# Proportions

## Exercise 13.2

Question 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

Question 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?

No. of winners | 1 | 2 | 4 | 5 | 8 | 10 | 20 |

Prize for each winner (in Rs.) | 100,000 | 50,000 | ? | ? | ? | ? | ? |

**Answer:** Here the constant `k = xy = 1,00,000`

Hence, `x_3\y_3= 1,00,000`

Or, `4xx\y_3=100000`

Or, `y_3=(100000)/(4)=25000`

Similarly, `y_4=(100000)/(5)=20000`

`y_5=(100000)/(8)=12500`

`y_6=(100000)/(10)=10000`

`y_7=(100000)/(20)=5000`

Question 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.

No. of spokes | 4 | 6 | 8 | 10 | 12 |

Angle between a pair of consecutive spokes | 90^{o} | 60^{o} | ? | ? |

**Answer:** Here, the constant `k=4xx90°=360°`

Hence, angle for 8 spokes `=(360°)/(8)=45°`

Angle for 10 spokes `=(360°)/(10)=36°`

Angle for 12 spokes `=(360°)/(12)=30°`

(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°)/(15)=24°`

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

**Answer:** `(360°)/text(Spokes)=40°`

Or, `text(Spokes)=(360°)/(40°)=9`

Question 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:** `x_1=24` and `y_1 = 5`

`x_2 = 20` and `y_2 = ?`

`x_1y_1 = x_2y_2`

Or, `24xx5=20xxy_2`

Or, `y_2=(24xx5)/(20)=6`

Question 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:** Since 20 animals can eat for 6 days

Hence, 1 animal can eat for `6xx20` days

Hence, 30 animals can eat for `(6xx20)/(30)=4` days

Question 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:** Since 3 persons can do the job in 4 days

Hence, 1 person can do the job in `3xx4` days

Hence, 4 persons can do the job in `(3xx4)/(4)=3` days

Question 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 `=(25xx12)/(20=15`

(Use `x_1y_1=x_2y_2`)

Question 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?

**Answer:** `42xx63=54xx\text(machines)`

Or, `text(machines)=(42xx63)/(54)=49`

Question 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:** `2xx60=t\xx80`

Or, `t=(2xx60)/(80)=1/5` hours

Question 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?

**Answer:** `2xx3=1xx\text(days)`

Or, `text(days)=(2xx3)/(1)=6`

(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.

Question 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?

**Answer:** `text(Period)_1xx\text(Duration)_1= text(Period)_2xx\text(Duration)_2`

Or, `8xx45=9xx\text(Duration)_2`

Or, `text(Duration)_2=(8xx45)/(9)=40` minutes