Chapter 1Directed Numbers and the Number Line
Question1
If 30 minutes means 30 minutes before the rocket launch, what does +0.25 minutes mean?
A.0.25 minutes before the launch
B.15 seconds after the launch
C.25 seconds after the launch
D.None of the above
Answer: B
Skill: Concept of Directed Numbers
Solution
0.25 minutes = 15 seconds
∴+0.25 minutes means 15 seconds after the launch.
Question2
If 200 calories means Cherry uses up 200 calories more than she takes in, what does +180 calories means?
A.Cherry uses up the same calories as she takes in.
B.Cherry uses up 180 calories less than she takes in.
C.Cherry takes in 180 calories more than she uses up.
D.None of the above
Answer: C
Skill: Concept of Directed Numbers
Solution
+180 calories means Cherry uses up 180 calories less than she takes in. It is the same as Cherry takes in 180 calories more than she uses up.
Try a similar problem:If +200 calories means Marytakes in200 calories more than she uses up, what does 120 calories means?
A.Mary uses up 120 calories morethanshe takes in.
B.Mary uses up 120 calories less than she takes in.
C.Mary takes in 80 calories more than she uses up.
D.Mary takes in 80 calories less than she uses up.
Answer: C
Solution
120 calories means Mary takes in 120 calories less than she uses up. It is the same as Mary uses up 120 calories more than she takes in.
Question3
Which of the following is NOT true?
A.All negative numbers are less than zero.
B.All directed numbers are negative numbers.
C.All negative numbers are less than any positive numbers.
D.The opposite number of a negative number is a positive number.
Answer: B
Skill: Concept of Directed Numbers
Solution
Positive numbers are also directed numbers.
Question4
If the length of a sample is longer than the standard length by more than 4 mm or shorter than the standard length by more than 2.5 mm, then the sample will be discarded. The following table shows the comparison between the lengths of 8 samples and the standard length. (+1 mm means 1 mm longer than the standard.)
A / B / C / D / E / F / G / H+5.1 mm / +3.9 mm / 1.2 mm / +2.8 mm / 4.4 mm / 5.3 mm / +4.1 mm / 0.7 mm
How many of the above 8 samples will be discarded?
A.3
B.4
C.5
D.6
Answer: B
Skill: The Number Line
Solution
Sample A: +5.1 > +4
Sample G: +4.1 > +4
Sample E: 4.4 < 2.5
Sample F: 5.3 < 2.5
The lengths of samples A and G are longer than the standard length by more than
4 mm. The lengths of samples E and F are shorter than the standard length by more than 2.5 mm.
∴4 Samples will be discarded.
Question 5
Refer to the given number line.
Which of the following has a positive value?
A.b + e
B.de
C.a + b
D.a c
Answer: B
Skill: Addition and Subtraction of Directed Numbers
Solution
For option B:
Since d is a positive number and e is a negative number, de has a positive value.
Try a similar problem:Refer to the given number line.
Which of the following has a negative value?
A.be
B.b+e
C.da
D.a c
Answer: D
Solution
For option D:
Since a is a negative number and c is a negative number, and a is smaller than c, ac has a negative value.
Question6
Which of the following must be true?
I.If we subtract a positive number from a positive number, the result is a positive number.
II.If we subtract a negative number from a negative number, the result is a negative number.
III.If we subtract a negative number from a positive number, the result is a positive number.
IV.If we subtract a positive number from a negative number, the result is a negative number.
A.I and II only
B.III and IV only
C.All of the above
D.None of the above
Answer: B
Skill: Addition and Subtraction of Directed Numbers
Solution
For I: Consider (+5) (+6).
(+5) (+6) = +5 6 = 1
∴I is not true.
For II:Consider (5) (6).
(5) (6) = 5 + 6 = +1
∴II is not true.
For III:We have (a) = +a, where a is a negative number.
∴III is true.
For IV:We have (+a) = a, where +a is a positive number.
∴IV is true.
Question7
If a is a negative number, which of the following is NOT true?
A.(+8) + (a) > (+3) + (a)
B.(a) (+1) < (a) (+0.1)
C.(+2) + (a) = (a) (2)
D.(a) + (7.7) > (a) + (6.7)
Answer: D
Skill: Addition and Subtraction of Directed Numbers
Solution
For option D:
(a) + (7.7) = (a) 7.7
(a) + (6.7) = (a) 6.7
∴(a) + (7.7) < (a) + (6.7)
Question 8
Which of the following expressions has a value different from the others?
A.(+1) + (1) (+1) + (1) (+1)
B.(+1) + (1) (+1) + [(1) (+1)
C.(+1) + (1) [(+1) + (1) (+1)
D.(+1) + [(1) (+1) + (1) (+1)
Answer: C
Skill: Addition and Subtraction of Directed Numbers
Solution
For option A:
(+1) + (1) (+1) + (1) (+1)
= +1 1 1 1 1
= 3
For option B:
(+1) + (1) (+1) + [(1) (+1)]
= +1 1 1 + (1 1)
= 1 + (2)
= 1 2
= 3
For option C:
(+1) + (1) [(+1) + (1) (+1)]
= +1 1 (+1 1 1)
= 0 (1)
= 0 + 1
= +1
For option D:
(+1) + [(1) (+1) + (1) (+1)]
= +1 + (1 1 1 1)
= +1 + (4)
= +1 4
= 3
Since (+1) + (1) [(+1) + (1) (+1)] has a value different from the others, the answer is option C.
Question9
There are 4 questions in a questionnaire about room service. For each question, the customer's satisfaction is represented by +2, +1, 0, 1 or 2. +2 means 'very satisfied', +1 means 'satisfied', 0 means 'average', 1 means 'dissatisfied' and 2 means 'very dissatisfied'.
The following table shows the feedback from 4 customers.
Question 1 / Question 2 / Question 3 / Question 4Customer A / 0 / 1 / +2 / +1
Customer B / +1 / 0 / +1 / 0
Customer C / 2 / +2 / +2 / 0
Customer D / +2 / 2 / +1 / +2
Which customer is the most satisfied?
A.Customer A
B.Customer B
C.Customer C
D.Customer D
Answer: D
Skill: Addition and Subtraction of Directed Numbers
Solution
Customer A: (0) + (1) + (+2) + (+1) = 0 1 + 2 + 1 = +2
Customer B: (+1) + (0) + (+1) + (0) = +1 + 0 + 1 + 0 = +2
Customer C: (2) + (+2) + (+2) + (0) = 2 + 2 + 2 + 0 = +2
Customer D: (+2) + (2) + (+1) + (+2) = +2 2 + 1 + 2 = +3
Therefore, customer D is the most satisfied.
Question 10
David aims to read 10 articles every day. Suppose that +1 article means he has read
1 article more than his target, and the records in the past four days are 3, +5, 0 and 1 articles respectively. How many articles has he read in the past four days?
A.39
B.41
C.45
D.49
Answer: B
Skill: Addition and Subtraction of Directed Numbers
Solution
1st day: 10 + (3) = 7
2nd day: 10 + (+5) = 15
3rd day: 10 + (0) = 10
4th day: 10 + (1) = 9
7 + 15 + 10 + 9 = 41
∴David has read 41 articles in the past four days.
Try a similar problem:Tom aims to run 1000metres every day. Suppose that +1 metre means he has run
1 metre more than his target, and the records in the past four days are +200, 50,0 and 70metresrespectively. How many metres has he run in the past four days?
A.3090
B.4080
C.4510
D.4930
Answer: B
Solution
1st day: 1000 + (+200) = 1200
2nd day: 1000 + (50) = 950
3rd day: 1000 + (0) = 1000
4th day: 1000 + (70) = 930
1200 + 950 + 1000 + 930 = 4080
∴Tom has run 4080metres in the past four days.
Question11
Which of the following expressions has a value different from the others?
A.(1) (1) (1) (1)
B.(+1) (+1) (+1) (+1)
C.(+1) (1) (+1) (1)
D.(1) (+1) (1) (1)
Answer: D
Skill: Multiplication and Division of Directed Numbers
Solution
For option A:(1) (1) (1) (1) = +1
For option B:(+1) (+1) (+1) (+1) = +1
For option C:(+1) (1) (+1) (1) = +1
For option D:(1) (+1) (1) (1) = 1
Since (1) (+1) (1) (1) has a value different from the others, the answer is option D.
Question12
(2)2 (32) =
A.24
B.+24
C.36
D.+36
Answer: C
Skill: Multiplication and Division of Directed Numbers
Solution
Try a similar problem:(4)2 (22) =
A.64
B.48
C.+48
D.+64
Answer: A
Solution
Question13
(25) (4) (+16) (5) = ______
Answer: 20
Skill: Multiplication and Division of Directed Numbers
Solution
Question 14
= ______
Answer: 1
Skill: Multiplication and Division of Directed Numbers
Show me an example: Example4_eng.ppt
Solution
Question 15
A water tank is leaking water, its water level decreases 200 cm every hour. At
08:00a.m., the water depth is 3 m.The water depth at 10:00 a.m. is ______m.
Answer: 2.6
Skill: Mixed Operations of Directed Numbers
Solution
Question16
(3) (+9) (+12) (6) = ______
Answer: -25
Skill: Mixed Operations of Directed Numbers
Solution
Try a similar problem:(2) (+8) (+6) (3) = ______
Answer: 14
Solution
Question17
A.+3
B.+6
C.7
D.14
Answer: A
Skill: Mixed Operations of Directed Numbers
Solution
Question18
(12) (+36) [(3) (+2)] =
A.6
B.+6
C.12
D.+12
Answer: A
Skill: Mixed Operations of Directed Numbers
Show me an example: Example5_eng.ppt
Solution
Question19
Multiply +3 by 13 and then add the result to the product of 5 and +7 equals ______.
Answer: 74
Skill: Mixed Operations of Directed Numbers
Solution
Question20
In a cooking competition, every participant has to make 20 pancakes. 10 points are awarded for each qualified pancake, 3 points are deducted for each unqualified pancake and 5 points are deducted for each unfinished pancake. If Alex makes
7 qualified pancakes and 5 unqualified pancakes, how many points does he get?
A.15
B.25
C.55
D.65
Answer: A
Skill: Mixed Operations of Directed Numbers
Solution
Since Alex only makes 7 + 5 = 12 pancakes, there are 20 12 = 8 unfinished pancakes.
(+10) 7 + (3) 5 + (5) 8
= (+70) + (15) + (40)
= +70 15 40
= +15
∴Alex gets 15 points.