8 Math Direct And Inverse Proportion

Finish Line & Beyond

Direct & Inverse Proportions Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. That is if

x = k [k is a positive number], then x and y are said to vary directly. In y

such a case if y1, y2 are the values of y corresponding to the values x1, x2 of x respectively then

x1 x 2 = y1 y 2

2. Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy = k, then x and y are said to vary inversely. In this case if y1, y2 are the values of y corresponding to the values x1, x2 of x respectively then x1 y1 = x2 y2 or,

x1 y1 = x2 y 2

Exercise 1 1. Following 4 hours 8 hours 12 hours 24 hours

are the car parking charges near a railway station upto Rs 60 Rs 100 Rs 140 Rs 180

Check if the parking charges are in direct proportion to the parking time. Answer: The charges are not increasing in direct proportion to the parking time because

x1 x 2 4 8 ≠ ⇒ ≠ y1 y 2 60 100 2. 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 Parts of base

1 8

4 ?

7 ?

12 ?

20 ?

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Finish Line & Beyond x1 1 = y1 8 1 x2 4 = = 8 y2 y2 y 2 = 4 × 8 = 32 y 3 = 7 × 8 = 56 y 4 = 12 × 8 = 96 y 5 = 20 × 8 = 160

Answer: k=

⇒ ⇒ ⇒ ⇒ ⇒

3. 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: x1=1 and y1 = 75 X2 = ? and y2 = 1800

x1 x 2 = y1 y 2 1 x2 ⇒ = 75 1800 1800 ⇒ x2 = = 24 75 4. A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours? Answer: x1=840 and y1=6 X2=? And y2=5

x1 x 2 840 x 2 840 × 5 = ⇒ = ⇒ x2 = = 700 y1 y 2 6 5 6 Alternate method:

 In 6 hours the machine fills 840 bottles 840 ∴ In 1 hour the machine fills bottles 6 840 ∴ In 5 hours the machine fills × 5 = 700 bottles 6 5. A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm. What is the actual length of the bacteria? If the photograph is enlarged 20,000 times only, what would be its enlarged length?

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Finish Line & Beyond 5 cms 50000 5 × 20000 = 2 cms Size after 20,000 times enlargement: 50000 Answer: Original Size of the bacteria =

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? Answer: x1= 9 cm and y1 = ? X2= 12 and y2 = 28

x1 x 2 9 12 9 × 28 = ⇒ = ⇒ y1 = = 21 cms y1 y 2 y1 28 12 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?

 2 kg sugar contains 9 × 106 crystals 9 × 106 ∴ 1 kg sugar contains crystals 2 9 × 106 ∴ 5 kg sugar contains × 5 = 45 × 53 = 360 crystals 2 9 × 106 ∴ 1.2 kg sugar contains × 1.2 = 10.8 × 53 = 572.4 crystals 2 Answer:

8. 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: x1=1 and y1 = 18 X2 = ? and y2 = 72

x1 x 2 1 x2 72 = ⇒ = ⇒ x2 = = 4 cm y1 y 2 18 72 18 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 5m long. Answer: x1 = 5.6, x2 = 10.5 and x3 = ? Y1 = 3.2, y2 = ? and y3 = 5

x1 x 2 x3 = = y1 y 2 y3

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Finish Line & Beyond 5.6 10.5 = 3.2 y2 10.5 × 3.2 ⇒ y2 = = 6 cm 5.6 5.6 x3 = (ii) ⇒ 3.2 5 5.6 × 5 ⇒ x3 = = 8.75 m 3.2 (i)

⇒

10. A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours? Answer: x1= 14 and y1 = 25 minutes X2 = ? and y2 = 300 minutes

x1 x 2 = y1 y 2 14 x2 14 × 300 ⇒ = ⇒ x2 = = 168 kms 25 300 25

Exercise 2 1. Which of the following are in inverse proportion? (i) The number of workers on a job and the time to complete the job. (ii) The time taken for a journey and the distance travelled in a uniform speed. (iii) Area of cultivated land and the crop harvested. (iv) The time taken for a fixed journey and the speed of the vehicle. (v) The population of a country and the area of land per person. Answer: (i), (iv) and (v) are in inverse proportion 2. In a Television game show, the prize money of Rs 1,00,000 is to be divided equally amongst the winners. Complete the following table and find whether the prize money given to an individual winner is directly or inversely proportional to the number of winners? Number of winners Prize for each winner (in Rs)

1 1,00,000

2 50,000

4 ?

5 ?

8 ?

Answer: Here the constant k = xy = 1,00,000 Hence, x3y3= 1,00,000

⇒ 4 × y 3 = 100000

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10 ?

20 ?

Finish Line & Beyond ⇒ y3 =

100000 = 25000 4

Similarly,

100000 = 5 100000 y5 = = 8 100000 y6 = = 10 100000 y7 = = 20 y4 =

20000 12500 10000 5000

3. Rehman is making a wheel using spokes. He wants to fix equal spokes in such a way that the angles between any pair of consecutive spokes are equal. Help him by completing the following table. Numberof spokes Angle between a pair of consecutive spokes

4 90°

6 60°

8

10

12

Answer: Here the constant k = 4 × 90° = 360° Hence, Angle for 8 spokes =

360° = 45° 8

360° = 36° 10 360° = 30° Angle for 12 spokes = 12 Angle for 10 spokes =

(i) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion? Answer: As with increasing number of spokes the angle is decreasing so they are in inverse proportion. (ii) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes. Answer:

360° = 24° 15

(iii) How many spokes would be needed, if the angle between a pair of consecutive spokes is 40°?

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Finish Line & Beyond 360° = 40° Spokes 360° ⇒ spokes = = 9 40°

Answer:

4. If a box of sweets is divided among 24 children, they will get 5 sweets each. How many would each get, if the number of the children is reduced by 4? Answer: x1=24 and y1 = 5 X2 = 20 and y2 = ? X1y1=x2y2

⇒ 24 × 5 = 20 × y 2 24 × 5 ⇒ y2 = = 6 20 5. A farmer has enough food to feed 20 animals in his cattle for 6 days. How long would the food last if there were 10 more animals in his cattle? Answer:  20 animals can eat for 6 days ∴ 1 animal can eat for 6 × 20 days 6 × 20 ∴ 30 animals can eat for = 4 days 30 6. A contractor estimates that 3 persons could rewire Jasminder’s house in 4 days. If, he uses 4 persons instead of three, how long should they take to complete the job? Answer:  3 persons can do the job in 4 days ∴ 1 person can do the job in 3× 4 days

∴ 4 persons can do the job in

3× 4 = 3 days 4

7. A batch of bottles were packed in 25 boxes with 12 bottles in each box. If the same batch is packed using 20 bottles in each box, how many boxes would be filled? Answer: Number of Boxes =

25 × 12 = 15 (use x1y1=x2y2) 20

8. A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days?

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Finish Line & Beyond Answer: 42 × 63 = 54 × machines

⇒ Machines =

42 × 63 = 49 54

9. A car takes 2 hours to reach a destination by travelling at the speed of 60 km/h. How long will it take when the car travels at the speed of 80 km/h? Answer: 2 × 60 = t × 80

⇒ t=

2 × 60 = 1.5 hours 80

10. Two persons could fit new windows in a house in 3 days. (i) One of the persons fell ill before the work started. How long would the job take now?

2 × 3 = 1 × days 2× 3 ⇒ days = = 6 1

Answer:

(ii) How many persons would be needed to fit the windows in one day? Answer: Using same calculation 6 persons are required to fit the windows in 1 day. 11. A school has 8 periods a day each of 45 minutes duration. How long would each period be, if the school has 9 periods a day, assuming the number of school hours to be the same?

period1 × duration1 = period 2 × duration 2 8 × 45 = 40 minutes ⇒ 8 × 45 = 9 × duration 2 ⇒ duration 2 = 9

Answer:

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