问题1：通过以下方式验证属性：x×y = y×x
(i)x = -1 / 3，y = 2/7
解决方案：
x × y = (-1/3) × 2/7
= (-1 × 2)/(3 × 7)
= -2/21
y × x = 2/7 × (-1/3)
= (2 × -1)/(7 × 3)
= -2/21
Hence, x × y = y × x, property is verified
(ii)x = -3 / 5，y = -11/13
解决方案：
x × y = (-3/5) × (-11/3)
= (-3 × -11)/(5 × 13)
= 33/65
y × x = (-11/13) × (-3/5)
= (-11 × -3)/(13 × 5)
= 33/65
Hence, x × y = y × x, property is verified
(iii)x = 2，y = 7 / -8
解决方案：
x × y = 2/1 × (-7/8)
= (2 × -7)/(1 × 8)
2 is the common factor
= -7/4
y × x = (-7/8) × (2/1)
= (-7 × 2)/(8 × 1)
2 is the common factor
= -7/4
Hence, x × y = y × x, property is verified
(iv)x = 0，y = -15/8
解决方案：
x × y = 0 × (-15/8)
= 0
y × x = (-15/8) × 0
= 0
Hence, x × y = y × x, property is verified
问题2：通过以下方式验证属性：x×(y×z)=(x×y)×z
(i)x = -7/3，y = 12/5，z = 4/9
解决方案：
x × (y × z)
= -7/3 × (12/5 × 4/9)
= -7/3 × ((12 × 4)/(5 × 9))
3 is the common factor of 12 and 9
= -7/3 × ((4 × 4)/(5 × 3))
= -7/3 × (16/15)
= (-7 × 16)/(3 × 15)
= -112/45
(x × y) × z
= (-7/3 × 12/5) × 4/9
= ((-7 × 12)/(3 × 5)) × 4/9
= ((-7 × 4)/(5)) × 4/9
= -28/5 × 4/9
= -112/45
Hence, x × (y × z) = (x × y) × z Property is verified
(ii)x = 0，y = -3 / 5，z = -9/4
解决方案：
x × (y × z)
= 0 × (-3/5 × -9/4)
= 0 × (27/20)
= 0(Any number multiplied with zero is zero)
(x × y) × z
= (0 × -3/5) × -9/4
= 0 × -9/4
= 0(Any number multiplied with zero os zero)
Hence, x × (y × z) = (x × y) × z Property is verified
(iii)x = 1/2，y = 5 / -4，z = -7/5
解决方案：
x × (y × z)
= 1/2 × (-5/4 × -7/5)
= 1/2 × ((-5 × -7)/(4 × 5))
Common factor 5
= 1/2 × (7/4)
= 7/8
(x × y) × z
= (1/2 × -5/4) × -7/5
= -5/8 × -7/5
Common factor 5
= 7/8
Hence, x × (y × z) = (x × y) × z Property is verified
(iv)x = 5/7，y = -12 / 13，z = -7/18
解决方案：
x × (y × z)
= 5/7 × (-12/13 × -7/18)
= 5/7 × ((-12 × -7)/(13 × 18))
Common factor 6 of 12 and 18
= 5/7 × ((-2 × -7)/(13 × 3))
= 5/7 × (14/39)
= (5 × 14)/(7 × 39)
7 is the common factor of 7 and 14
= 5 × 2/39
= 10/39
(x × y) × z
= (5/7 × -12/13) × -7/18
= ((5 × -12)/(7 × 13)) × -7/18
= (5 × -12 × -7)/(7 × 13 × 18)
Common factor 7 and 6
= (5 × -2 × -1)/(1 × 13 × 3)
= 10/39
Hence, x × (y × z) = (x × y) × z Property is verified
问题3：通过以下操作验证属性：x×(y + z)= x×y + x×z
(i)x = -3 / 7，y = 12/13，z = -5/6
解决方案：
x × (y + z)
= -3/7 × (12/13 + -5/6)
LCM of 13 and 6 is 78
= -3/7 × ((12 × 6 – 5 × 13)/78)
= -3/7 × ((72 – 65)/78)
= -3/7 × 7/78
= -3 × 7/7 × 78
Common factor 7 and 3
= -1/26
x × y + x × z
= -3/7 × 12/13 + -3/7 × -5/6
= (-3 × 12)/(7 × 13) + (-3 × -5)/(6 × 7)
= -36/91 + 15/42
= (-36 × 6 + 15 × 13)/546
= 196 – 216/546
= -21/546
= -1/26
Hence, the property x × (y + z) = x × y + x × z is verified
(ii)x = -12 / 5，y = -15 / 4，z = 8/3
解决方案：
x × (y + z)
= -12/5 × (-15/4 + 8/3)
LCM is 12
= -12/5 × ((-15 × 3 + 8 × 4)/12)
= -12/5 × ((-45 + 32)/12)
= -12/5 × (-13)/12
= (-12 × -13)/(5 × 12)
12 is the common factor
= 13/5
x × y + x × z
= -12/5 × -15/4 + -12/5 × 8/3
= (-12 × -15)/(5 × 4) + (-12 × 8)/(5 × 3)
Common factor 4 and 5, 3
= 9 + -32/5
LCM is 5
= (9 × 5 – 32)/5
= 45 – 32/5
= 13/5
Hence, the property x × (y + z) = x × y + x × z is verified
(iii)x = -8 / 3，y = 5/6，z = -13/12
解决方案：
x × (y + z)
= -8/3 × (5/6 + -13/12)
LCM is 12
= -8/3 × (5 × 2 – 13)/(12)
= -8/3 × (10 – 13)/12
= -8/3 × (-3/12)
= (-8 × -3)/(3 × 12)
Common factor 4 and 3
= 2/3
x × y + x × z
= -8/3 × 5/6 + -8/3 × -13/12
= (-8 × 5)/(3 × 6) + (-8 × -13)/(3 × 12)
Common factor 2 and 4
= (-4 × 5)/(3 × 3) + (-2 × -13)/(3 × 3)
= -20/9 + 26/9
= (-20 + 26)/9
= 6/9
Common factor is 3
= 2/3
Hence, the property x × (y + z) = x × y + x × z is verified
iv)x = -3 / 4，y = -5 / 2，z = 7/6
解决方案：
x × (y + z)
= -3/4 × (-5/2 + 7/6)
LCM is 6
= -3/4(-5 × 3 + 7)/6
= -3/4 × (-15 + 7)/6
= -3/4 × -8/6
= (-3 × -8)/(4 × 6)
Common factor 3 and 4
= 1
x × y × + x × z
= -3/4 × -5/2 + -3/4 × 7/6
= (-3 × -5)/(4 × 2) + (-3 × 7)/(4 × 6)
= 15/8 + -7/8
= (15 – 7)/8
= 8/8
= 1
Hence, Hence, the property x × (y + z) = x × y + x × z is verified
问题4：使用有理数相乘而不是加法的分布来简化：
(i)3/5×((35/24)+(10/1))
解决方案：
= 3/5 × 35/24 + 3/5 × 10/1
= (3 × 35)/(5 × 24) + (3 × 10)/(5 × 1)
Common factor is 5 and 3
= (1 × 7)/(1 × 8) + (3 × 2)/(1)
= 7/8 + 6/1
LCM is 8
= (7 + 6 × 8)/8
= 7 + 48/8
= 55/8
(ii)-5/4×((8/5)+(16/5))
解决方案：
= -5/4×8/5+-5/4×16/5
= (-5×8)/(4×5)+(-5×16)/(4×5)
Common factor is 4, 5
= -2/1+-4/1
= -6
(iii)2/7×((7/16)-(21/4))
解决方案：
= 2/7×7/16-2/7×21/4
= (2×7)/(7×16)-(2×21)/(7×4)
Common factor 2 and 7
= 1/8-3/2
LCM is 8
= (1-3×4)/8
= (1-12)/8
= -11/8
(iv)3/4×((8/9)-40)
解决方案：
= 3/4×8/9-3/4×40
= (3×8)/(4×9)-(3×40)/(4×1)
Common factor 3 and 4
= 2/3-30/1
LCM is 3
= (2-90)/3
= -88/3
问题5：找出以下每个有理数的乘法逆(倒数)：
(i)9
解决方案：
Multiplicative inverse of 9/1 is 1/9
(ii)-7
解决方案：
Multiplicative inverse of -7/1 is 1/-7 or -1/7
(iii)12/5
解决方案：
Multiplicative inverse of 12/5 is 5/12
(iv)-7/9
解决方案：
Multiplicative inverse of -7/9 is 9/-7 or -9/7
(v)-3 / -5
解决方案：
Multiplicative inverse of -3/-5 is -5/-3 or 5/3
(vi)2/3×9/4
解决方案：
(2×9)/(3×4)
2 is common factor of 2 and 4, 3 is common factor of 3 and 9
=3/2
Multiplicative inverse is 2/3
(vii)-5/8×16/15
解决方案：
(-5×16)/(8×15)
5 is the common factor of 5 and 15, 8 is the common factor of 8 and 16
=-2/3
Multiplicative inverse is 3/-2 or -3/2
(viii)-2×-3/5
解决方案：
=(-2×-3)/(1×5)
=6/5
Multiplicative inverse is 5/6
(ix)-1
解决方案：
Multiplicative inverse is -1
(x)0/3
解决方案：
Multiplicative inverse is 3/0 which does not exist
(xi)1
解决方案：
Multiplicative inverse is 1
问题6：命名有理数乘法的性质，其说明如下：
(i)-5/16×8/15 = 8/15×-5/16
解决方案：
a×b=b×a
This is commutative property
(ii)-17/5×9 = 9×-17/5
解决方案：
a×b=b×a
This is commutative property
(iii)7/4×(-8/3 + -13/12)= 7/4×-8/3 + 7/4×-13/12
解决方案：
a×(b+c)=a×b+a×c
This is distributive property of multiplication over addition
(iv)-5/9×(4/15×-9/8)=(-5/9×4/15)×-9/8
解决方案：
a×(b×c)=(a×b)×c
This is associative property of multiplication
(v)13 / -17×1 = 13 / -17 = 1×13 / -17
解决方案：
a×1=a=1×a
This is multiplicative identity
(vi)-11/16×16 / -11 = 1
解决方案：
a×1/a=1
This is multiplicative inverse
(vii)2/13×0 = 0 = 0×2/13
解决方案：
a×0=0=0×a
Any number multiplied with 0 is 0
(viii)-3/2×5/4 + -3/2×-7/6 = -3/2×(5/4 + -7/6)
解决方案：
a×b+a×c=a×(b+c)
This is distributive law of multiplication over addition
问题7：填空：
(i)两个正有理数的乘积始终为…
解决方案：
The product of two positive rational numbers is always positive.
(ii)正有理数与负有理数的乘积始终为…。
解决方案：
The product of a positive rational number and a negative rational number is always negative
(iii)两个负有理数的乘积始终为…
解决方案：
The product of two negative rational numbers is always positive
(iv)正有理数的倒数是…
解决方案：
The reciprocal of a positive rational numbers is positive
(v)负有理数的倒数是…
解决方案：
The reciprocal of a negative rational numbers is negative
(vi)零有……。互惠的。
解决方案：
Zero has no reciprocal.
(vii)有理数及其倒数的乘积是…
解决方案：
The product of a rational number and its reciprocal is 1
(viii)数字…和…是他们自己的倒数。
解决方案：
The numbers 1 and -1 are their own reciprocals.
(ix)如果a是b的倒数，则b的倒数是。
解决方案：
If a is reciprocal of b, then the reciprocal of b is a
(x)数字0是…任意数字的倒数。
解决方案：
The number 0 is not the reciprocal of any number.
(xi)1 / a的倒数，a≠0是…
解决方案：
Reciprocal of 1/a, a ≠ 0 is a
(xii)(17×12) -1 = 17 -1 ×…
解决方案：
(17×12)-1 = 17-1 × 12-1
问题8：填写空白：
(i)-4×79 = 79×…
解决方案：
-4 × 79= 79 × -4
By using commutative property.
(ii)5/11×-3/8 = -3/8×…
解决方案：
5/11 × -3/8 = -3/8 × 5/11
By using commutative property.
(iii)1/2×(3/4 + -5/12)= 1/2×…+…×-5/12
解决方案：
1/2 × (3/4 + -5/12) = 1/2 × 3/4 + 1/2 × -5/12
By using distributive property.
(iv)-4/5×(5/7 + -8/9)=(-4/5×…)+ -4/5×-8/9
解决方案：
-4/5 × (5/7 + -8/9) = (-4/5 × 5/7) + -4/5 × -8/9
By using distributive property.