Numbers and Operations Practice Quiz Answer and Explanation Sheet

1. C 2. B 3. A 4. B 5. C 6. D 7. D 8. B 9. E 10.B 11.E 12.A 13.D 14.C 15.A 16.E 17.A 18.B 19.C 20.D 21.B 22.C 23.D 24.D 25.D

1. Explanation for Correct Answer C : Choice (C) is correct. This problem can be solved by dividing the total number of square feet to be painted by the number of square feet that can be painted with 1 gallon: Since you are asked to find the least whole number of gallons that must be purchased, you would have to buy the next highest whole number of gallons, which is 11 gallons.

2. Explanation for Correct Answer B : Choice (B) is correct. Since x lies within all three circles, x is the number of people who purchased all three flavors. Since 15 people purchased strawberry ice cream, it follows that 6+2+3+x=15 Solving this equation yields x=4. From the diagram, people purchased chocolate ice cream. Since x=4 it follows that people purchased chocolate ice cream. 3. Explanation for Correct Answer A :

Choice (A) is correct. The ratios provide two equations: and . To find the ratio of to which can be written as you can multiply by . That yields

Therefore, the ratio of ac to be is 1:10

4. Explanation for Correct Answer B : Choice (B) is correct. You can rewrite this number as

since and Thus, is the fourth power of the integer

5. Explanation for Correct Answer C : I. A number is even if it is divisible by The product of an even integer with an even integer is divisible by so this product is also even. Thus, the set I has the given property. II. A prime number is a positive integer that does not have any factors other than 1 and itself. The product of two primes will have those two primes as factors. (For example, the product of 3 and 5 is 15 The number 15 is not prime, because it can be factored as 3 x 5) Thus, set II does not have the given property. II. The product of two positive numbers is positive. Thus, set III has the given property. ~Therefore, only sets I and III have the given property.

6. Explanation for Correct Answer D: The correct answer is D.) 67. If you do not want to compute all 100 numbers at the start of the sequence, you need to determine if there is a pattern of odd and even numbers. Starting with the first number in the sequence, write down whether each number is odd (O) or even (E); this gives the sequence E, O, E, E, O, E, . . . This pattern of E, O, E will repeat, because an odd number plus an even number plus 1is even, and the sum of two even numbers plus 1 is odd. You can see that for every 3 numbers in this sequence, 2 are even. Therefore, 66 of the first 99 numbers in the sequence are even. The 100th number will also be even, because the E, O, E, pattern starts with E. Therefore, there are 67 even numbers among the first 100 numbers in this sequence. 7. Explanation for Correct Answer D : Choice (D) is correct. One way to make this round trip is to travel from Quarryton to Richfield to Bayview to Quarryton. There are 3 roads to Richfield, then 2 roads to Bayview, and then 1 road to Quarryton; this gives a total of possible ways of making the round trip going through Richfield first. The other way to make the round trip is to travel from Quarryton to Bayview to Richfield to Quarryton. There is road to Bayview, then 2 roads to Richfield, and then 3 roads to Quarryton; this gives a total of possible ways of making the round trip going through Bayview first. Therefore, there are possible routes for the round trip.

8. Explanation for Correct Answer B :

Choice (B) is correct. The highway portion of the trip will be miles.

Since the car uses 30 miles per gallon on the highway, it will use gallons of gas on the highway portion of the trip. The remainder of the trip, 300 – 240 =60 miles, will be in the city, so the car will use gallons in the city. Therefore, the car will use 8 + 3 =11 gallons of gas altogether.

9. Explanation for Correct Answer E : Choice (E) is correct. There are a total of flowers, with in each vase, and they are to be moved so that the ratio in the vases is Since this means the flowers are to be arranged so that of them are in the first vase, are in the second, and are in the third. Thus, there should be flowers in the first vase, in the second, and in the third. The easiest way to accomplish this is to move flowers from the third vase the first vase. Since at least flowers must be removed from the third vase, it is impossible to achieve the ratio without moving at least flowers.

10.Explanation for Correct Answer B : Choice (B) is correct. You are given that so multiplying the inequality by the positive number gives Furthermore, multiplying by again gives Combining these last two inequalities yields Since the square of any real number is positive, it follows that and so Therefore,

11.Explanation for Correct Answer E : Choice (E) is correct. In order for to be an integer, has to be an even integer. If is an even integer, then must be an odd integer. The odd positive integers that are less than are and . Thus, there are five positive integers less than for which is an integer.

12.Explanation for Correct Answer A : Choice (A) is correct. Jamal ran feet, which is equal to yards. Thus, the ratio of Jamal's distance to Lonnie's distance is yards to yards, which is the same as

13.Explanation for Correct Answer D : Choice (D) is correct. For a fraction to be positive, the numerator and denominator must have the same sign, either both positive or both negative. The arrow corresponding to points to the value of is and

14.Explanation for Correct Answer C :

Choice (C) is correct. Since Pat removed of the cartons and Lee removed together they removed of the cartons. Let represent the number of cartons originally in the truck. Since of the cartons were removed, remain.

Since there are cartons left, it follows that Solving this equation for gives Thus, there were originally cartons in the truck.

15.Explanation for Correct Answer A : Choice (A) is correct. To find out how many times the apples are placed in all six baskets, divide 74 apples by 6 baskets. This gives 12 apples per basket, with 2 apples yet to be placed. Since the apples are put into the baskets in order, the first of the 2 remaining apples will be placed in basket one. Thus, the second one, which is the 74th apple, will be placed in basket two.

16.Explanation for Correct Answer E : Choice (E) is correct. The problem states that the pole is 60 feet long, with 12 feet below ground, so 48 feet of the pole is above ground. Therefore, of the pole is above ground.

17.Explanation for Correct Answer A : Choice (A) is correct. Since st3u4 > 0 none of s, t, or u are equal to 0. Therefore, t2 > 0. and u4 > 0 so, t2u4 > 0. Dividing both sides of the inequality st3u4>0by the positive number t2u2 yields st > 0.

18.Explanation for Correct Answer B : Choice (B) is correct. If each student participated in only 1 activity, there would be students in the homeroom. However, 5 of the students participate in 2 activities. Therefore, 5 students are counted twice in the sum. Since these students must be counted only once, the number of students in the homeroom is

19.Explanation for Correct Answer C : Choice (C) is correct. The segment from to is marked with tick marks indicating

equal segments, and point is at the tick mark that is of the way from to

Therefore, is the coordinate of

20.Explanation for Correct Answer D : Choice (D) is correct. Set and set If is a member of and is a member of the greatest possible value of is the greatest member of minus the least member of or

21.Explanation for Correct Answer B : The correct answer is . Maria and Diane together received out of the total votes. This leaves a total of votes for the second- and third place finishers, Kevin and Carlos. Since Kevin finished second, he received more votes than Carlos. Therefore, the minimum number of votes Kevin could have received is (If Kevin received votes, then Carlos would have received votes, which would put him in third place, as stated.)

22. Explanation for Correct Answer C : The correct answer is . In the enrollment at the college was male students and female students, for some positive integer Current enrollment has more male students and an unchanged number of female students. Therefore, current total enrollment is male students and female students. Since the current ratio of males to females is to the number of males equals the number of females, or Solving for you find that Therefore, there are male students and female students, for a total of students.

23.Explanation for Correct Answer D : The correct answer is . For convenience, call the cabins and Name a straight path between two cabins by giving the cabins that it connects; for example, the paths from cabin to the other cabins are and There are a total of names for the straight paths between the cabins. However, each path has exactly two different names in this scheme; for example,

and are names for the same path. Therefore, there are just paths.

24.Explanation for Correct Answer D : Choice (D) is correct. is a divisor of both and since and

25.Explanation for Correct Answer D : Choice (D) is correct. Since the ramp rises vertically foot for every feet of horizontal distance, the ramp is at a height of feet at a horizontal distance of feet from its starting point, where Cross multiplying gives or