Chapter 13: Direct and Inverse Proportions

Let the number of bottles filled in five hours be x.

Here, the ratio of hours and bottles is in direct proportion.

6x = 5×840

x = 5×840/6 = 700

Hence, the machine will fill 700 bottles in five hours.

5. 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?

Let the enlarged length of bacteria be x.

Actual length of bacteria = 5/50000 = 1/10000  cm = 10^-4  cm

Here, the length and enlarged length of bacteria are in direct proportion.

x= 2cm

Hence, the enlarged length of bacteria is 2 cm.

6. 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?

Let the length of the model ship be x.

Here, the length of the mast and the actual length of the ship are in direct proportion.

x = 21 cm

Hence, the length of the model ship is 21 cm.

7. Suppose 2 kg of sugar contains 9×10⁶ crystals. How many sugar crystals are there in

(i) 5 kg of sugar? (ii) 1.2 kg of sugar?

(i) Let sugar crystals be x.

Here, the weight of sugar and the number of crystals are in direct proportion.

=

Hence, the number of sugar crystals is

(ii)  Let sugar crystals be x.

Here, the weight of sugar and the number of crystals are in direct proportion.

=

Hence, the number of sugar crystals is  5.4×10⁶.

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

Let the distance covered in the map be x.

Here, the actual distance and distance covered in the map are in direct proportion.

x = 4 cm

Hence, the distance covered on the map is 4 cm.

9. 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 5 m long.

Here, the height of the pole and the length of the shadow are in direct proportion.

And 1 m = 100 cm

5 m 60 cm = 5×100+60 = 560 cm

3 m 20 cm = 3×100+20 = 320 cm

10 m 50 cm = 10×100+50 = 1050 cm

5 m = 5×100 = 500 cm

(i) Let the length of the shadow of another pole be x.

x= 600 cm = 6m

Hence, the length of the shadow of another pole is 6 m.

(ii) Let the height of the pole be x.

= 875 cm = 8 m 75 cm

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

10. A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?

Let the distance covered in 5 hours be x km.

1 hour = 60 minutes

Therefore, 5 hours = 5×60 = 300 minutes

Here, the distance covered and time are in direct proportion.

25x = 300(14)

x = 168

Therefore, the truck can travel 168 km in 5 hours.