A tricky Sentence Correction question
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Stage II | Sentence Correction | Q 104
Like many successful authors, Salman Rushdie’s first novel, Grimus, about a Native American who receives the gift of immortality, was an abject failure.

A. Like many successful authors, Salman Rushdie’s first novel, Grimus, about
B. As have many successful authors, the first of Salman Rushdie’s novels, Grimus, about
C. Just as with many successful authors, the first of Salman Rushdie’s novels, Grimus, about
D. Just like many successful authors, Salman Rushdie’s first novel, Grimus, on
E. As did many successful authors, Salman Rushdie’s first novel, Grimus, on

Question: Aren't all the 5 options incorrect? I sense an error in the modifier.

Solution:
A. Incorrect comparison between novel and authors
B. 'Incorrect comparison between novel and authors. 'have' indicates that authors have done something, but then it continues with 'the novel'.
C. Correct. "Just as with my brother, my job is well paying too." This is the structure this option follows and there is nothing wrong. Note that without a 'with', it will be an incorrect option. I must agree though that this is a little akward in usage, but the most correct in this context.
D. Incorrect comparison between novel and authors
E. Incorrect comparison between novel and authors

Correct choice would be C.

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A very interesting Quant problem on simple arithmetic...
=================================================
1,234
1,243
1,423
........
........
+4,321
=========

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1,2,3 and 4 exactly once in each integer. What is the sum of the 24 integers ?

(A) 24,000
(B) 26,664
(C) 40,440
(D) 60,000
(E) 66,660

-->
Since there are 4 digits, we can arrange them in 4! = 24 ways. Lets assume there are only 3 digits - 1,2,3. In how many ways can we arrange them? 3!= 6. They would be

123
132
231
213
312
321

Now, lets attach 4 to each of them. The 6 numbers now are...

1234
1324
2314
2134
3124
3214

We can repeat the process with 1, 2, and 3 to be attached and we will get another three sets of 6 numbers. So, overall, we have 24 numbers.

When we split the 24 numbers in such 4 sets of 6 numbers, we see that in the units digits of these 24 numbers, 1 is going to appear 6 times, 2 is going to appear 6 times, and so on. The same will happen in the 10's digit, 100's digit, and 1000's digit. This is all there is to this problem!

So when we add these 24 numbers -->
We first add the units digits, which is going to be 6*(1+2+3+4) = 60. So we put down 0 and carry over 6. The sum so far is _____0.
We then add the tens digits, which is going to be 6*(1+2+3+4) = 60. We add the carry 6 and we get 66. So we put down 6 and carry over 6. The sum so far is ______60.
We then add the hundreds digits, which is going to be 6*(1+2+3+4) = 60. We add the carry 6 and we get 66. So we put down 6 and carry over 6. The sum so far is ______660.
We then add the thousands digits, which is going to be 6*(1+2+3+4) = 60. We add the carry 6 and we get 66. So we put down 66. There is no need for a carry here. The sum so far is 66660.

Bingo, 66660 happens to be option E! Magic? Math!

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From a well shuffled pack of 52 cards, three cards are drawn at random. Find the probability of drawing an ace,
a king and a jack.

==========================
Evaluate all possibilities if you were to 'manipulate behind the scenes' and make the described event.

1. The first card to be drawn can be A, K or J. That means we have 3 options.

Options so far = 3

Lets say we choose A.

2. Then, once we choose an option, we have 4 cards for the chosen option. i.e. A's of all 4 suits.

Options so far = 3 * 4

3. The second card to be drawn can be two remaining ranks i.e. K and J in our case. That means we have 2 options.

Options so far = 3 * 4 * 2

Lets say we choose K.

4. Then, once we choose an option, we have 4 cards for the chosen option. i.e. K's of all 4 suits.

Options so far = 3 * 4 * 2 * 4

5. For the third card, we have only the last rank to choose. i.e. J in our case.

Options so far = 3 * 4 * 2 * 4 * 1

6. Then, we have 4 cards for the last option. i.e. J's of all 4 suits.

Options so far = 3 * 4 * 2 * 4 * 1 * 4

All this will happen with 52C3 = 52*51*50 options in totality.

So, the required probability = (3 * 4 * 2 * 4 * 1 * 4)/(52*51*50 ) = 16/5525.

==========================

If j and k are integers and j^2/k is odd, which of the following must be true?

(A) j and k are both even
(B) j = k
(C) If j is even, k is even
(D) j is divisible by k
(E) j^2 > k

--->
Lets see how to solve this.
j^2/ k = odd. i.e. j^2 = k times an odd number.

Now lets see the answer choices. We will try to figure out of the choices must be true'. If we are able find a case in which they are not true, we will skip that answer choice.

(A) j and k are both even
--> Putting j=1, j^ 2= 1 = 1*1. So k=1. Both j and k can be odd. Incorrect choice.

(B) j = k
--> Putting j=9, j^ 2= 81 = 27*3. So k can be either 27 or 3. Either way, j and k are not the same. Incorrect choice.

(C) If j is even, k is even
--> Putting j=even, j^ 2= even = k * some odd number. Since (k*some odd number) = even, k will have to be even! This holds true for any value of j. Must be true. Correct choice.

(D) j is divisible by k
--> Putting j=9, j^ 2= 81 = 27*3. So k can be either 27 or 3. When k=27, j=9 is not divisible by k=27. Incorrect choice.

(E) j^2 > k
--> Putting j=1, j^ 2= 1 = 1*1. So k = 1. But j^2=1 > k =1 does not hold true. Incorrect choice.

So finally, the answer is C.

If set S consists of the positive integers w, x, y, and z, is the
range of the numbers in S greater than 6 ?
(1) No two numbers in set S are consecutive.
(2) None of the numbers in set S are multiples of 3.

Question:
My answer was E since I also assumed that all numbers can be equal. It is not given that all numbers are distinct. However, this consideration in not taken in solution. Can you please explain why it is incorrect to assume that all can be equal.

-->
That is a good question! Lets see what is implied by "If set S consists of the positive integers w, x, y, and z"..
If all the number are the same, lets say 1, We have the set as S = {1,1,1,1} But then, we don't repeat numbers in a set! So the set actually is S ={1}. But this set has only ONE item...whereas the question says that w,x,y,z are the four items the set has. This means that all these numbers CANNOT be the same...

If it wasn't a set, you thinking of the possibility of all numbers being equal is a good way to tackle the question

According to a survey of graduating medical students conducted by the Association of American Medical Colleges, minority graduates are nearly four times more likely than are other graduates in planning to practice in socioeconomically deprived areas.
(A) minority graduates are nearly four times more likely than are other graduates in planning to practice
(B) minority graduates are nearly four times more likely than other graduates who plan on practicing
(C) minority graduates are nearly four times as likely as other graduates to plan on practicing
(D) it is nearly four times more likely that minority graduates rather than other graduates will plan to practice
(E) it is nearly four times as likely for minority graduates than other graduates to plan to practice

--> Doubt: I marked A as the answer although C was tempting too but 'planned on' is idiomatically incorrect. But the correct answer is C. Please resolve the doubt.

--> Explanation:

A) Minority graduates are four times more likely than are other graduates.
'are' is not required. Without the 'are' the meaning is clear. 'are' or such helping verb will required in case the meaning is ambiguous.
E.g.
I like Math more than you.

We have two meanings here:
a) I like Math more than you DO.
OR
b) I like Math more than I like you.

In this case, the 'do' is required to avoid ambiguity. At the same time, note that "I like Math more than History" has no ambiguity, so it is correct as is.

In our question, there is no ambiguity, so 'are' is not required.

B) 'who plan on' changes the meaning...
C) 'Planning on' is accepted in American usage. No errors, so the best answer.
D) 'it' has no clear precedent. 'rather than' is awkwardly placed.
E) 'it' has no clear precedent.

Q: 60% Ms, 40% Fs are married, what is % married population.

I get different answers each time I take different numbers.

For example if I take

1. 100 Ms and 100 Fs . The total married people come to 100 which is 50% of total population 200

2. 600 Ms and 400 Fs. The total married people become 520 which is 52% of total population 1000
--> 520? There are only 400 Fs...

3. 2000 Ms and 3000 Fs, The total married people come to 2400 which is 48% of total population 5000
--> 2400 exceeds the number of males again...

Can you please explain how to derive the solution.

--> The logic is like this.
Lets think in percentages. m is percentage of males and f is percentage of females.
m + f = 100....(I)
60% Ms and 40% Fs are married. i.e. married to each other.
0.6 m = 0.4 f
i.e. 3m = 2f ...(II)

From I,
m + f = 100
i.e.
3m+ 3f = 300
Substituting from II
2f + 3f = 300
So, f = 60 % of population
So, m = 40 % of population

So how many males are married? 60%, as given = 0.6* 40 % of population = 24% of total population.
So how many females are married? 40%, as given = 0.4* 60 % of population = 24% of total population.
So, from the total population, 24%, who are males, are married to 24%, who are females. Note that this number will always come out to be the same.
So, in all, 48% of the population is married.
E.g. Out of 100 people in the town, 24 are married men and 24 are married women, making it 48 married people.

Doubt with this question, the answer given is B, how can it be possible ?
-->

A) 5H72 is divisible by 4. Does not help. Since 100 is divisible by 4, it doesn't matter what we have after the ten's digit.
72 will always be divisible by 4 and hence H can have any value.

B) 5H72 is divisible by 9. We know that 72 is divisible by 9, so 5H should also be divisible by 9. There is only one value of H, H=4, that makes 5H divisible by 9.
Sufficient.

Answer is B.

I have a confusion. Which one of the idioms are correct - so as to or so that?

---> Both are correct in their own right, depending on the sentence structure.
I studied well so as to graduate with first class.
I studied well so that I could graduate with first class.

Note the difference in sentence structures. Both mean the same, but the first one is just a phrase, while the other one is a clause, with a subject and verb of its own. Apart from this, there is no difference,

Three hundred guests is as many as even the most ambitious host or hostess should invite, and this number of guests is advisable only when circumstances actually demand it.

A) as many as even

B) so many as even

C) so many that even

D) as many that even

E) even so many that

Can you please help me to understand the correct usage of "so many that" in SC.
-->
Basically, ' so many that' would be only in the following context.

I was so much hungry that I could eat a horse.
I am showing that I was very hungry. So much that <I give an example for a comparison.>
Note that this structure is basically made up of two parts
I was hungry - Fact
so much that ... - a baseline for comparison.

This is the only structure in which we can use 'so many/much that...'

In the above Q Option C, this structure does not hold. Option B and D are plain wrong because there is no such thing as 'so many as' or 'as many that'. In option D, 'even' modifies 'host' but is incorrectly placed. The best answer is Option A.

Reiko drove from point A to point B at a constant speed, and then returned to A along the same route at a different constant speed. Did Reiko travel from A to B at a speed greater than 40 miles per hour?
(1) Reiko's average speed for the entire round trip, excluding the time spent at point B, was 80 miles per hour.
(2) It took Reiko 20 more minutes to drive from A to B than to make the return trip.

-->
Lets say the distance between A and B is d. While going from A to B, the speed was s1 and while returning, the speed was s2.
The question is whether s1 is greater than 40 m/hr.

(1) Total average speed = 80 m/hr.
i.e. 80 = Total distance/total time = 2d/(time from A to B + time from B to A)

Time from A to B = d/s1
Time from B to A = d/s2

So, 80 = 2d/(d/s1 + d/s2) = 2*s1*s2/(s1+s2).
i.e. 40 = s1*s2/(s1+s2)

Now, note that s2/(s1+s2) will always be less than 1. So,
40 = s1 * (something less than 1)
That means that s1 will have to be more than 40. The same goes with s2. That means, both s1 and s2 will have to be more than 40.

SUFFICIENT.

A simpler, more logical approach:
Lets say the distance is 80 miles. If we go from A to B with 40m/hr, it takes us 2 hours.
Now, if the average speed is supposed to be 80 m/hr, to cover 2*80 miles, it means that the total time would have to be 2 hours. But, if we take take the speed of 40 m/hr while going, we have already used 2 hours! So we have ZERO time to return. Even if we come back with speed of light, we will still consume SOME time and the average will never be 80 m/hr. So that means, while going, WE HAVE TO GO FASTER THAN 40m/hr.


(s) Time taken from A to B = 20 mins + Time taken from B to A
i.e. d/s1 = (1/3) + d/s2

Again, we cannot determine whether s1 is greater than 40 because we don't know the value of d.

INSUFFICIENT.


Option A is the correct choice.

I have doubt regarding the Q41 , Q51 , Q75 & Q78 ( Sentence Correction ) from the Stage - 3 ( new book ) .

--->
Q41

Japan received huge sums of capital from the United States after the Second World War, using it to help
build a modern industrial system.
(A) Japan received huge sums of capital from the United States after the Second World War, using it to help build
--> 'it' and 'sums' don't go together. 'using' would mean that there is some cause-effect relationship. This is not correct.
(B) Japan received huge sums of capital from the United States after the Second World War and used it to help in
building
--> 'it' and 'sums' don't go together.
(C) Japan used the huge sums of capital it received from the United States after the Second World War to help
build
--> Nothing wrong here.
(D) Japan’s huge sums of capital received from the United States after the Second World War were used to help it
in building
--> 'Japan’s huge sums of capital' is illogical. 'used to help it in building' is wrong.
(E) Receiving huge sums of capital from the United States after the Second World War, Japan used it to help build
--> 'it' and 'sums' don't go together.


Q51
Margaret Courtney-Clarke has traveled to remote dwellings in the Transvaal to photograph the art of
Ndebele women, whose murals are brilliantly colored, their geometrical symmetries embellished with old
and new iconography and in a style that varies from woman to woman and house to house.

(A) whose murals are brilliantly colored, their geometrical symmetries embellished with old and new iconography
and in a style that varies from woman to woman and house to house
--> Slightly complicated to read, but nothing wrong here. After 'colored', it talks about murals.
(B) whose murals are brilliantly colored, their geometrical symmetries are embellished with old and new
iconography, and their style is varying among women and houses
--> 'are embellished' and 'is varying' are incorrect.
(C) whose murals are brilliantly colored, their geometrical symmetries are embellished with old and new
iconography, and they are in styles that vary from woman to woman and house to house
--> 'are embellished' is incorrect. 'they' has ambiguous reference to women or mural.
(D) with murals brilliantly colored, their geometrical symmetries embellished with old and new iconography, and
their style varies among women and houses
--> 'women, with murals colored' is awkward. 'style varies among women and houses' is not be best expression.
(E) with murals that are brilliantly colored, their geometrical symmetries embellished with old and new
iconography, and their styles vary among women and houses
--> 'women, with murals colored' is awkward. ' 'vary' is wrong tense.

Q75
Published during the late eighteenth century, Diderot’s factual Encyclopedia and his friend Voltaire’s fictional
Candide were the cause of such a sensational scandal, and both men prudently chose to embark on
extended vacations in nearby Austria.

(A) Diderot’s factual Encyclopedia and his friend Voltaire’s fictional Candide were the cause of such a sensational
scandal, and
--> 'such a sensation' has not closing. 'his' referes to Diderot but there is no Diderot in the sentence. All we have is 'Diderot 's Encyclopedia'.
(B) Diderot and his friend Voltaire’s caused such a sensational scandal with their factual Encyclopedia and fictional
Candide, respectively, that
--> Incorrect modifier for Diderot. It must a book that should come after comma. Very awkward sentence too.
(C) Diderot’s factual Encyclopedia and his friend Voltaire’s fictional Candide were the cause of a scandal so
sensational that
--> 'his' referes to Diderot but there is no Diderot in the sentence. All we have is 'Diderot 's Encyclopedia'.
(D) the scandal caused by Diderot’s factual Encyclopedia and his friend Voltaire’s fictional Candide was so
sensational
--> Incorrect modifier. It must a book that should come after comma. 'his' referes to Diderot but there is no Diderot in the sentence. All we have is 'Diderot 's Encyclopedia'.
(E) a factual Encyclopedia by Diderot and the fictional Candide, by his friend Voltaire, caused a sensational scandal,
--> Best choice,


Q78

Rembrandt so treasured his collection of Islamic portraits that when forced to sell them in order to raise
money, he first made copies of more than twenty.
(A) he first made copies of more than twenty
--> Best choice. 'he first' is correct because that is meaning of the sentence. When he was forced to X, he first did Y...
(B) first he made copies of more than twenty
--> 'first' is misplaced.
(C) more than twenty were copied
--> Passive voice. The clause should start with 'he'.
(D) copies of more than twenty were made
--> Passive voice. The clause should start with 'he'.
(E) he copies more than twenty of them first

--> Wrong tense. 'first' is misplaced.

A few CR questions...

STAGE 1 CR:
Chapter 2
Q7
In recent years many cabinetmakers have been winning acclaim as artists. But since furniture must be useful, cabinetmakers must exercise their craft with an eye to the practical utility of their product. For this reason, cabinetmaking is not art.
Which of the following is an assumption that supports drawing the conclusion above from the reason given for that conclusion?
-->
Furniture must be useful --> Furniture makers are not artists.
Missing link: How is practical use related to art?
An assumption here is that there is no intersection of art and practical utility. i.e. something practical cannot be artistic and vice versa.
A. Some furniture is made to be placed in museums, where it will not be used by anyone.
--> Where some furniture is used is out of scope.
B. Some cabinetmakers are more concerned than others with the practical utility of the products they produce.
--> Degree of concern does not help us in any way.
C. Cabinetmakers should be more concerned with the practical utility of their products than they currently are.
--> ‘should be’ is an expectation. It does not talk anything about the art and utility intersection.
D. An object is not an art object if its maker pays attention to the objects practical utility.
--> Bingo. Says exactly what we need. Now negate this and you will see that the argument does not hold. That means this is an assumption.
E. Artists are not concerned with the monetary value of their products.
--> Out of scope.



Q9
Correctly measuring the productivity of service workers is complex. Consider, for example, postal workers: they are often said to be more productive if more letters are delivered per postal worker. But is this really true? What if more letters are lost or delayed per worker at the same time that more are delivered?
The objection implied above to the productivity measure described is based on doubts about the truth of which of the following statements?
-->
The argument is like this.
Postal workers are said to have more productivity if they deliver more letters à But what about delay and loss of letters? à This productivity measurement is not correct.
Clearly, the objection says that ‘delivery of letters is not the only thing that should be considered in calculating productivity’.
We need an answer choice that brings up this missing link.
(A) Postal workers are representative of service workers in general.
--> Does not matter. Does not talk about the missing link.
(B) The delivery of letters is the primary activity of the postal service.
--> What other activities postal service does is out of scope.
(C) Productivity should be ascribed to categories of workers, not to individuals.
--> We are not talking about categories vs individuals. Out of scope.
(D) The quality of services rendered can appropriately be ignored in computing productivity.
--> Yes, delay and loss are aspects related to quality of service. The quality aspect is completely ignored in the argument and that is the objection the author has. Correct choice.
(E) The number of letters delivered is relevant to measuring the productivity of postal workers.
--> The objection is not based on this. Does not fill the missing link.



Q10

“Fast cycle time” is a strategy of designing a manufacturing organization to eliminate bottlenecks and delays in production. Not only does it speed up production, but it also assures quality. The reason is that the bottlenecks and delays cannot be eliminated unless all work is done right the first time.
The claim about quality made above rests on a questionable presupposition that
-->
FCT is a strategy to eliminate bottleneck and delays à speed and quality à It works because bottlenecks and delays cannot be eliminated unless all work is done right the first time.
i.e. when everything is done right in the first time, there is no bottleneck and delay.
What is the relation between ‘doing things right the first time’ and ‘FCT’. The argument implies that if things are on FCT, then everything is done right, and then there are no bottlenecks or delays.
Lets see which option talks about this.
(A) any flaw in work on a product would cause a bottleneck or delay and so would be prevented from occurring on a “fast cycle” production line
--> Right. This says that if there is any flaw, it would mean that FCT is not possible. This way, ‘doing things right the first time’ and FCT are connected.
(B) the strategy of “fast cycle time” would require fundamental rethinking of product design
--> Implementation of FCT is out of scope.
(C) the primary goal of the organization is to produce a product of unexcelled quality, rather than to generate profits for stockholders
--> Out of scope.
(D) “fast cycle time” could be achieved by shaving time off each of the component processes in production cycle
--> How FCT can be achieved’ is not relevant.
(E) “fast cycle time” is a concept in business strategy that has not yet been put into practice in a factory
--> Out of scope.


Chapter 5
Q1
Defense Department analysts worry that the ability of the United States to wage a prolonged war would be seriously endangered if the machine-tool manufacturing base shrinks further. Before the Defense Department publicly connected this security issue with the import quota issue, however, the machine-tool industry raised the national security issue in its petition for import quotas.
Which of the following, if true, contributes most to an explanation of the machine-tool industry’s raising the issue above regarding national security?
-->
Defense department: We cannot make long term war if machine-tool industry is shrinking
Machine-tool industry said the same thing earlier, while petitioning for import quota.
The question is , why did the industry sound more concerned about national security?
If we can show that the machine-industry benefitted in any way from sounding this concern, we have our answer.
A. When the aircraft industries retooled, they provided a large amount of work for tool builders.
-->How does this matter? Out of scope.
B. The Defense Department is only marginally concerned with the effects of foreign competition on the machine-tool industry.
-->This may be true, but does not explain why machine-tool industry is concerned about national security.
C. The machine-tool industry encountered difficulty in obtaining governmental protection against imports on grounds other than defense.
--> Yes. The machine-tool industry is trying to get some protection against foreign competition from the government. They cannot get it without raising issue about security. This explains the industry’s concern.
D. A few weapons important for defense consist of parts that do not require extensive machining.
-->Out of scope.
E. Several federal government programs have been designed which will enable domestic machine-tool manufacturing firms to compete successfully with foreign toolmakers.
--> Does not provide explanation for the industry’s concern.

Some more CR Qs...

Chapter 6
Q3
The price the government pays for standard weapons purchased from military contractors is determined by a pricing method called “historical costing.” Historical costing allows contractors to protect their profits by adding a percentage increase, based on the current rate of inflation, to the previous year’s contractual price.

Which of the following statements, if true, is the best basis for a criticism of historical costing as an economically sound pricing method for military contracts?
-->
The argument explains how a pricing works. We need to criticize this practice. That is, we need to show that inflation based pricing is not economically sound.
The government might continue to pay for past inefficient use of funds.
--> Yes, if the previous price was incorrect, the government is continuing with the same incorrectness in future years too.
(B) The rate of inflation has varied considerably over the past twenty years.
-->That is not a flaw in the pricing.
(C) The contractual price will be greatly affected by the cost of materials used for the products.
-->This is what inflation-based pricing means. Inflation-based pricing will include change of cost of materials.
(D) Many taxpayers question the amount of money the government spends on military contracts.
--> Out of scope.
(E) The pricing method based on historical costing might not encourage the development of innovative weapons.
-->Out of scope. Does not talk about a flaw in pricing.


Chapter 8
Q10
A recent report determined that although only three percent of drivers on Maryland highways equipped their vehicles with radar detectors, thirty-three percent of all vehicles ticketed for exceeding the speed limit were equipped with them. Clearly, drivers who equip their vehicles with radar detectors are more likely to exceed the speed limit regularly than are drivers who do not.
The conclusion drawn above depends on which of the following assumptions?

-->Lets understand the scenario first.
Before the radars came in the market:
There are 100 drivers in Maryland, some were caught speeding.
After the radars came in the market, 3 of the drivers bought the radars.
Now, there are 100 drivers in Maryland, and these 3 drivers account for 33% speed violators. i.e. there are only 4.3% drivers caught speeding.
What is the catch here? Either the number of drivers caught speeding is low, and/or the concentration of radar-users is high in them. We need something that will explain this.

(A) Drivers who equip their vehicles with radar detectors are less likely to be ticketed for exceeding the speed limit than are drivers who do not.
-->This is exactly opposite of what the facts say.
(B) Drivers who are ticketed for exceeding the speed limit are more likely to exceed the speed limit regularly than are drivers who are not ticketed.
--> ’regularly’ is the key term here. So the radar-drivers are more likely to be caught repeatedly, and hence they constitute higher percentage. They number of radar-drivers is low, but they are repeatedly caught and hence account for 33%.
(C) The number of vehicles that were ticketed for exceeding the speed limit was greater than the number of vehicles that were equipped with radar detectors.
--> This relation does not help us in any way.
(D) Many of the vehicles that were ticketed for exceeding the speed limit were ticketed more than once in the time period covered by the report.
--> This relation does not help us in any way.
(E) Drivers on Maryland highways exceeded the speed limit more often than did drivers on other state highways not covered in the report.
--> Out of scope.

Two problems on Aliigation.
================================================== =
Q1: Freely available water was mixed with alcohol; if the cost of the alcohol is $100 per litre, the solution was sold for $81 per litre, and the profit margin was 50%, in what ratio were water and alcohol were mixed
-->
Profit margin was 50%, so the cost of the solution is 81-x/x = 1/2 , x= 54

As per Alligation ,

0 100
54
46 54
i.e.
23 : 27


Lets verify this.
So 23 litres of water and 27 litres of alcohol were mixed to make 50 litres of solution.

Total Cost = 23*0 + 27*100 = 2700
Cost per litre = 2700/50 = 54.
Selling Price with 50% margin = 54 * 1.5 = 81.
Fits well!

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Q2: Two ores, A and B, contained Iron and Copper in ratio 3:1 and 1:4. When the two were mixed, resultant ore had Iron and Copper in ratio 1:1. If the resultant ore had 110 kg Copper, what quantity of ore A was used.

--->
Quality of copper in the A, B and resultant ore

1/4 4/5

1/2

3/10 1/4

6 : 5 (A : B in resultant ore)

Resultant ore contains 110 Kg copper that means , total quantity of resultant ore is 220 kg
Hence, quantity of A is 220 x 6/11 = 120 kg

Lets verify this.
We mix A = 120 kg (IronA = 90 kg, CopperA = 30 kg) and B = 100 kg (IronB = 20 kg, CopperB = 80 kg)

Total Final Quantity = 120 + 100 = 220 kg
Total Final Iron = 90 + 20 = 110
Total Final Copper = 30 + 80 = 110
Iron:Copper Ratio in the Final Quantity = 1:1.
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