问题1：使用适当的属性查找。
(i)-2/3×3/5 + 5/2 – 3/5×1/6
(ii)2/5×(-3/7)– 1/6×3/2 + 1/14×2/5
解决方案：
(i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6
Given equation: -2/3 × 3/5 + 5/2 – 3/5 × 1/6
By regrouping we get,
= -2/3 × 3/5 – 3/5 × 1/6 + 5/2
= 3/5 (-2/3 – 1/6)+ 5/2 [taking 3/5 as common]
= 3/5 ((-2×2/3×2 -1×1/6×1 )+ 5/2 [by using distributive property]
= 3/5 ((-4-1)/6)+ 5/2
= 3/5 ((–5)/6)+ 5/2
= – 15/30 + 5/2 [Dividing -15 and 30 by 2 we get -1/2]
= – 1/2 + 5/2
= 4/2
= 2
Therefore,
-2/3 × 3/5 + 5/2 – 3/5 × 1/6 = 2
(ii) 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
Given equation: 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
By regrouping we get,
= 2/5 × (-3/7) + 1/14 × 2/5 – (1/6 × 3/2)
= 2/5 × (-3/7 + 1/14) – 3/12
= 2/5 × ((-6 + 1)/14) – 3/12 [by using distributive property]
= 2/5 × ((-5)/14)) – 1/4
= (-10/70) – 1/4 [Dividing -10 and 70 by 10 we get -1/7]
= -1/7 – 1/4
= (-4 -7)/28
= -11/28
Therefore,
2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5 = -11/28
问题2：写下以下各项的加法逆
(i)2/8
(ii)-5/9
(iii)-6 / -5
(iv)2 / -9
(v)19 / -16
解决方案：
We know that the additive inverse of x will be -x,
(i) 2/8
Given: 2/8
Additive inverse of 2/8 will be -2/8
(ii) -5/9
Given: -5/9
Additive inverse of -5/9 will be 5/9
(iii) -6/-5
Given: -6/-5
-6/-5 = 6/5 [Dividing both by -1 ]
Additive inverse of 6/5 will be -6/5
(iv) 2/-9
Given: 2/-9
2/-9 = -2/9
Additive inverse of -2/9 will be 2/9
(v) 19/-16
Given: 19/-16
19/-16 = -19/16
Additive inverse of -19/16 will be 19/16
问题3：验证：-(-x)= x。
(i)x = 11/15
(ii)x = -13/17
解决方案：
(i) x = 11/15
Given, x = 11/15
Since, additive inverse of x will be -x
Therefore, the additive inverse of 11/15 will be -11/15 (as 11/15 + (-11/15) = 0)
We can also represent the following as 11/15 = -(-11/15)
Thus, -x = -11/15
-(-x) = -(-11/15) = (11/15) = x
Hence, verified: -(-x) = x
(ii) -13/17
Given, x = -13/17
Since, additive inverse of x will be -x as x + (-x) = 0
Therefore, the additive inverse of -13/17 will be 13/17 as 13/17 + (-13/17) = 0
We can also represent the following as 13/17 = -(-13/17)
Thus, -x = -13/17
-(-x) = -(-13/17) = (13/17) = x
Hence, verified: -(-x) = x
问题4：找出
(i)-13
(ii)-13/19
(iii)1/5
(iv)-5/8×(-3/7)
(v)-1×(-2/5)
(vi)-1
解决方案：
We know that the multiplicative inverse of x will be 1/x as a × 1/a = 1
(i) -13
Given: -13
The multiplicative inverse of -13 will be -1/13
(ii) -13/19
Given: -13/19
The multiplicative inverse of -13/19 will be -19/13
(iii) 1/5
Given: 1/5
The multiplicative inverse of 1/5 will be 5
(iv) -5/8 × (-3/7)
Given: -5/8 × (-3/7)
-5/8 × (-3/7) = 15/56
The multiplicative inverse of 15/56 will be 56/15
(v) -1 × (-2/5)
Given: -1 × (-2/5)
-1 × (-2/5) = 2/5
The multiplicative inverse of 2/5 will be 5/2
(vi) -1
Given: -1
The multiplicative inverse of -1 will be -1
问题5：命名以下每个属性中使用的乘法属性。
(i)-4/5×1 = 1×(-4/5)= -4/5
(ii)-13/17×(-2/7)= -2/7×(-13/17)
(iii)-19/29×29 / -19 = 1
解决方案：
(i) -4/5 × 1 = 1 × (-4/5) = -4/5
Given: -4/5 × 1 = 1 × (-4/5) = -4/5
It is representing the property of multiplicative identity.
(ii) -13/17 × (-2/7) = -2/7 × (-13/17)
Given: -13/17 × (-2/7) = -2/7 × (-13/17)
It is representing the property of commutativity.
(iii) -19/29 × 29/-19 = 1
Given: -19/29 × 29/-19 = 1
It is representing the property of multiplicative inverse
问题6：将6/13乘以-7/16的倒数
解决方案：
Given: 6/13 × (Reciprocal of -7/16)
Since, reciprocal of -7/16 = 16/-7 = -16/7
Therefore,
6/13 × (-16/7) = -96/91
问题7：告诉您什么属性可以将1/3×(6×4/3)计算为(1/3×6)×4/3
解决方案：
Given: 1/3 × (6 × 4/3) = (1/3 × 6) × 4/3
Here, the product of their multiplication does not change. Hence, Associativity Property is used in the given equation.
问题8：8/9是-1 1/8的乘法逆吗?为什么或者为什么不?
解决方案：
Given: -1 1/8 which is equal to -9/8
Since it is the multiplication inverse, therefore the product should be 1.
8/9 × (-9/8) = -1 ≠ 1
Hence, 8/9 is not the multiplication inverse of -1 1/8
问题9：如果0.3等于3 1/3的乘法逆?为什么或者为什么不?
解决方案：
Give: 3 1/3 = 10/3
Since it is the multiplication inverse, therefore the product should be 1.
0.3 × 10/3 = 3/3 = 1
Hence, 0.3 is the multiplicative inverse of 3 1/3.
问题10：写
(i)没有倒数的有理数。
(ii)等于其倒数的有理数。
(iii)等于负数的有理数。
解决方案：
(i) The rational number that does not have a reciprocal.
Since, 0 = 0/1
Therefore, the reciprocal of 0 = 1/0, which is not defined.
Hence, the rational number that does not have a reciprocal is 0.
(ii) The rational numbers that are equal to their reciprocals.
Since, 1 = 1/1
Therefore, the reciprocal of 1 = 1/1 = 1
Similarly,
-1 = -1/1
Therefore, the reciprocal of -1 = -1/1 = -1
Hence, the rational numbers that are equal to their reciprocals are 1 and -1
(iii) The rational number that is equal to its negative.
Since negative of 0 = -0 = 0
Therefore, the rational number that is equal to its negative is 0.
问题11：填写空白。
(i)零具有__________倒数。
(ii)__________和__________是他们自己的倒数
(iii)– 5的倒数是__________
(iv)1 / x的倒数，其中x≠0是__________。
(v)两个有理数的乘积始终是__________。
(vi)正有理数的倒数是__________。
解决方案：
(i) Zero has no reciprocal.
(ii) The numbers -1 and 1 are their own reciprocals
(iii) The reciprocal of – 5 is -1/5.
(iv) Reciprocal of 1/x, where x ≠ 0 is x.
(v) The product of two rational numbers is always a rational number.
(vi) The reciprocal of a positive rational number is positive.