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Sam owns a football accessory shop. At present, the number of items in

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New post Updated on: 18 Jul 2019, 09:09
Quote:
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

A. 200
B. 216
C. 220
D. 230
E. 260


yr0: 2000
yr1: 2000*11/10
yr2: 2000*121/100=2420 items and 11 stores

condition that no store has more than 20% items of any other store;
so lets make 10 stores with the same items and 1 store with 20% more;
10x+1.2x=2420
56x/5=2420
x=2420*5/56
now the store with no more than 20% is 1.2x
1.2x=2420*5/56,… 1.2x=(aprox.) 260

Answer (E).

Originally posted by exc4libur on 17 Jul 2019, 09:37.
Last edited by exc4libur on 18 Jul 2019, 09:09, edited 1 time in total.
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 09:46
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

A. 200
B. 216
C. 220
D. 230
E. 260

no of items in the shop after 2 years = 2000 x 1.1 x 1.1 = 20 x 11 x 11

he adds 10 more shops so total is 11 shops so average items per shop = 20 x 11 x 11 / 11 = 20 x 11 = 220 items

so options A, B, are ruled out.

for c, yeah ans can be easily greater than 220 ie 221 and which is easily greater than 219 for one of the shops ie ( 221, 219 , 220 x 9)
so better leave that as of now without checking.

remaining we can check for 260 now for upper limit

if one shop is 260 then remaining 10 shops has to have 216 items to maintain the higher level. but 1.2 x 216 is 259.2 which is less than 260. ( but it cant be greater than 259.2 as per the question).

so e is ruled out. even 259 is possible. so between d and c, definitely d is greater. so D is the answer.
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 09:58
1
Quote:
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?


Number of items in the shop in the beginning is N = 2000
Increase in number of items per year r = 10%
Time: 2 years
After 2 years, 10 more shops are opened, and the items are spread between the shops.
We need to find maximum of items in a shop, if it is maximum 20% more than in any other shop.

First, let us find the number of items in the shop after 2 years: \(N = 2000 * (1 + 0.1) * (1 + 0.1) = 2000 * 1.1^2 = 2420\)
Now, as we know that 10 more shops are opened, the total number of shops is 10 + 1 = 11.
The number of items in 1 shop will be maximized, if all other shops have the same amount of items, and the one particular shop has 20% more items. Thus, the number of items in this shop will be 120%. In this case, to find the number of items in 1 ordinary shop (s), we need to divide the total number of items on the number of shops taking into account that one of the shops has 20% more items than others.
\(s = 2420 / (10 + 1 * 1.2) = 2420 / 11.2 = 216\)
As we found the approximate number of items in 1 shop, to find the number of items in a shop with maximum items, we need to multiply it by 120%:
\(216 * 1.2 = 260\)

Answer: E
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 10:04
so total books considering 20% each year increase in 2 years will be 2000+ 200 + 220=2420.Total no of shops=11
so if consider all equal its 2420/11 i.e. 220. Now a shop can have not more than 20% from any other.When we consider 260 it goes more than 20% as 20% of 220 is 44, if 1 hs 260 other will be less than 220 by more than 44.

Therefore D is correct.
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 10:07
Total items = 2000

Increase of 10% per year

After 1st year 1.10 x 2000 = 2200
After 2nd year 1.10 x 2200 = 2420

Total shops after 2 years = 1 + 10 = 11
Average item per shop = 2420/11 = 220

2420 - 220 = 2200

20% of 2200 = 440

220 < 440

Thus each shop will have a maximum item of 220

Hence answer choice C.
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 10:12
1
IMO-E

Current No. of shops=2000
Annual % increase= 10%

Therefore , after 2 years No. of shop= 2000 x 1.1 x 1.1= 2420

Sam decides to split into 10 shops.

Condition: Any shop content < 20% greater than other= 1.2 x other shop content

To maximise content of one shop, let 9 shops each has y items
10th Shop x items
x < 1.2 y

Also, x+9y=2420
9y + 1.2y>2420
10.2 y > 2420
y> 237

Max possible item > 237
Only E) 260 satisfies...
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 10:15
1
IMO E

Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop? 

A. 200 
B. 216 
C. 220 
D. 230 
E. 260 

Given, initial no. of items is 2000
After one year the no. of items is 2200
After two years the no. of items is 2420
And no. of shops is 1+10=11
So by the given condition, we know the number of items in each shop has to be > 2420/11, i.e., 220
So option A, B and C are out
Between D & E, we can get only 260 approximately fits the condition.
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New post 17 Jul 2019, 10:20
1
After 2 years, the number of items in Sam's shop is \(2,000 * \frac{11}{10}* \frac{11}{10} = 2,420\) and the total number of shops is \(1 + 10 = 11\) shops.

Let the minimum number and maximum number of total items in a shop be \(a\) and \(1.2a\) , respectively.

To find the maximum possible value of \(1.2a\), assume that only \(1\) shop has \((1.2a)\) items, while the rest have \((a)\) items each:
\(2,420 = (1.2a)*1 + (a)*10\) --> \(2,420 = 11.2a\) --> \(a = 216.07\) --> \(1.2a = 259.3 \approx{260}\)

Let's check the maximum/ minimum ratio: \(\frac{260}{216} = 1.204 \approx{1.20}\) --> ACCEPTABLE

Thus, the approximate maximum possible number of items in a shop \(= 1.2a \approx{260} -\) Correct answer is (E)
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 11:12
1
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

At present - 2000 items, next year 2200, after two years 2420 items
among 11 shops - equal distribution ==> 220 items
So, 220*120/100 ==> 264 items. So, E
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 11:15
Given,

1. Total number of items in Sam's shop = 2000
2. Percentage increase in number of items in Sam's shop every year = 10%
=> Total number of items in Sam's shop after 2 years = 2420
3. Total number of shops after 2 years = 11
4. Number of items of no shop is greater than 20% of the total number of items of any other shop.

To find,

Approximate maximum possible number of items in a shop.

Now if we see then we find that 2420 can be equally divided into 11 shops by giving 220 items to each shop.
=> At 220 items in each shop, all conditions are met. But note that this is not the maximum.
=> Hence maximum will be any number greater than 220.

Let us check the options -

A. 200 --> As the maximum needs to be greater than 220, this is wrong.

B. 216 --> As the maximum needs to be greater than 220, this is wrong.

C. 220 --> As the maximum needs to be greater than 220, this is wrong.

D. 230 --> This is a possibility, let us check this further.

Now 230 is 10 more than the average number of items per shop which is 220. Now, if we balance the increase of 10 by reducing the same number of items in another shop and making it to be 210 and keeping every other shop constant at 220, all our conditions are met. Hence this is a possible answer but note that it is still not a maximum as for 210, the maximum can go up to 252.

E. 260 --> This is a possibility, let us check this further.

Now 260 is 40 more than the average number of items per shop which is 220. Now, if we balance the increase of 40 by reducing the same number of items in anoher shop and making it to be 180 and keeping every other shop constant at 220, then our main condition of 260 not being more by 20% of other shops fails. Hence this is not the correct answer.

If we try to find the maximum then we can find it by using the following equation -->

220 + x = 220 - x + 0.2 * (220 - x)
=> 2.2 *x = 44
=> x = 20

Hence the maximum is at 240.

Given our options we have,

Answer: D
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Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post Updated on: 18 Jul 2019, 03:56
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

Given:
# number of items after 2 years = 2000 * 1.1 * 1.1 = 2000 * 1.21 = 2420
# number of shops after 2 years = 10 + 1 =11
total items of no shop are greater than 20% of the total items of any other shop means Nmax <= 1.2*Nmin, where Nmax = maximum number of items in a shop & Nmin = minimum number of items in a shop
Need to find out the maximum number of items in a shop.


A. 200 --> 230> 200
B. 216 --> 230> 216
C. 220 --> 230> 220
D. 230 --> correct: let say Nmax = 230, other 10 shops has Nmin number of items each, so Nmin = (2420 - 230)/10 = 219, & 230 < 1.2 * 219 = 262.8
E. 260 --> wrong: let say Nmax = 260, other 10 shops has Nmin number of items each, so Nmin = (2420 - 260)/10 = 216, but 260 > 1.2 * 216 = 259.2

Originally posted by hiranmay on 17 Jul 2019, 11:29.
Last edited by hiranmay on 18 Jul 2019, 03:56, edited 1 time in total.
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 12:06
1
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

Red portion of the question seems confusing. My interpretation is - If shop with max items has 120 items, no other shop should be below 100.

Also Note That: QUESTION SAYS APPROXIMATE MAX ITEMS. So It is ok if we dont get an Exact Answer which in this case we dont. We get a decimal answer.

Solution:
Year 0 = 2000 items
Year 2 = \(2000 *1.1^2\) = 2420 items

Total number of shops = 10+1 = 11

We have to distribute 2420 items to 11 shops such that shop with max items is not more than 20% higher than EACH of the remaining shops.

We need to find what is the max items a shop can have which fulfils this condition. We can do this by ensuring 10 shops have exact same items and 11th shop has 20% more. Any other distribution strategy will not give us max possible items of a shop.

Let items of each of 10 shops be X
So 11th shop has 1.2 X .... (We have to find 1.2X)

10X + 1.2X = 2420

11.2X = 2420

1.2X = 1.2 (2420)/11.2

1.2X = Approx 260...... The exact answer is 259.xxx and number of items cannot be subdivided, but question asks for approximate answer.





ANSWER: E. 260
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 12:07
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

After two years, total number of items = 2420
Total shops = 11
maximum possible number of items in a shop = 2420 / 11 = 220

So, the correct answer choice is (C)
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 12:08
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

Total no. of items in shop = 2000.
Total no. of items in shop after 1 year = 1.1*2000 = 2200 (10%increase)
Total no. of items in shop after 2nd year = 1.1*2200 = 2420 (10%increase)
Now Sam opens 10 more shops, so total shops = 11.

Now Sam distributes the items(2420) in 11 shops in such a manner that total items of no shop are greater than 20% of the total items of any other shop.
Lets say a shop has 200 items. 20% of 200 = 40. Therefore no other shop should have items more than 240.

If items are equally distributed among 11 shops then each shop will have 220 items.

Consider one shop has 230 and one has 210. Rest other being 220. In this case the difference between max. items and min. items is 230 - 210 = 20. If we take 20% of 210 = 42. This scenario is possible as the the shop having max. items is less than 20% of the items with the shop having min. items. Therefore, a shop can have 230 items.

Consider one shop has 260 and one has 180. Rest others being 220. In this case the difference between max. items and min. items is 260-180=80. If we take 20%of 180 = 36. This scenario is not possible as the shop with max. items is more than 20% of the items with shop with min. items. Therefore a shop cannot have 260 items.

Hence from the answer choices, the approximate maximum possible number of items in a shop = 230.


A. 200
B. 216
C. 220
D. 230
E. 260

Answer Choice: D
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 12:53
1
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

No. of items in his shop = 2000
after 1 yr total items = 2000 + 200 = 2200
after 2 yrs total items = 2200 + 220 = 2420

He decides to open more 10 shops, so he will have a total of 11 shops.

If he distributes an equal number of items then
2420 / 11 = 220 items per shop.

We will have to a minimum number of items in each shop and max in one shop

We can eliminate option A 200, B 216, C 220 immediately because minimum items are 220.

So we have 2 options 230 and 260.

If we consider,
216 items in each shop: 216 * 10 = 2160
2420 - 2160 = 260

But 210 * 120 % = 259.2
and the shop cannot have more than 20 % so 259 items can be there but not 260.

any number less than 216 will have a lesser value of 120% than 260.
So from remaining options of 230 and 260
We are asked for approx number of items. and as 230 and 260 has a difference of 30 number of items,
we can conclude the answer to be 260.

Answer: E

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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 13:10
1
Total number of items Sam has in the shop= 2000*1.1*1.1=2420

Since we want to maximize the number of products in one shop and total items of no shop are greater than 20% of the total items of any other shop, we need to divide equal number of items in 10 shops and 11th shop has 20% more items than any other shop.

10*x+1.2x=2420
1.2x= (2420*1.2)/11.2
1.2x= 260(approximately)

the approximate maximum possible number of items in a shop= 260

IMO E
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Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post Updated on: 18 Jul 2019, 06:09
Total items in shop currently =2000
Yearly projection for increment =10%
Hence after two years new stock level = 1.1x1.1x2000 =2420 items

Ten additional shops were opened after two years, implying total shops now is 11.
A requirement of sharing the total items into the eleven shops is that no shop must contain more than 20% more items than any other shop.
With this constraint, the maximum possible items in a shop can only occur if ten shops have the same number of items while one has 20% more items.
If x is the minimum items in the ten shops, then 10x+1.2x=2420
11.2x=2420
x=216
Maximum possible =1.2x216=259.2
This is approximately 260.

Hence E.

Posted from my mobile device

Originally posted by eakabuah on 17 Jul 2019, 14:15.
Last edited by eakabuah on 18 Jul 2019, 06:09, edited 1 time in total.
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 14:33
1
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

A. 200
B. 216
C. 220
D. 230
E. 260



First of all, we need to figure out the total number of items after 2 years.
1,1 *1,1 = 1,21.
2000*1,21 = 2420 items

lets assume, that there are 10 shops with identical number of items - X and 1 shop with +20% - 1,2X

so we have 10X+1,2X = 2420
11,2X=2420
X is approximately 216
1,2 X is approximately 260

The answer is E
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 14:53
Answer D
Stock at end of 2 years: 2420
Now distribution of stock in 11 shops, 1 current shop and 10 new:
2420/11 : 220 if question ask equal distribution, But question is asking maximum no of items
So test 230 and 260 :
Consider 230 given to 8 shops : 230* 8 = 1840
Left : 580 /3 : 193,
193+20% = 231.5, So clearly other shops has less than 20% of shops with 193 items

260 : will not give such solution,
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Re: Sam owns a football accessory shop. At present, the number of items in  [#permalink]

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New post 17 Jul 2019, 15:23
Sam owns a football accessory shop. At present, the number of items in his shop is 2000 and every year, he plans to increase it by 10% every year. After 2 years, he decides to open 10 more shops and distributes the total number of items he has in such a way that total items of no shop are greater than 20% of the total items of any other shop. What is the approximate maximum possible number of items in a shop?

A. 200
B. 216
C. 220
D. 230
E. 260

2000 items in the shop at 10% increase per year, there will be 2200 after 1 year and 2420 after 2 years.
Now there are 10 more shops, so total of 11 shops to split the 2420 items.

The split is such that no shop carries more than 20% of 2420. So, the maximum possible number of items in any shop would be 20% of 2420.
Since there are 11 shops, each shop should carry a maximum of 220 items in order to sell all 2420 items.

Correct answer choice C
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Re: Sam owns a football accessory shop. At present, the number of items in   [#permalink] 17 Jul 2019, 15:23

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