问题1：解释直接变化的概念。
解决方案：
If the values of two quantities depend on each other in such a way that if we increase the value of one quantity the value of the other quantity also increase, similarly if we decrease the value of one quantity the value of other quantity also decreases, therefore if the ratio between the two variables remains constant, it is said to be in direct variation.
问题2：以下哪些数量彼此直接不同?
(i)物品数量(x)及其价格(y)。
(ii)物品的重量(x)及其成本(y)。
(iii)距离x和时间y，速度保持不变。
(iv)工资(y)和工作小时数(x)。
(v)覆盖的速度(x)和时间(y)保持相同)。
(vi)土地面积(x)及其成本(y)。
解决方案：
(i) Number of articles (x) and their price (y)
The number of articles is directly proportional to their price, therefore, when the number of articles increase, then the cost of article will also increase. So it is a case of direct proportion.
(ii) Weight of articles (x) and their cost (y).
The weight (x) of the articles is directly proportional to their cost (y), therefore, If weight of the article is increasing then the cost of article will also increase. So it is a case of direct proportion.
(iii) Distance x and time y, speed remaining the same.
On increasing the distance between objects, the time required to cover them will also increase, therefore, on constant speed, time increases when distance increases. So it is a case of direct proportion
(iv) Wages (y) and number of hours (x) of work.
If the workers work for more hours, they will be paid more wages, therefore, wages increases if the number of working hours increase. So it is a case of direct proportion
(v) Speed (x) and time (y) distance covered remaining the same.
Time is inversely proportional to distance , that is, keeping the same distance, the time taken will reduce if speed is increased, here one quantity is decreasing when we are increasing the other. So it is not a case of direct proportion.
(vi) Area of a land (x) and its cost (y).
On increasing the area of the land available, its cost will also increase and multiply. So it is a case of direct proportion
问题3：以下x和y表中的哪些直接不同?
(一世)
| a | 7 | 9 | 13 | 21 | 25 |
| b | 21 | 27 | 39 | 63 | 75 |
(ii)
| a | 10 | 20 | 30 | 40 | 46 |
| b | 5 | 10 | 15 | 20 | 23 |
(iii)
| a | 2 | 3 | 4 | 5 | 6 |
| b | 6 | 9 | 12 | 17 | 20 |
(iv)
| a | 12 | 22 | 32 | 42 | 52 |
| b | 13 | 23 | 33 | 43 | 53 |
解决方案：
(i) Directly proportional.
If we clearly notice the values in the table, then the value in column b is thrice the value of column a. Therefore, the rows ‘a’ and ‘b’ are directly proportional, in this case.
(ii) Directly proportional.
If we clearly notice the values in the table, then the value in column b is half the value of column a. Therefore, ‘a’ and ‘b’ are directly proportional, in this case.
(iii) Not directly proportional.
If we clearly notice the values in the table, then the value in column b is thrice the value of column a, only in first three columns and not in others. Therefore, ‘a’ and ‘b’ are not directly proportional, in this case.
(iv) Not directly proportional.
If we clearly notice the values in the table, then the value in column b differ by different constant amounts with respect to values in column a. Therefore, ‘a’ and ‘b’ are not directly proportional, in this case.
问题4：在以下各项中填入空白，以使陈述正确：
(i)据说有两个数量不同……。彼此之间，如果它们以相同值的比率保持相同的方式一起增加(减少)。
(ii)如果对于某个正数k，………= k，则x和y彼此直接变化。
(iii)如果u = 3v，则u和v会变化……。彼此。
解决方案：
(i) directly
(ii) k = x/y where k is a positive number.
(iii) directly
问题5.假设x与y直接变化，请完成下表。
(一世)
x 2.5……15
y 5 8 12…
(ii)
x 5…10 35 25…
y 8 12…………32
(iii)
x 6 8 10…20
y 15 20…40…
(iv)
x 4 9……3…
y 16…48 36…4
(v)
x 3 5 7 9
y…20 28…
解决方案：
(i)
We know k = x/y
2.5/5 = x1/8
By cross-multiplying
8(2.5) = 5×1
20 = 5x1
x1 = 20/5
= 4
We know k = x/y
4/8 = x2/12
By cross-multiplying
12(4) = 8x2
48 = 8x2
x2 = 48/8
= 6
We know k = x/y
6/12 = 15/y1
By cross-multiplying
6y1 = 15(12)
6y1 = 180
y1 = 180/6
= 30
x 2.5 4 6 15
y 5 8 12 30
(ii)
We know k = x/y
5/8 = x1/12
By cross-multiplying
12(5) = 8x1
60 = 8x1
x1 = 60/8
= 7.5
We know k = x/y
7.5/12 = 10/y1
By cross-multiplying
7.5y1 = 10(12)
7.5y1 = 120
y1 = 120/7.5
= 16
We know k = x/y
10/16 = 35/y2
By cross-multiplying
10y2 = 35(16)
10y2 = 560
y2 = 560/10
= 56
We know k = x/y
35/56 = 25/y3
By cross-multiplying
35y3 = 56(25)
35y3 = 1400
y3 = 1400/35
= 40
We know k = x/y
25/40 = x2/32
By cross-multiplying
25(32) = 40x2
800 = 40x2
x2 = 800/40
= 20
x 5 7.5 10 35 25 20
y 8 12 16 56 40 32
(iii)
We know k = x/y
8/20 = 10/y1
By cross-multiplying
8y1 = 10(20)
8y1 = 200
y1 = 200/8
= 25
We know k = x/y
10/25 = x1/40
By cross-multiplying
10(40) = 25x1
400 = 25x1
x1 = 400/25
= 16
We know k = x/y
16/40 = 20/y2
By cross-multiplying
16y2 = 20(40)
16y2 = 800
y2 = 800/16
= 50
x 6 8 10 16 20
y 15 20 25 40 50
(iv)
We know k = x/y
4/16 = 9/y1
By cross-multiplying
4y1 = 9(16)
= 144
y1 = 144/4
= 36
We know k = x/y
9/36 = x1/48
By cross-multiplying
9(48) = 36x1
432 = 36x1
x1 = 432/36
= 12
We know k = x/y
12/48 = x2/36
By cross-multiplying
12(36) = 48x2
432 = 48x2
x2 = 432/48
= 9
We know k = x/y
9/36 = 3/y2
By cross-multiplying
9y2 = 3(36)
= 108
y2 = 108/9
= 12
We know k = x/y
3/12 = x3/4
By cross-multiplying
3(4) = 12x3
12 = 12x3
x3 = 12/12
= 1
x 4 9 12 9 3 1
y 16 36 48 36 12 4
(v)
We know k = x/y
3/y1 = 5/20
By cross-multiplying
3(20) = 5y1
60 = 5y1
y1 = 60/5
= 12
We know k = x/y
7/28 = 9/y2
By cross-multiplying
7y2 = 9(28)
= 252
y2 = 252/7
= 36
x 3 5 7 9
y 12 20 28 36
问题6.从下表中找到变化常数：
| x | 3 | 4 | 7 | 9 |
| y | 12 | 20 | 28 | 36 |
设置表并解决以下问题。使用单一方法来验证答案。
解决方案：
Dividing the second column values by first column values, y/x we get the corresponding ratios
| C1 | C2 | C3 | C4 | |
| y/x | 12/3 = 4 | 20/5 = 4 | 28/7 = 4 | 36/9 = 4 |
Therefore, for all the columns y is four times x.
∴ The constant of variation in the given table is x/y = 1/4.
问题7.罗希特(Rohit)以156卢比的价格购买了12个寄存器，并找到了7个这样的寄存器的成本。
解决方案：
Cost of 12 registers = Rs 156
Cost of 1 register = Rs 156/12
= Rs 13 (Taking 12 to RHS)
Cost of 7 registers = Cost of 1 register * number of req. registers
=> Rs 13 × 7
=> Rs 91
Therefore, 7 registers cost Rs 91.
问题8.阿努帕玛(Anupama)花125分钟步行100米。她在315分钟内将覆盖什么距离?
解决方案：
Distance covered in 125 minutes = 100 metre
Distance covered in 1 minute = 100 m / 125 min
Now distance covered in 315 minutes = Distance covered in 1 minute x 315 minutes
=> distance covered in 315 minutes = (100/125) x 315
Solving the equation,
=> distance covered in 315 minutes = (31500/125) =252
∴ The distance covered in 315min is 252 meters.
问题9.如果93 m的某种塑料板的价格为1395卢比，那么购买105 m的这种塑料板将花费多少?
解决方案：
Cost of 93m plastic sheet = Rs 1395
Cost of 1m plastic sheet = Rs 1395/93
Cost of 105m plastic sheet = Cost of 1m plastic sheet x required length of plastic sheet
=> Cost of 105m plastic sheet = Rs (1395/93) x 105
=> Cost of 105m plastic sheet = 146475/93
= Rs 1575
∴ The cost of 105m plastic sheet is Rs 1575
问题10. Suneeta在一小时内输入1080个单词。她的GWAM(毛钱/分钟)是多少?
解决方案：
No of words typed in one hour (60 minutes) = 1080 (because 1 hour = 60 min)
No of words typed in 1 minute = 1080 /60
Solving, we get
=>No of words typed in 1 minute =18
∴ Number of words typed in one minute is 18