# Inverse Variation

### Inverse Variation or Inverse Proportion:

Two quantities are said to vary inversely the increase (or decrease) in one quantity causes the decrease (or increase) in the other quantity.

### Examples on Inverse Variation or Inverse Proportion:

(i) The time taken to finish a piece of work varies inversely as the number of men at work. (More men at work, less is the time taken to finish it)

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

### Solved worked-out problems on Inverse Variation:

1. If 35 men can reap a field in 8 days; in how many days can 20 men reap the same field?

Solution:

35 men can reap the field in 8 days

1 man can reap the field in (35 × 8) days   [less men, more days]

20 men can reap the field in (35 × 8)/20 days  [more men, more days]

= 14 days

Hence, 20 men can reap the field in 14 days.

2. A fort had provisions for 300 men for 90 days. After 20 days, 50 men left the fort. How long would the food last at the same rate?

Solution:

Remaining number of men = (300 - 50) = 250.

Remaining number of days = (90 - 20) days = 70 days.

300 men had provisions for 70 days

1 man had provisions for (300 × 70) days   [less men, more days]

250 men had provisions for (300 × 70)/250 days
[more men, more days]

= 84 days.

Hence, the remaining food will last for 84 days.

### More examples on Inverse Variation word problems:

3. 6 oxen or 8 cows can graze a field in 28 days. How long would 9 oxen and 2 cows take to graze the same field?

Solution:

6 oxen = 8 cows

⇒ 1 ox = 8/6 cows

⇒ 9 oxen ≡ (8/6 × 9) cows = 12 cows

⇒ (9 oxen + 2 cows) ≡ (12 cows + 2 cows) = 14 cows

Now, 8 cows can graze the field in 28 days

1 cow can graze the field in (28 × 8) days   [less cows, more days]

14 cows can graze the field in (28 × 8)/14 days
[more cows, less days]

= 16 days

Hence, 9 oxen and 2 cows can graze the field in 16 days.

4. 6 typists working 5 hours a day can type the manuscript of a book in 16 days. How many days will 4 typists take to do the same job, each working 6 hours a day?

Solution:

6 typists working 5 hours a day can finish the job in 16 days

6 typists working 1 hour a day can finish it in (16 × 5) days
[less hours per day, more days]

1 typist working 1 hour a day can finish it in (16 × 5 x 6)/6 days
[less typists, more days]

1 typist working 6 hours a day can finish it in 6 days
[more hours per day, less days]

4 typists working 6 hours a day can finish it (16 × 5 × 6)/(6 × 4) days

= 20 days.

Hence, 4 typists working 6 hours a day can finish the job in 20 days.

Ratio and Proportion (Direct & Inverse Variation)

Direct Variation

Inverse Variation

Practice Test on Direct Variation and Inverse Variation

Ratio and Proportion - Worksheets

Worksheet on Direct Variation

Worksheet on Inverse Variation