41. A bag contains 5 black marbles, 10 blue marbles, and 6 red marbles.
What are the odds of picking a black marble from this bag?
a. 5/11
b. 5/10
c. 5/16
d. 5/21
e. 5/15
42. 90% of students in a class have brown eyes. If 30% of those with brown eyes also have black hair, what percent of the total number of students have brown eyes and black hair?
a. 60%
b. 27%
c. 45%
d. 120%
e. 30%
43. George rides his bike and jogs for exercise. If he rides for 3 1/2 miles and jogs for 2 2/3 miles each day, how much is his total daily mileage for biking and jogging?
a. 5 miles
b. 5 1/2 miles
c. 6 1/6 miles
d. 6 2/3 miles
e. 6 5/6 miles
44. A class takes a bus trip that has a total of 540 miles round-trip. If the total trip is to be 6 days, how many miles do they need to average per day ?
a. 54
b. 9
c. 90
d. 45
e. 60
45. A rectangle as a length of 14 centimeters. If the perimeter of this rectangle is 44 centimeters, what is the width of the rectangle?
a. 8 centimeters
b. 14 centimeters
c. 22 centimeters
d. 28 centimeters
e. 30 centimeters
46. The lengths of the 2 smallest sides of a triangle are 6 inches and 12 inches. Which of the following is a possible length of the third side ?
a. 18 inches
b. 15 inches
c. 6 inches
d. 10 inches
e. 24 inches
47. A floor in a room has an area of 440 square feet. What percent of the floor area is covered by a rectangular rug with the dimensions of 11 feet x 5 feet?
a. 10%
b. 12.5 %
c. 25%
d. 11.5 %
e. 20%
48. A tank holds 23 3/4 gallons of gasoline. If 10 1/3 gallons are removed, how many gallons of gasoline are left in the tank?
a. 13 1/2
b. 12 1/3
c. 13 1/12
d. 13 5/12
e. 13 7/12
49. What is the perimeter of a pentagon if all sides have a length of 6.5 inches ?
a. 30.5 inches
b. 31.5 inches
c. 32.5 inches
d. 33.5 inches
e. 34.5 inches
50. The largest side of a triangle is 12 feet. Its smallest side is 7 feet. If the unknown side is three feet shorter than the longest side, what is the perimeter of this triangle?
a. 16 feet
b. 19 feet.
c. 5 feet
d. 26 feet
e. 28 feet
ANSWERS BELOW:
------------------------------
ANSWERS
------------------------------
Detailed Explanations of Answers, and Alternative Ways of Solving Selected
Problems.
1. We are simply looking for 2 numbers that when added together, we
get 115. Also, one number is 47 more than the other. Let x represent the
smaller number, then x + 47 would be the larger, so we have the equation:
x + x + 47 = 115
Combine the x with the other x gives:
2x + 47 = 115
Subtract 47 from both sides.
2x = 68
Divide each side by 2 gives:
x = 34
Thus, x = 34 which represents the smaller number. That is the one we
were looking for.
Correct answer is c.
We could have let x represent the larger number. Our equation would
have been x + x - 47 = 115. Solve it in a similar manner, and x would be
81. But we should realize we have not found the answer yet. We need to
check to see we are solving for the smaller number. Thus, 81 - 47 = 34
would get us the right answer.
Alternative way to solve #1:
We can eliminate answers b and e because they are too much. Work backwards
to find the correct answer. trying 68, we know the other number would have
to be 47 in order to add up to 115. But 68 is only 21 more than 47, not
47 like the problem states. Trying c, 34, gives the other number as 81
to yield a sum of 115. 34 is indeed 47 less than 81, so c MUST be the answer.
We can check answer d, 81, but be careful. 81 + 34 = 115, and 34 is 47
less than 81, but 81 is the larger number. We were looking for the smaller.
2. She needs a total of 32 x 4 or 128 circles. 6 can be cut from 1 paper, so divide 128 by 6 and you get 21 and a small remainder. Forget the remainder, you KNOW you will have to use 1 more paper for make up for that remainder. The answer will be 21 + 1 or 22. 21 may look attractive, but where would the little extra come from?
3. Multiply 3 x 30 = 90. It will take 90 minutes. But the answers are in hours, 60 minutes to an hour, so the answer must be 1 hour 30 minutes.
4. You should know that a square has all sides equal. Multiply 2 sides together to get the area. Since the sides will be the same number, we are looking for a number multiplied by itself to get 36. The only number that satisfies that is 6. ( 6 x 6 = 36). Each side is 6 inches, therefore the perimeter is 6 + 6 + 6 + 6 = 24 inches.
5. 4 out of 9 is the fraction 4/9. Multiply 540 x 4, then divide your
answer by 9. You get 240. This represents the number of students attending
college. (Always reread the question to make sure you are answering what
is asked for.)
The number not attending college would be 540 - 240 = 300. The answer
is c.
You could have also recognized that you need to find 5/9 of 540, solved similarly as above.
Alternative way to solve #5:
4/9 represents a little less than half. So, the number attending college
is a little less than half. Half of 540 is 270. The number attending college
will be slightly lower than this. Likewise, the number not attending college
is a little higher than 270. The only logical answer would be 300.
5/9 represents the number not attending college. 5/9 is about 55% Finding 55% of 540 would be easy and very close to 300.
*Remember: You do not need to get the exact answer. Just be close enough to logically choose the answer from the 5 choices.
6. Multiply .25 x 272 to get 68.
Alternative way of solving #6:
You should know that 25% means 1/4. So, I am looking for a number that
when multiplied by 4, I get 272. The only answer that works is 68 x 4 =
272. However, I knew it was going to end in an 8 because that is the only
number I can multiply by 4 to get a 2 on the end. (272 ends in 2 and 4
x 8 = 32). 68 is the only number on the list that ends in 8. 68 MUST be
the answer.
7. Multiply .36 x 650 to get 234.
Alternative way of solving #7:
10% of 650 is 65. So, 30% = 65 + 65 + 65 or 195.
5 % of 650 = 32.5, so 35% = 195 + 32.5 or 227.5. (I could logically
pick 234 because it is only slightly higher than 227.5)
1% of 650 = 6.5, so add 6.5 + 227.5 = 234.
8. If John rode his bike, the distance MUST be greater than 9 miles. Look for an answer that is more than 9 miles. The key words in the question are "could be true." All we need is one that fits, not necessarily the real distance he actually traveled. Answer d is the only one that fits him riding the bike.
9. The discount was 27 - 21.60 or 5.4.
10% os 2.7, 20% would be 5.4. That is the answer, 20% Remember, these
questions are supposed to be done without a calculator!
10. The total of the given items with prices is .85 + 1.75 + .45 or 3.05. Her real total was 3.80. So, subtract 3.80 - 3.05 to get .75, the cost of the chips. (Yes, I did eliminate the dollar sign in my calculations. It is meaningless to the answer.)
11. On questions like this, remember:
You will never multiply the quantities together.
You will never multiply the prices together.
The above cautions help you eliminate answers a, b, and d. We are left
with c or e. Answer e is obviously wrong. The only one left is c.
Alternative way to solve #11:
Do a quick calculation as to what the original equation equals. It
is $20. Look for an expression that gives $20 dollars. And use the 2 rules
as above. The only one close between c and e is c.
12. Add the scores and divide the total by 5. It is 430 ÷ 6 or 86. This is the only way one should do problems like this.
13. Change the fractions to a common denominator of 8. You get 6/8 + 7/8 or 13/8 which reduces to 1 and 5/8. Add the 1 to 4 and the 5 to get 10. The answer is e. 10 5/8.
Alternative way to solve #13:
We already have 4 + 5 or 9. 3/4 and 7/8 are both more than 1/2, so
I can take those halves, add them together to get another whole, so we
now have 10. That leaves 3/4 - 2/4 or 1/4, and 7/8 - 4/8 or 3/8. I know
the answer is 10 and 1/4 + 3/8 . I should recognize that 1/4 = 2/8, so
the answer will have 5/8 in it. The answer must be e. I could have also
logically concluded that 1/4 + 3/8 is not 7/8 and it is more than 1/2,
so the 10 5/8 must be the answer.
14. Let x be the number, then the equation is 5 + 3x = 23.
Subtract 5 from each side gives:
3x = 18
Dividing each side by 3 gives:
x = 6.
Alternative way to solve #14:
Work backward with the answers. Start with 6. 3 x 6 = 18, 5 more is
23. 6 must be the answer. I could eliminate the others in a similar fashion
if the first one did not work out. Once I have found an answer that works,
I can stop. It MUST be the answer. No need to waste time on the rest.
15. I like putting the smaller number first and get the range. I need a number larger than .0037 and smaller than .056. I compare all answers and need to find one that satisfies both rules. Answer a is larger than .056, so it is out. Answer b, c, and d are all smaller than .0037, so they are out. The only one left is e.
16. The 6 is in the hundreds place. Look at the number immediately to the right. If it is 5 or more, the 6 becomes a 7. If it is 4 or less, it remains a 6. Therefore, the only 2 possible answers are 105,600 or 105,700. All places to the right of where I am rounding off will be zeroes. To the right of 6 is a 7, so I round up. The answer is 105,700.
17. Multiply .06 x 355.50 or 21.33. Add 21.33 + 355.50 to get 376.83.
Alternative way of solving #17.
1% of 355.50 = 3.5550. Add 6 of these together and add it to 355.55
to get 376.83.
18. 10,100 - 9,100 = 1000. Be careful about answer d. It looks attractive.
19.
2x + 4 = 22
Subtracting 4 from each side gives:
2x = 18.
Divide both sides by 2 gives:
x = 9
Alternative way of solving #19:
We could work backwards using all answers. Multiply each answer by
2, then add 4. If we get 22, then we have then right answer. The only one
that works is 9.
20. We should realize that even numbers are 2 more than previous even
number. Consecutive even numbers are 4 and 6, 12 and 14, etc. Let x be
the smallest, then x + 2 is the next one. This gives the equation:
x + x + 2 = 26
Combine the x and x gives
2x + 2 = 26
Subtract 2 from each side gives:
2x = 24
Dividing each side by 2 gives 12.
We need to ask ourselves, is this the smaller or larger? If is the
smaller, then 12 + 14 should equal 26. It does, so 12 is the answer. Notice
how 14 is an answer choice.
Alternative way of solving #20.
Work backwards. Take the next even number and add it to the answer
choice. Answer a and e can be eliminated because they are odd numbers.
The only one that works is 12. Be careful about the 14. Many times the
test will include the other numbers you get and be very attractive.
21. Add each category separately.
Chocolate Chip = 5 x 1.25 = $6.25
Plain = 4 x .75 = $3.00
With Sprinkles = 6 x 1.05 = $6.30
Total = 6.25 + 3.00 + 6.30 = $15.55
I could have also recognized the total will end in a 5. Answer a is too small. Answer d would be the only one that fits. It MUST be the right answer.
22. To find 7/9 divide 7 by 9 to get a decimal of about .77. The range I am looking for is between .45 and .77. Answer a is too low, answer c is too high. Before we look at the fractions, we can look at the other answers. Answer e, .68 is perfect. It MUST be the answer. Answer b should logically be seen as too small. 3/7 is close. But 3 divided by 7 is about .43, a little too small. You do need to know how to convert a fraction to a decimal and visa versa.
23. Just look for 2 numbers that multiply to 36. The only one that works is answer c, 4 x 9.
24. I need to look at 35 divided by 4, the number of students at one time, to see how many "sessions" I need. 35 divided by 4 is 8, plus a remainder of 3. Multiply 8 x 10 to get the total minutes of 32 students. This equals 80 minutes, or 1 hour 20 minutes. But we need to account for the 3 extra students. This will also take 10 minutes, no matter that we have less than 4 students. The correct answer is d, 1 hour 30 minutes.
Alternative way to solve #24:
Make a chart or table.
10 min = 4 students
20 min = 8 students
30 min = 12 students
....and so on. I could even now jump ahead to:
60 min = 24 students.
I continue until I have had all 35 students accounted for.
25. Don't confuse area with perimeter. Perimeter is the distance all
the way around. In a rectangle, opposites sides are equal. The length of
the rectangle in the problem is twice as long as the width. The width is
9, so the length is 18. The perimeter is all 4 sides added to each other:
9 + 18 + 9 + 18 = 54. Correct answer: d.
26. He has taken 4 tests so far. One more will make a total of 5. He needs a total of 450 points to have an average score of 90 because 450 ÷ 5 = 90. His scores so far total 85 + 89 + 84 + 94 = 352. He needs 450 - 352 or 98 points. The lowest score he can get is a 98. The score of 100 looks attractive because it will indeed push his average over 90. But, we are looking for the lowest score. The answer is c, 90.
27. You need to assume and note that the movies start 2 hours and 10 minutes after the previous one starts. It would be logical that the next one would start 2 hours and 10 minutes after 6:35 pm. The correct time would be 8:45 pm, answer a.
28. Area of a rectangle is length times width. Therefore, we are looking for 2 numbers when multiplied, you get 60. The only answer that fits is answer d, 5 x 12. We can verify the sides add up to 34: 5 + 12 + 5 + 12 = 34.
29. We need to notice the answers are all in feet. There are 12 inches in a foot, so we will need to convert inches to feet at the end. 5 in x 34 = 170 inches. Dividing 170 by 12 gives an answer of 14 and a remainder of 2. Correct answer is e, 14 feet 2 inches.
30. Bob cannot buy a partial box. Therefore, if he needs less than a multiple of 6, he will need to but another full box. 39 divided by 6 is 6 and one half. So, he needs to but 7 boxes. Correct answer is b.
31. We can just divide 42 by 210. I should recognize that 42 is around 1/5 or 20%. In fact, it is exactly .20 or 20%. The answer is d, 20. Remember, all problems are to be done without a calculator. Picking the answer may be just logic.
32. We need to realize that 21.56 represents 70% of the original price. Hence, 21.56 divided by 7 will equal 10%. It is easy to divide 21.56 by 7. It is 3.08. Therefore the original price is 21.56 +3.08 + 3.08 + 3.08, which is $30.08 Correct answer is a. I could have also estimated the addition to be about 30. I could have picked a without hesitation.
Alternative way to solve #32.
Algebraic equation would be: x -.3x = 21.56, where x = the original
price, and then solved for x.
Also, I could have realized that 30% is about 1/3, therefore 21.56
represents about 2/3 of the original price. 2/3 is equal to 1/3 + 1/3,
so cutting 21.56 in half would have given me about 30%, or 10.78. Adding
this to 21.56 would give a rough answer of 32.34. But I know the real answer
will be only slightly less, so I can pick a with confidence.
33. The volume of any rectangular solid is length X width X height. If the solid is a cube, all 3 edges are equal. Therefore, we need a number that when cubed equals 125. The only number that works is 5, because 5 x 5 x 5 = 125. The answer is d.
34. We need to know Joan's age NOW. The problem gives her sister's age as 25 now, so count back ten years and we get an age of 15 for her. But 10 years ago Joan was twice her age. Therefore, Joan was 15 x 2 or 30 years old ten years ago. Joan would have to be ten years older today, or 30 + 10 = 40. Correct answer is d.
35. We need to find and equation that multiplies the total by .12, then ADDS it to the original price. The only answers that add it to the original price are b and e. But b only multiples 250 by .12, not the new total. Therefore, e must be correct.
36. Substitute - 2 for x gives the equation:
y = 2(- 2) + 3 So,
y = - 4 + 3 and,
y = - 1
Correct answer is b.
37. We need to be able to read the sentence correctly. We are adding
6 to an number that gets multiplied by 2.
The correct algebraic equation is:
6 + 2x = 28. Subtract 6 from each side,
2x = 22. Divide both sides by x,
x = 11.
Correct answer is d.
Alternative way of solving #37.
Work backward. Double each answers, the add 6. If you get 28, you have
the right answer. The only one that works is 11.
38. We need to realize all products will total 100%. Adding the given figures, we get 37 + 32 + 25 = 95. We need 100 - 95, or 5% more to total 100. Correct answer is d.
39. We need to convert 2 hours and 3 minutes to minutes, because the rate is given in minutes. 2 hours and 3 minutes = 123 minutes. Multiply 123 by 9 to get 1107 gallons. Correct answer is a.
40. At 63 mph, 189 miles would take 189 divided by 63. 189 divided by 63 is 3. So, the trip will take 3 hours. Subtract 1:40 from 3:00, we get 1:20. Correct answer is c, 1 hour 20 minutes.
41. There are 21 total marbles, 5 that are black. The odds are 5 out of 21, or 5/21.
42. We are looking at .90 x .30, which is 27. Correct answer is b.
Alternative way to solve #42:
We are not given a total, but it does not matter. We only need a percentage
of the total. All numbers behave the same, so let's work with an easy total
of 100 students. That means that 90 of them have brown eyes. I now need
to calculate 30% of 90. It is roughly a third, so the closest answer would
be b. Doing it another alternative way, I know that 10% of 90 = 9, so 30%
must be 9 + 9 + 9, which is 27.
43. Since we are adding, we only need to put the fractional part over common denominators. That would be sixths. 1/2 = 2/6, and 2/3 = 4/6, so 3/6 + 4/6 = 7/6. That gives me 1 whole more, plus 1/6. So the total is 6 1/6. But I also could have noticed that the only answer that has 1/6 is c. It must be the correct answer.
Alternative way of solving #43:
1/2 + 2/3 will logically be more than 1, so the answer will be slightly
more than 6. 2/3 is way too much for the fractional part, so c is the only
logical choice.
44. Simply take 540 divided by 6 to get 90 miles. Answer is c.
45. I know opposite sides in a rectangle are equal. If one side is 14, the opposite side is also 14. That makes a total of 28 so far. But the perimeter is 44, so I subtract 44 - 28 to get 16. This 16 represents the sum of the other 2 sides. They are equal, so the only number that works is 8 because 8 +8 = 16. Therefore, the answer is a.
46. The sum of the two smallest sides MUST add up to more than the longest side. (I'll let you figure out why as a good number theory exercise.) Therefore, since 6 + 12 is 18, the longest side must be shorter than 18. But it cannot be 12 or less because we know 12 represents a side shorter than the longest side. The number must be between 12 and 18. The only number that works is answer b, 15.
47. 11 x 5 equals 55, so 55/440 of the room is covered. This fraction can be reduced to 11/88, then to 1/8. 1/8 is exactly 12.5% (Learn your eighths!). Correct answer is b.
48. 3/4 is larger than 1/3, so we don't have to worry about borrowing. The common denominator between 3 and 4 is 12, so 3/4 = 9/12 and 1/3 = 4/12. 9/12 - 4/12 = 5/12 and 23 - 10 = 13, so the answer is d, 13 5/12.
Alternate way of solving #48:
Let's look at 3/4 - 1/3. If we know that 3/4 = .75, and 1/3 = .33,
then 3/4 - 1/3 = about .75 - .33 or .42. Therefore, 23 3/4 - 10 1/3 is
very close to 13 and .42. We know it has a 13, so answer b is out. We also
know it is for sure less than 1/2 or .5. That eliminates anything at or
above 1/2. This eliminates answers a, c and e. We are now left with choice
d. It MUST be the correct answer. Learn your fourths and thirds!
49. We should know that a pentagon has 5 sides. If each side is 6.5, we simply have to multiply 6.5 by 5. Don't let decimals scare you. If you never learned any rule, split it up! Take 6 x 5 to get 30. Then do .5 (or even think 1/2) times 5. It is 2.5. Therefore, the answer is 30 + 2.5 = 32.5, answer c.
50. The missing side is 3 feet shorter than the longest side, or 12
- 3. That makes it 9. The perimeter would be 12 + 9 + 7 = 28, answer e.
Test has been edited, but not all editing is perfect.
Please notify of any errors.
Copyright June 2002 Stephen Carr