8类NCERT解决方案–第13章正反比例–练习13.1

问题1.以下是火车站附近的停车收费标准。
4小时—₹60
8小时—₹100
12小时—₹140
24小时—₹180
检查停车费是否与停车时间成正比。

解决方案：

Table for the given question –

The ratio of time and respective parking charge can be calculated as:

4/60 = 1/15

8/100 = 2/25

12/140 = 3/35

24/180 = 2/15

We can clearly see that 1/15 ≠ 2/25 ≠ 3/35 ≠ 2/15

So, the parking charges are not direct proportions to the parking time.

涂料的问题2.一种混合物是通过与8份的碱的混合红色颜料的1份制备。在下表中，找到需要添加的基础部分。

Number of hours	4	8	12	24
Parking Cost	60	100	140	180

解决方案：

The mixture is prepared by adding 1 part of red pigment to 8 parts of the base, if red pigment parts increase, then correspondingly parts of the base will also increase, so they are in Direct proportion.

Parts of red pigment	1	4	7	12	20
Parts of base	8	a1	a2	a3	a4

As the red pigment is in direct proportion with parts of the base, 

i) 1/8 = 4/a1

So, a1 = 32

ii) 1/8 = 7/a2

So, a2 = 56

iii) 1/8 = 12/a3

So, a3 = 96

iv) 1/8 = 20/a4

So, a4 = 160

Hence, the required parts of the bases are 

Parts of red pigment	1	4	7	12	20
Parts of base	8	32	56	96	160

问题3。在上面的问题2中，如果1份红色颜料需要75毫升的碱，那么我们应该将1800毫升的碱与多少红色颜料混合?

解决方案：

Let the parts of red pigment required to mix with 1800 mL of the base = a 

Parts of red pigment	1	a
Parts of base	75	1800

Since the red pigment is in Direct proportion with parts of the base, 

So, 1/75 mL = a/1800 mL

a = 1800/75 = 24

Hence, 24 parts of red pigment are required to mix with 1800 mL of a base to form a mixture.

问题4.软饮料工厂的一台机器在六个小时内填充了840个瓶子。五个小时后它将充满几瓶?

解决方案：

Let the number of bottles filled = a 

Number of Bottles	840	a
Time in hours	6	5

Since the time taken to fill the bottles is directly proportional to the number of bottles filled,

So, 840/6 = a/5

a = (840 × 5)/6

a = 700

Hence, 700 bottles will be filled in 5 hours.

问题5.如图所示，放大50,000倍的细菌照片的长度为5厘米。细菌的实际长度是多少?如果仅将照片放大2万倍，那么放大的长度将是多少?

解决方案：

Let the actual length of bacteria is a cm and enlarged length is b cm

Length of Bacteria (cm)	5	a	b
Number of times photograph is enlarged	50,000	1	20,000

Since the length of bacteria is directly proportional to the number of times the photograph is enlarged,

So, 5/50000 = a/1

a = 1/10000 = 10^-4

The actual length is 10^-4 cm

And, 5/50000 = b/20000

b = 2 cm

The enlarged length is 2 cm

Hence, the actual length of Bacteria is 10^-4 cm, and the length of bacteria is 2 cm when the photograph is enlarger 20,000 times.

问题6：在船舶模型中，桅杆高9厘米，而实际船舶的桅杆高12 m。如果船长是28 m，那么模型船要多久?

解决方案：

Let the required length of the model ship is a cm

	Mast Length	Ship Length
Model Ship	9 cm	a cm
Actual Ship	12 m	28 m

Since the mast length and ship length of both model and actual ship are directly proportional 

So, 9/12 = a/28

a = (28 X 9)/12 = 21 cm

Hence, the length of the model ship is 21 cm, if the length of the actual ship is 28 m.

问题7.假设2公斤糖中含有9×10 ^6个晶体。 (i)5公斤糖有多少个糖晶体? (ii)1.2公斤糖?

解决方案：

i) Let the number of sugar crystals is a

Weight of Sugar (kg)	2	5
Number of Sugar crystals	9 X 106	a

Since the weight of Sugar is directly proportional to the number of sugar crystals,

So, 2/9 X 10⁶ = 5/a

a = 2.25 X 10⁷ Crystals

Hence, the number of sugar crystals is 2.25 X 10⁷ in 5 kg of sugar.

ii) Let the number of sugar crystals is b

Weight of Sugar (kg)	2	1.2
Number of Sugar crystals	9 X 106	b

Since the weight of Sugar is directly proportional to the number of sugar crystals,

So, So, 2/9 X 10⁶ = 1.2/b

b = 5.4 X 10⁶ Crystals

Hence, the number of sugar crystals is 5.4 X 10⁶ in 1.2 kg of sugar.

问题8. Rashmi的路线图的比例为1 cm，代表18 km。她在公路上行驶了72公里。她在地图上覆盖的距离是多少?

解决方案：

Let the distance covered on the map is a cm

Distance covered on the road (km)	18	72
Distance covered on the map (cm)	1	a

Since distance covered on road is directly proportional to the distance covered on the map

So, 18/1 = 72/a

a = 4 cm

Hence, Rashmi covers only 4 cm on the map, while she drives 72 km on the road.

问题9.一根5 m 60厘米高的垂直杆投下3 m 20厘米长的阴影。同时找到(i)由另一根杆子投射的阴影的长度10 m高50 cm(ii)另一根杆子投射的阴影的高度5 m长

解决方案：

i) Let the length of the pole’s shadow is a m

Height of Pole (m)	5.60	10.50
Length of Pole’s Shadow (m)	3.20	a

As we know more is the height of the pole, more will be the length of its shadow, it means the height of the pole is directly proportional to the length of the shadow.

So, 5.60/3.20 = 10.50/a

a = 6 m

Hence, the length of the pole shadow is 6 m, when the height of the pole is 10 m 50 cm.

ii) Let the height of the pole is b m

Height of Pole (m)	5.60	b
Length of Pole’s Shadow (m)	3.20	5

As we know more is the height of the pole, more will be the length of its shadow, it means the height of the pole is directly proportional to the length of the shadow.

So, 5.60/3.20 = b/5

b = 8.75 m or 8 m 75 cm

Hence, the height of the pole is 8 m 75 cm.

问题10.一辆载重卡车在25分钟内行驶14公里。如果速度保持不变，则5小时内可以行驶多远?

解决方案：

Let the truck travels a km

As 1 hour = 60 minutes 

So, 5 hours = 300 minutes 

Distance travelled (km)	14	a
Time taken (minutes)	25	300

Since the distance travelled by truck is directly proportional to the time taken,

So, 14/25 = a/300

a = 168 km 

Hence, the truck travels 168 km in 5 hours.