Problem Solving

At the beginning of 2010, 60% of the population of Town X lived in the south and the rest lived in the north. During 2010, the population of Town X grew by 5.5%. If the population in the south grew by 4.5%, by how much did the population in the north grow?

A)1%
B)3.5%
C)6.5%
D)7%
E)13.75%

Guys ,

Pls suggest .

Let the no. of population is 100

So in south = 60 and north=40

Population increase by 5.5% so = 100(1+5.5/100)

It gives = 105.5

In south it grew 4.5% = 60(1+4.5/100) so it gives = 62.7

Then in North it will be = 105.5 - 62.7= 42.8

Now here i have one question

Should i do like this

A) North percentage change = 42.8 - 40/40 *100

OR

B) North percentage = 42.8/40*100

if i do like A then my ans is correct.

Pls suggest

j_shreyans wrote:At the beginning of 2010, 60% of the population of Town X lived in the south and the rest lived in the north. During 2010, the population of Town X grew by 5.5%. If the population in the south grew by 4.5%, by how much did the population in the north grow?

A)1%
B)3.5%
C)6.5%
D)7%
E)13.75%

Let the total population = 1000.
Since the total population grew by 5.5%, the increase in the total population = (5.5/100)(1000) = 55.

Since 60% lived in the south, the number living in the south = (60/100)(1000) = 600.
Since the population in the south grew by 4.5%, the increase in the southern population = (4.5/100)(600) = 27.

Since 600 of the 1000 people lived in the south, the number living in the north = 1000-600 = 400.
Since the total population increased by 55, while the southern population increased by 27, the increase in the northern population = 55-27 = 28.
Percent increase in the northern population = (increase in the northern population)/(total northern population) = 28/400 = 7/100 = 7%.

The correct answer is D.

% change = 100 * (New - Old)/Old, so you want to do it the first way.

2010:
South: .6x
North: .4x

2011:
South: .6x * (1.045) = .627x
Total = 1.055x
North = 1.055x - .627x = .428x

Our % change is thus

.428x - .4x
----------- * 100
.4x

which reduces to (2.8/4)%, or 7%.

Of course, there's also an easier way!

If the two populations were EXACTLY THE SAME in 2010, then the overall increase would be weighted 50/50: in other words, if the South grew by 4.5% and the total growth was 5.5%, the growth in the North would be 6.5%. (After all, the average of 6.5 and 4.5 = 5.5!)

However, the populations were NOT equal: there were more people in the South. Hence the growth in the North has LESS of an impact on the average, and must thus be slightly HIGHER than 6.5 to compensate. D is the only answer in range, so that's our choice.

Start: P = South 0.6P + North 0.4P
Later: 1.055P = 1.045*0.6P + g*0.4P
So, 1.055 = 0.627 + 0.4g
Solving for g gives:
0.4g = 1.055 - 0.627 = 0.428
So g = 0.428/0.4 = 1.07 = 107%
Hence increase = 7%
ANSWER = D

Hi, what I did was:
5.5 = (4.5*0.60+n*0.40), so, n = (5.5-2.7)/0.4, so, n = 7. so, ans. (D)

And there's always the trusty number line. If the population growth in the South was 4.5% and the total population growth was 5.5%, we can designate the population growth in the North as x, and depict that information like so:

4.5----5.5------x

If 60% of the population is in the South, 40% must be in the north, giving us a 6:4 or 3:2 ratio. We're weighted towards the South, so we can call the gap from south to total '2t' and the gap from total to North '3t.'

If 2t is the gap from 4.5 to 5.5, then 2t = 5.5 - 4.5, and t = .5. If t =.5, then 3t = 1.5. So the North is 1.5 away from the total.

5.5 + 1.5 = 7.

j_shreyans wrote:
Should i do like this

A) North percentage change = 42.8 - 40/40 *100

OR

B) North percentage = 42.8/40*100

if i do like A then my ans is correct.

Pls suggest

You should do like A, because they ask for growth, which means percentage change: (N-O)/O)*100.

Your second option,(N/O)*100 is not the growth, but rather the new as a percentage of the old: . They do sometimes ask for such quantities, but they call them "X as a percentage of Y" or "What percent of Y is X?"

Regards,
Mike Miagi
www.google.com/+MikeMiagi

Hi ,

It is given that Town X grew by 5.5% and south grew by 4.5% then north must be grew by 1%.

Can i take like above?

Please advise.

Thanks,

Shreyans

It is given that Town X grew by 5.5% and south grew by 4.5% then north must be grew by 1%.

To see that we're not simply adding the growth of North and South to get the total growth rate, take a very simple scenario.
Say there are 100 people in the North and 100 people in the South. And then say we have an increase of 1% in the North and an increase of 5% in the South.

Here's the initial situation:
North: 100
South: 100
Total: 200

And after our change:
North: 101
South: 105
Total: 206

So our total percent change is ](206-200)/200] * 100 = 3%. Notice that we're not adding the growth rates for North and South, but rather, taking the average, which falls in between the percent change for North and South individually.

The same is true for this question. If South growth =4.5% and Total growth = 5.5%, North growth must exceed 5.5%, because the total growth represents the weighted average of the percent changes for North and South