Unit 1 Test Rationales
1. Answer A is incorrect because a number does not have to be even to be divisible by 3. Ex: 24 ÷ 3 = 8
Answer B is incorrect because a number does not have to be odd to be divisible by 3. Ex: 30 ÷ 3 = 10
Answer C is correct because to be divisible by a number it must be a multiple of that number. 13 is not a multiple of 3.
Answer D is incorrect because it should have said 3 is not a factor of 13 to be correct.
2. Answer A is correct because a composite number must have more than two factors.
Answer B is incorrect because a number divisible by only 1 and itself is prime.
Answer C is incorrect because a number can be composite and not divisible by 2. Ex: 9 is composite.
Answer D is incorrect because a number can be odd and composite. Ex: 9 is odd and composite.
3. Answer A is incorrect because the rule is incomplete.
Answer B is incorrect because a number with a ones digit of 1 is not divisible by 5. Ex: 11 is not divisible by 5.
Answer C is correct because a number is divisible by 5 if the ones digit is a 0 or a 5. Ex: 40 or 45 are divisible by 5.
Answer D is incorrect because a number will not be divisible by 5 with a ones digit of 1.
4. Answer A is incorrect because it is a correct divisibility rule for 6. A number divisible by 2 and 3 will be divisible by 6.
Answer B is correct because it is not a correct divisibility rule. 45 is divisible by 5 but not by 10.
Answer C is incorrect because it provides correct divisibility rule information for 2. If a number ends in 6, it is even and as a result
divisible by 2.
Answer D is incorrect because it is the correct divisibility rule for 9. The sum of 333 is 9 which is a multiple of 9 and as a result 333
can be divided by 9.
5. Factors of a number are numbers that can divide evenly into the number with no remainder. The factors of 48 are 1, 2, 3, 4, 7, 16,
24, and 48, so the one that is NOT a factor is C (5).
6. Prime factorizations must contain all prime numbers. 2x3x5x7 and 2x7x15 both equal 210, but 15 is not prime, so the answer is D
(2x3x5x7).
3
7. In order for a fraction to be equivalent to , you would have to multiply the numerator and denominator by the same number
(Property of One). A.
3
4
x
2
2
=
6
8
B.
3
4
x
4
9
9
=
27
36
C.
3
4
x
𝟑
𝟒
=
9
16
D.
3
4
x
12
12
=
36
48
If you look at choice C, you will notice that the
numerator and denominator are multiplied by different numbers, so it cannot be equivalent to
3
4
.
8. The greatest common factor of two numbers is the largest number that is a factor of both of the numbers. The factors of 12 are 1,
2, 3, 4, 6 and 12. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. They have the factors 1, 2, 3, 4, 6, and 12 in common. The
largest factor they have in common (GCF) is 12 (choice C).
9. There are two ways to solve this problem.
You can create a chart and list the multiples of 15 and 18. Once the chart is completed, the least common multiple will be the smallest
multiple found in both lists.
x1
x2
x3
x4
x5
x6
15
30
45
60
75
90
18
32
54
72
90
108
The other way you can solve this problem is by using use prime factorization. You will find the greatest common factor. You then
multiply the greatest common factor by the remaining factors.
15- 3 x 5
18- 3 x 3 x 2
GCF =3
Multiply 3 x 5 x 2 x 3
10. When comparing and ordering fractions you need to use a common denominator. The common denominator for this problem is
16. Change all the fractions using the property of one, multiplying the bottom by numerator by the same number you multiply the
1 4 8 2
denominator by. The fractions with a common denominator are ,
.
The correct answer is D.
16 16 16 16
11. These fractions all have a common numerator. We learned that if the numerators were the same that the fraction with the largest
integer as the denominator will be the smallest fraction. So the ordered set with the denominators in order from largest to smallest is
1/9, 1/8, 1/6.
12. For this problem with varying numerators and denominators, I must find the least common denominator (8 for A and B or 140 for
C and D) and then convert the numbers to equivalent fractions. Letter A converted is 1/8, 2/8, 3/8, 4/8, 6/8. These are in order.
13. Correct Response D
The given number line is divided into five equal sections between 4 and 5. Since the improper fraction
correct answer.
Incorrect Response A
4
Point D represents 4 .
21
5
5
=4
1
5
NOT 4
21
5
=4
5
1
equals 4 , D is the
5
4
5
Incorrect Response B
Point A represents a number less than 3.
Incorrect Response C
Point B represents 4.
21
21
5
=4
1
5
4
5
14. Correct Response D
The order of the decimals should be least to greatest. To compare the decimals, first line up the decimals and then compare
each place value. All of the numbers start with one. When you compare the tenths digits, the smallest number will be 1.05
followed by 1.1. Because 1.32 and 1.33 both have a three in the tenths place you need to compare the digit in the
hundredths place. Two is smaller than three so 1.32 will be smaller than 1.33. The final order will be 1.05, 1.11, 1.32, 1.33.
Incorrect Response A
The question asks for the decimals to be placed in order from least to greatest. The first number listed in choice A is 1.32.
That is the largest number in value so it is incorrect.
Incorrect Response B
This choice lists 1.1 as being smaller than 1.05. Zero tenths is smaller than 1 tenth so 1.05 should listed before 1.1.
Therefore, response B is not correct.
Incorrect Response C
The first two numbers listed are correct, so you need to compare the next two numbers: 1.33 and 1.32. Two-hundredths is
smaller than 3 hundredths, so 1.32 should be listed before 1.33. Therefore, response C is not correct.
15. Correct Response C
3
To change 8 to an improper fraction you multiply the denominator (4) by the whole number (8) to get 32. Then add the
4
numerator (3) to get 35. 8
3
4
=
35
4
3
3
as an improper fraction so choice A is equivalent to 8 . Thirty-two fourths is equal to 8 so
choice C is NOT equivalent to 8 . Three-fourths is equal to 0.75 as a decimal so 8
3
4
3
3
4
4
= 8.75. Choice B is also equivalent to 8
. Because A and B are both equivalent to 8 , choice D is not a correct option.
4
4
16. Correct Response A
First, divide 1 by 8 to get the decimal form of 1/8. This gives a table length of 75.125 inches. To round to the nearest tenth,
look at the digit in the next place to the right, the hundredths place. If that digit is 5 or greater, round up. Otherwise, leave
the tenths digit as it is and drop all digits to the right. The digit 2 is in the hundredths place. This is less than 5, so the 1 in
the tenths place stays as it is and all the following digits are dropped. Thus, 75.125 inches rounds to 75.1 inches to the
nearest tenth of an inch.
Incorrect Response B
75.13 inches is the length rounded to the nearest hundredth of an inch. The question asks for the length rounded to nearest
tenth of an inch.
Incorrect Response C
75.13 inches is the length rounded to the nearest hundredth of an inch. The question asks for the length rounded to the
nearest tenth of an inch.
Incorrect Response D
This choice may result from incorrectly rounding 75.125 up to 75.2, rather than down to 75.1. When rounding a number, look
at the digit to the right of the rounding place, not at the digit to the left.
17. Correct Response D. The figure has one large square shaded and
of the second square shaded.
Since each large square represents one unit, the total shaded area represents the number
Incorrect Response A.
.
is the portion shaded of the second square. However, the question asks for the number
represented by the entire shaded region, which includes the first square.
Incorrect Response B.
is the number represented by the unshaded region. However, the question asks for the
number represented by the shaded region.
Incorrect Response C.
is equivalent to
. Since the entire region represents two units,
cannot be the number
represented by the shaded region.
18. Incorrect Response A. 2.2 is equal to the mixed number 2
fraction is not in simplest terms.
Correct Response B.2.2 is equal to the mixed number 2
term, which is
11
5
2
2
10.
. Changing 2
. Changing 2
10.
2
10.
2
10.
to an improper fraction is
to an improper fraction is
22
10
22
10
, but the improper
, but it must be in simplest
.
Incorrect Response C.
7
= 1.4
5
Incorrect Response D.
5
11
= 0.45454545
19. Correct Response A. The absolute value of a number is its distance from 0 and the distance of the integer -9 from
zero (on the left or negative side of zero on a number line) is 9. This is the largest distance from zero of all the values given.
Incorrect Response B. The absolute value of a number is its distance from 0 and -9 is further away from 0 (to the right of
0) than 3 is.
Incorrect Response C. The absolute value of a number is its distance from 0 and -9 is further away from 0 (to the left of 0)
than -2 is.
Incorrect Response D. The absolute value of a number is its distance from 0 and -9 is further away from 0 (to the right of
0) than 8 is.
20. Incorrect Response A. The numbers are not listed from least to greatest because -12 is less than -8.
Incorrect Response B. The numbers are listed from greatest to least, not least to greatest.
Incorrect Response C. The numbers are not listed from least to greatest, they are listed in order by their
from least to greatest.
Correct Response D. The numbers are in order from least to greatest.
absolute value