# 8th Class Mathematics Direct and Indirect Variations Inverse Variations

## Inverse Variations

Category : 8th Class

### Inverse Variation

Two quantities are said to be vary inversely if increase in one quantity results in decrease in the other quantity and vice versa.

(i) The time taken to finish a piece of work varies inversely as the number of men at work varies, (more men take less time to finish the job)

(ii) The speed varies inversely as the more time taken to cover a distance (more is the speed less is the time taken to cover a distance.

• In case of direct variation the ratio of the two variables is always constant,
• In case of indirect variation the product of the two variables is always constant.
• The graph of the direct variation is always a straight line in the first quadrant.

• Two quantities vary directly if the rate of increase of one quantity also increases the other quantity.
• Two quantities varies inversely if the rate of increase of one quantity decreases the other.

If 4 women or 3 men earn Rs. 960 in a day, then the earning of 11 women and 7 men in a day will be:

(a) Rs. 4880

(b) Rs. 2200

(c) Rs. 1860

(d) Rs. 1480

(e) None of these

Explanation:

One day earning of 4 women or 3 men = Rs. 480

Therefore, one day earning of 1 women or 1 men $=Rs.\frac{960}{4}or\,Rs.\frac{960}{3}$

One day earning of 11 women or 7 men $=Rs.\frac{960}{4}\times 11\,or\,Rs.\frac{960}{3}\times 7$

= Rs. 2640 or Rs. 2140

Therefore, total earning of 11 women and 7 men for one day is Rs. 2440.

A transport company transports goods from one place to another and charges certain cost for it. A merchant asked to transport 2 quintals of rice from one city to another at a distance of 125 km. The transportation cost of 160 kg of rice from one city to another is Rs. 60. Find the amount the merchant has to pay for the transportation of the entire 2 quintals of rice.

(a) Rs. 75

(b) Rs. 70

(c) Rs. 100

(d) Rs. 125

(e) None of these

Harry wants to mix the flour of two different rates so that he can sell at the rate he wants. In what proportion he must mixes the flour at Rs. 16.6 per kg with a flour at Rs. 16.45 per kg so that the mixture can be sold at the rate of Rs. 16.54 per kg.

(a) 1:3

(b) 2:3

(c) 1:2

(d) 3:2

(e) None of these

James is a milk dealer and usually uses to mix water in milk. He has certain quantity of milk and mixes it with 16 liters of water is worth Rs. 0.9 per liters. If pure milk is worth Rs. 1.08 per liters, how much milk is there in the mixture?

(a) 50 liters

(b) 60 liters

(c) 70 liters

(d) 80 liters

(e) None of these

Richard has to go from Delhi to Mumbai for shopping. He buys 16 bus tickets for himself and his company for Rs. 450. Each first class ticket cost Rs. 50 and each second class ticket cost Rs. 15. Find the cost of another lot of 16 ticket in which the present number of first arid second class tickets are interchanged:

(a) Rs. 480

(b) Rs. 590

(c) Rs. 360

(d) Rs. 485

A university has its own hostel for its students. It provides flooding and lodging to the students. Due to festive season some of the students are left for their home and 100 students stays in the hostel. There is a food provision for 20 days for these students. How long the food will last if 25 more students decide to stay back in the hostel?

(a) 12 days

(b) 13 days

(c) 16 days

(d) 14 days

(e) 16days

Explanation:

Initially the number of students = 100.

Provision for food = 20 days.

Finally number of students = 125.

Number of days food will last $=\frac{100\times 20}{125}=16\,days$

There are two neighbors, one has 6 oxen and other has 8 cows, and they have common field for the grazing of the cattle. They use the field for their cattle turn wise for feeding them. The 6 oxen or the 8 cows can graze the field in 25 days. In how much time will 1 ox and 2 cows graze the same field if left together?

(a) 40 days

(b) 60 days

(c) 50 days

(d) 55 days

(e) None of these

Explanation:

According to the question,

Time taken by 8 cows = Time taken by 6 oxen.

Time taken by 1 cow = Time taken by $\frac{6}{8}$oxen.

Time taken by 2 cows $=\frac{6}{8}\times 2=\frac{3}{2}$oxen.

1 ox and 2 cows = 1 oxen + $\frac{3}{2}$oxen $=\frac{2+3}{2}=\frac{5}{2}$ oxen.

Now, 6 oxen can graze the field in 25 days

1 ox can graze the field in $~\text{25}\times \text{6}=\text{15}0$ days.

$\frac{5}{2}$ oxen (1 oxen and 2 cows) can graze the field in $\frac{150}{2.5}$= 60 days. $\left( \frac{5}{2}=2.5 \right)$

A gardener uses pipes to water his garden from the tank. For his entire garden he uses 6 pipes to water and it takes him 1 hour 20 minutes to completely empty the tank. Now if he uses only 5 pipes to water his garden, how long it will take him to empty the tank?

(a) 1 hour 30 min

(b) 1 hour 36 min

(c) 1 hour 45 min

(d) 1 hour 55 min

(e) None of these

A factory produces fast foods and many others eatables. It got an order from a government agency to manufacture biscuits for the school children. It has 6 machines for the work and has to complete the task in 9 days. The number of machines require to complete the work if they are asked to complete the same work in 18 days.

(a) 3 Machine

(b) 4 Machine

(c) 5 Machine

(d) 5 Machine

(e) None of these