therefore option C is correct. A unit check also confirms the formula: people divided by people per square kilometre gives square kilometres. The result can be checked by multiplying back: \(612.5\times720=441000\) people. The other options would produce different populations at the stated density, so they cannot be correct. The decimal density causes no ambiguity when handled by equivalent multiplication.

Answer: C, 650 persons per square kilometre. Density is P/A, so changes in population and area must be applied as a ratio, not simply added or subtracted. An 18% population increase gives a population factor of 1.18. A 5.6% area decrease leaves 94.4% of the old area, giving an area factor of 0.944. Thus the new density factor is 1.18/0.944 = 1.25. New density = 520 × 1.25 = 650. A is too low because population grew while area shrank. B is also below the calculated value. C is correct. D is too high because the increase is 25%, not about 31%. Memory cue: new density factor = population factor ÷ area factor.

Answer: C, 80%. Let the original population and area be P and A, so old density is P/A. A 6% population decrease leaves 0.94P. A 17.5% area increase makes the area 1.175A. The ratio of new density to old density is (0.94P/1.175A) ÷ (P/A) = 0.94/1.175 = 0.80. Therefore, 80% of the old density remains; equivalently, density falls by 20%. A is too low, B is lower than the exact ratio, and D is too high. C is correct because both changes reduce density: the numerator falls and the denominator rises. A common mistake is to subtract 6% from 17.5% and call the result the density change. Density changes through a ratio, so use factors.

Answer: B, 480 persons per square kilometre. Density is the ratio P/A. If the original population is P and area is A, the original density is P/A = 480. After equal 25% increases, population becomes 1.25P and area becomes 1.25A. The new density is (1.25P)/(1.25A) = P/A = 480. A is incorrect because neither quantity falls. C and D incorrectly increase density by applying the population increase without applying the same increase to area. B is correct because equal proportional changes cancel in a ratio. The actual population and area are unnecessary. Memory cue: if numerator and denominator are multiplied by the same non-zero factor, their ratio does not change.

With area unchanged, density changes in the same proportion as population because \(D=\frac{P}{A}\) and the denominator is fixed. The fall in density is \(1800-1440=360\) persons per square kilometre. To express this fall as a percentage of the original density, calculate \(\frac{360}{1800}\times100=20\%\). Since the area has not changed, the population has undergone the same percentage decrease.

Therefore, option B, 20 percent, is correct. The comparison must be made with the original value, 1800, not with the new value, 1440. A 25 percent answer would use an incorrect base. The unchanged area allows density to serve as a direct indicator of the proportional change in population.