NUMBERS AND OPERATIONS : SAT MATH QUESTIONS
NUMBERS AND OPERATIONS – REMAINDERS / DIVISIBILITY
Question
What is the remainder when 163 + 173 + 183 + 193 is divided by 70?
A. 0
B. 69
C. 1
D. 34
E. 47
Correct answer is 0. Choice A.
Explanatory Answer
= (16 + 19)(162 - 16 * 19 + 192) + (17 + 18)(172 - 17*18 + 182)
= 35 (162 - 16 * 19 + 192 + 172 - 17*18 + 182)
= 35 (even number).
When 35 is multiplied by an even number, the product will be divisible by 70.
Therefore, the remainder is ’0′.
Question
If 3 students are made to sit in a row, then 9 students do not have a seat. Alternatively, if 9 students are made to sit in a row, then 5 rows are empty. How many students have to be seated in a row so that an equal number of students sit in all the rows and all the students are seated?
A. 5
B. 4
C. 8
D. 7
E. 6
Correct answer is 4. Choice B.
Explanatory Answer
Arrangement I
When 3 students are made to sit in a row, 3n students will be seated.
In this arrangement 9 students do not have a seat.
Therefore, there are (3n + 9) students in the class
Arrangement II
When 9 students are made to sit in a row, then 5 rows are empty.
i.e., the students have been seated in n – 5 rows.
Therefore, there are 9*(n – 5) students in the class.
As the number of students in the class is a constant, irrespective of the arrangement, we get 3n + 9 = 9*(n – 5)
i.e., 3n + 9 = 9n – 45
or 6n = 54 or n = 9
There are 9 rows in the class.
As there are 9*(n – 5) students, we get 9*(9 – 5) = 36 students.
If 36 students have to be seated in 9 rows, such that equal number of students sit in each of the rows, then we have to seat 36/9 = 4 students to a row.
Question
Which of the following is the smallest positive integer that when divided by 4, 6 and 8 leaves a remainder of ’2′ in each case, but is perfectly divisible by 37?
A. 146
B. 666
C. 296
D. 74
E. 242
Correct answer is 74. Choice D.
Explanatory Answer
Let the least positive integer be “x”.
“x” leaves a remainder of 2 when divided by 4, 6 and 8.
Therefore, x – 2 will be divisible by all the three numbers viz., 4, 6 and 8.
The least value that x – 2 can take will be the LCM of 4, 6 and 8, which is 24.
The other values that x – 2 can take will be multiples of 24 as all the multiples of 24 will be multiples common to 4, 6 and 8.
Now, if x – 2 = 24, then x = 26. It is not a multiple of 37.
If x – 2 = 48, then x = 50. It is also not a multiple of 37.
If x – 2 = 72, then x = 74. It is a multiple of 37.
Therefore, the correct answer is 74, choice D.
How to approach this question in the test?
The best approach is to go from answer choices.
You know that the given number should be a multiple of 37 as it is divisible by 37.
- Look at choice A. 146 is not a multiple of 37. Eliminate it. So, is the case for 242. So, you have eliminated choices A and E.
- Now look at choice B. It is evident that 666 is divisible by 6. Therefore, it will not leave a remainder of 2. So, eliminate choice B.
- Choice C – the value suggested for x is 296. This number is divisible by 4 and will therefore, not leave a remainder of “2″ when divided by 4. So eliminate it.
PERMUTATION COMBINATION – REPLACEMENT AND ORDERING
Question
In how many ways can 5 letters be posted in 3 post boxes, if any number of letters can be posted in all of the three post boxes?
A. 5C3
B. 5P3
C. 53
D. 35
E. 15
Correct answer is 35. Choice D.
Explanatory Answer
The first letter can be posted in any of the 3 post boxes. Therefore, it has 3 choices.
The question explicitly states that any number of letters can be posted in all of the three boxes.
Hence, the second, the third, the fourth and the fifth letter can each be posted in any of the 3 post boxes.
Therefore, the total number of ways the 5 letters can be posted in 3 boxes is 3*3*3*3*3 = 35
Choice D is the correct choice.
CRITICAL READING – PASSAGE : SAMPLE QUESTION 1
PASSAGE
That well-imagined nightmare in which a bloodthirsty Tyrannosaurus rex is chasing the family car down a lonely road in the red-rock desert as the children scream and the gas gauge hovers on empty and the dinosaur gnashes at the rear bumper is just that: a bad dream. T. rex was a slowpoke. The most feared and revered of the dinosaurs did not have the leg strength to run very fast, if at all, according to a computer model developed by two experts in the mechanical movements of living creatures.
7. The model suggests the Cretaceous landscape was filled with large, lumbering creatures that any human with a fast car or bike or maybe even a quick sprint could outpace. The research brings the discipline of biomechanics to the long and at times contentious debate over just how fast the largest of the largest creatures ever to roam Earth could run. “Large animals need a larger fraction of their body mass as leg muscles in order to do the same things that smaller animals can do, but there is a limit to how large that fraction can be,” said John Hutchinson, co-author of the paper and a postdoctoral research fellow in the Biomechanical Engineering Division at Stanford University in Palo Alto, California.
15. At 13,228 pounds (6,000 kilograms), T. rex was over that limit, he said. So, too, were some of T. rex’s potential prey, such as Edmontosaurus (large duckbill) and Triceratops (horned dinosaur). Consequently, Hutchinson and his colleague Mariano Garcia, a mechanical engineer in Ithaca, New York, concluded that the large dinosaurs must have lumbered around at a much slower pace than suggested by some paleontologists and depicted in popular movies. A classic scene in the movie Jurassic Park, for example, shows T. rex chasing a car that’s traveling about 45 miles (72 kilometers) an hour. According to Hutchinson and Garcia’s model, that’s impossible. Eighty-six percent of T. rex’s body mass would have to be leg muscle for the behemoth dinosaur to run that fast. No creatures can have most of their body weight in their legs. It doesn’t leave enough room for a skeleton, muscles, and other body parts.
25. “An animal cannot be made 100 percent out of leg muscle,” said Hutchinson. “In fact, muscle of any kind normally is about one half of an animal’s mass, and supportive leg muscle is usually only 5 to 20 percent of an animal’s mass.”
Question 1
The passage is concerned with the theme that
(a) Movies are correct in portraying that large dinosaurs are very fast
(b) The general perception of large animals or dinosaurs being fast is incorrect
(c) T Rex was the fastest dinosaur that roamed the earth
(d) Jurrassic park was made by a paleontologist
(e) A large dinosaur had more than 50% of its body mass as leg muscle
Question 2
In line 9 the words “Contentious” means
(a) Disputative
(b) Callous
(c) Conclusive
(d) Incorrect
(e) Misleading
Question 3
On the subject of leg muscle the passage says that
(a) An animal can be made of 100% leg muscle
(b) Eighty Six percent of a T Rex’s body mass is leg muscle
(c) 5 to 20% of an animal’s mass is usually the leg muscle
(d) The larger the animal the smaller the leg muscle component of body mass
(e) The smaller the animal the larger the leg muscle component of body mass
Correct Answer
(1) e
(2) c
(3) c
Explanation to the answers
1. The correct answer is (b). The passage talks about the speed of big animals and this choice brings out the idea correctly.
The passage starts off by saying that movies are wrong about large dinosaurs being very fast animals and this stand rules out option (a) and ( c). Choice (d) is also incorrect because the passage does not talk anything about the makers of Jurrassic Park. The percentage of body mass is not correct in choice (e).
2. The correct choice is (a). “Disputative” comes closest to the meaning of “Contentious” since both indicate a clash or lack of agreement on a topic.
Here there is a debate or argument over the speed of large creatures and the passage does not arrive at any decision. Therefore choice ( c ) will not be correct since there is no conclusion in the passage. “Incorrect” and “Misleading” will not be the correct adjectives since both denote confusion and there is no confusion on the topic of debate. “Callous” means insensitive and has no relation to the given word.
3. The correct choice is ( c). The last line of the passage clearly supports this choice.
Line 25 of the passage rules out option (a). Lines 21, 22, 23 and 24 explain that eighty six percent of the T Rex’s body will have to be leg muscle for the animal to run at 45 miles an hour. This is merely a calculation and not the actual case. Choices (d) and (e) can be ruled out since they are baseless and not supported by the passage.
SENTENCE COMPLETION PRACTICE QUESTIONS
It is important that discussion of the issues relating to avian influenza should ____________ between the real problems caused by the spread of the disease within the bird population and the theoretical risk of a human __________ which might not occur.
(1) meander……epidemic
(2) integrate……epiphytotic
(3) differentiate……pandemic
(4) identify……endemic
(5) confuse……infection
Answer
The correct answer is choice (3)
The sentence talks about two events. One is the spread of avian influenza in the bird population and the other event is concerned with the humans. It is further indicated that the event connected to human beings is not a likely event. Therefore there is a clear distinction between the two events since one is likely to occur and the other is not. None of the words except for “differentiate” brings out this distinction. “Pandemic” means an epidemic spread over a wide geographical area and this is apt for the second blank because anything connected to humans is likely to spread all over the world. The first choice “Epidemic” will also be appropriate for the second blank but since it is clubbed with the word “Meander” which means to wander it will not be the correct choice.
Epiphytotic – epidemic among plants of a single kind especially over a wide area
Question 2
In Robin Hood’s time, few people could read or write and ____________ little was written down about the exploits of our hero.
(1) subsequently
(2) consequently
(3) accordingly
(4) surprisingly
(5) ludicrously
Answer
The correct answer is (2)
It is clear that the first part of the sentence has some effect on the second part. This is because the first part talks about people being unable to read and write and second part laments that there is very little written material available about the exploits of the hero, Robin Hood. “Consequently” is the word that brings out this cause effect relationship correctly.
Ludicrously – arousing laughter
Volatility has collapsed because policymakers have done an outstanding job __________ the global economy towards ________ growth.
(1) resurrecting…shaky
(2) refining…downward
(3) redirecting…stable
(4) reckoning…decisive
(5) reverting…unsettled
Answer
The correct answer is (3).
The first part of the sentence commends the policymakers for doing a good job with the use of the word “outstanding”. The reason for this good result is due to the action taken by them, which is explained in the second part of the sentence. The growth of the Global economy should have been positive for volatility to collapse. “Shaky”, “Downward”, “Unsettled” are words, which cannot be used for the second blank. The use of these words implies that there is negative growth which could have only contributed to more confusion. Therefore Option (a), (b) and (e) can be eliminated. This leaves us with option ( c ) and (d). In option (d) “Reckoning” implies that there is some calculation or counting involved. Therefore we can also eliminate option (d). Option ( c ) is the remaining choice which when read with the sentence gives the right feel to it.
Resurrecting – bring alive
Sure, it’s not always that simple..