**Direct and Inverse Proportions**An increase (↑) in one quantity brings about an increase (↑) in the other quantity and similarly a decrease (↓) in one quantity brings about a decrease (↓) in the other quantity, the these two quantities are in

*direct proportion*or

*direct variation*.

In direct proportion the ratio of two quantities (say x and y) is a constant called constant of proportionality.

⇒ y/x = k or y = k × x

**NCERT Chapter Solutions and other Q & A**__NCERT Ex. 13.1__

**Q1: Following are the car parking charges near a railway station upto**

**4 hours Rs 60**

**8 hours Rs 100**

**12 hours Rs 140**

**24 hours Rs 180**

**Check if the parking charges are in direct proportion to the parking time**

Answer: Calculating the ratio of parking charging wrt parting time:

No. of Hours (a) | Parking Charges(B) | Ratio (B/A) |
---|---|---|

4 | 60 | 60/4 = 15:1 |

8 | 100 | 100/8 =25:2 |

12 | 140 | 140/12 =35:3 |

24 | 180 | 180/24=15:2 |

Since the ratio of various parking charging vs parking time is not the same, the parking charges are NOT in direct proportion to the parking time.

**Q2: A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base.**

**In the following table, find the parts of base that need to be added.**

Parts of red pigment | 1 | 4 | 7 | 12 | 20 |

Parts of base | 8 | ... | ... | ... | ... |

Answer: With increase in parts of the red pigments, there will be corresponding increase in parts of the base, since the parts are in direct proportion.

Parts of red pigment | 1 | 4 | 7 | 12 | 20 |

Parts of base | 8 | p_{1} | p_{2} | p_{3} | p_{4} |

p_{1}_{ }8 ⇒ ------- = ----- 4 1 ⇒ p_{1}= 8 × 4 = 32 p_{2}_{ }8 ⇒ ------- = -------- 7 1 ⇒ p_{2}= 8 × 7 = 56 p_{3}8 ⇒ ------- = -------- 12 1 ⇒ p_{3}= 8 × 12 = 96 p_{4}8 ⇒ ------- = -------- 20 1 ⇒ p_{4}= 8 × 20= 160

**Q3: In Question 2 above, if 1 part of a red pigment requires 75 mL of base, how much red pigment should we mix with 1800 mL of base?**

Answer: Let the red pigment in 1800mL of base = p mL

red pigment in base are in direct proportion and are in ratio = 1:75

∴ p:1800 = 1:75

∴ p/1800 = 1/75

⇒ p = 1800/75 = 24

Thus 24 parts of red pigment should be mixed with 1800mL of base.

**Q4: A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?**

Answer: A machine in soft drink factory fills 840 bottles in six hours.

Less hours means less number of bottles and more hours means more bottles will be filled in.

∴ Number of hours is in direct proportion with number of bottles.

Let number of bottles to be filled in five hours = n

⇒ n:5 = 840:6

⇒ n/5 = 840/6

⇒ n = 840 × 5/6 = 700

Thus in five hours, 700 bottles will be filled in.

**Unitary Method:**

In 6 hours, the machine fills in = 840 bottles

In 1 hours, the machine fills in = 840/6

In 5 hours, the machine fills in = 840 × 5/6 = 700

Thus in five hours, 700 bottles will be filled in.

**Q5: A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm as shown in the diagram. What is the actual length of the bacteria? If the photograph is enlarged 20,000 times only, what would be its enlarged length?**

Answer: No. of times the photograph is enlarged, more will the length of the photograph.

⇒ Length of photograph is in direct proportion with Enlargement of the photograph.

⇒ Length of photograph(l) ∝ no. of times photo is enlarged.

⇒ 5cm ∝ 50000

or 5:50000 = x:20000

∴ x = 20000 × 5 / 50000 = 2 cm

∴ Photograph when enlarged 20,000 times will have length of 2cm.

**Q6: In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 12 m high. If the length of the ship is 28 m, how long is the model ship?**

Answer: The model dimensions are in direction proportion to actual dimensions of the ship.

⇒ length of mast in model:length of mast of ship = length of model ship(x) : length of actual ship

⇒ 9:12 = x:28

⇒ x = 28 × 9 /12 = 21 cm

Thus length of model ship is 21 cm.

**Q7: Suppose 2 kg of sugar contains 9 × 10**

^{6}crystals. How many sugar crystals are there in**(i) 5 kg of sugar?**

**(ii) 1.2 kg of sugar?**

Answer: More the sugar weighs, higher the number of sugar crystals.

⇒ number of sugar crystals ∝ weight of the sugar.

(i)

⇒ 2: 9 × 10

^{6}= 5:n

⇒ 2/(9 × 10

^{6}) = 5/n

⇒ n = 9 × 10

^{6}× 5 /2 = 22.5 × 10

^{6}= 2.25 × 10

^{7}... answer

(ii) Similarly,

2: 9 × 10

^{6}= 1.2:n

⇒ n = 9 × 10

^{6}× 5 /1.2 = 5.4 × 10

^{6}... answer

**Q8:Rashmi has a road map with a scale of 1 cm representing 18 km. She drives on a road for 72 km. What would be her distance covered in the map?**

Answer: Road map scale is in direct proportion to the actual road length.

Let the map length for 72km long road is x cm.

∴ 1cm:18km = x:72 km

⇒ 1/18 = x/72

⇒ x = 72/18 = 4 cm

Thus the map distance is 4 cm for the equivalent road distance of 72 km.

**Q9: A 5 m 60 cm high vertical pole casts a shadow 3 m 20 cm long. Find at the same time**

**(i) the length of the shadow cast by another pole 10 m 50 cm high**

**(ii) the height of a pole which casts a shadow 5m long.**

Answer: (i) Let the length of shadow cast by another pole = x m

Since Shadow length ∝ pole length

∴ 3.20 : 5.60 = x : 10.50

⇒ 3.20/5.60 = x / 10.50

⇒ x = 10.50 × 3.20 / 5.60 = 6

∴ Length of shadow of pole of length 10.50 m is 6 m

(ii) Let the height of the pole = y m

Since Shadow length ∝ pole length

∴ 3.20 : 5.60 = 5 : y

⇒ y = 5 × 5.60 / 3.20 = 8.75m

∴ Height of the pole is 8 metres and 75 centimetres.

**Q10: A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far**

can it travel in 5 hours?

can it travel in 5 hours?

Answer: Let the distance travelled by the truck in 5 hours =

*d*km

5 hours = 5 × 60 min = 300 min

Distance travelled ∝ Time

∴ 14 km :25 min = d : 300 min

⇒ 14/25 = d / 300

⇒ d = 14 × 300/25 = 168 km

Thus the distanced travelled in 5 hrs is 168 km.

__Miscellaneous Q & A__

**Q11: If the cost of 16 pencils is ₹ 48, find the cost of 4 pencils.**

Answer: If number of files decreases, cost also decreases. It means number of pencils and their cost are in direct proportion.

∴ 16 : 48 = 4 : x

⇒ 16/48 = 4/x

⇒ x = 48 × 4 / 16 = ₹ 12

.

good

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