**SAT Maths: What is Direct Variation?**

Let the cost of 1 pen be $12. By unitary method, we can find the cost of any number of pens. Clearly, the cost of 2 pens is $ 24; the cost of 3 such pens is $ 36; the cost of 5 such pens is $ 60; the cost of 8 such pens is $ 96 and so on…

**Number of pens (a) Cost in Dollars(b)**

1 12

2 24

3 36

5 60

8 96

We find that:

*The more the number of pens, the greater the cost; *

*Lesser the number of pens, lesser the cost.*

Also, the ratio **a/b = Number of pens/cost in dollars** = 1/12 (constant)

We say that the number of pens varies ** directly** as the cost in dollars.

We may also notice that the ratio of the number of pens is equal to the ratio of their costs.

1:2=12:24, 2:3= 24:36, 3:8= 36:96 etc.

**Direct Variation: Two quantities a and b are said to vary directly if the ratio a/b remains constant.**

The ratio a/b is called the **constant of variation**. In the above example, the constant of variation is 1/12.

**Important Rule of Direct Variation:**

** If two quantities a, b are in direct variation, the ratio of any two values of ‘a’ is equal to the ratio of the corresponding values of ‘b’.**

**Example 1: **

If the cost of 8 books is $ 300, find the cost of 18 such books.

Ans: Let the cost of 18 books be Rs x. Then we have:

**Number of Books: 8 18 **

**Cost of Books (in dollars) 300 x **

We know, the more the number of books, the more the cost.

So, it is a case of direct Variation.

Therefore ratio of the number of books = ratio of the cost in dollars.

8:18:: 300: x

Or 8 * x = 18 * 300

Or x = (18 * 300/8) = 675.

Therefore the cost of 18 books is Rs 675.

**Example 2: **

If 30 meters of cloth cost $ 1455, how many meters of it can be bought for $ 679?

Solution: Let the cloth bought for Rs 679 be x meters.

Money (in Dollars) 1455 679

Cloth (in Meters) 30 x

Clearly, less money will fetch less meters of cloth.

So, this is again a case of direct variation.

Therefore, the ratio of the cost in rupees = the ratio of the number of metres of cloth.

So, 1455:679: 30: x

1455*x = 679 * 30

X = (679*30/1455) = 14

Therefore the length of cloth that can be bought for $ 679 is 14 meters.

**Example 3: **

The length of the shadow of a 3-m-high pole at a certain time of the day is 3.6 meter. What is the height of another pole, whose shadow at that time is 54 meters long?

Solution: Let the required height of the pole be x meters.

Then,

Length of the Shadow (in meters) 3.6 54

Height of the pole (in meters) 3 x

Clearly, the longer the shadow, the greater the height of the pole.

This is a case of direct variation.

Therefore ratio of the lengths of the shadows = ratio of the corresponding heights of the poles.

So,

3.6:54:: 3: x

3.6x = 54*3

Or x = (54*3/3.6) = 45

Therefore, the height of the pole is 45 meters.

**What to Remember in Direct Variation: **

**If two quantities increase or decrease together in such a manner that the ratio of their corresponding values remains constant, we say that the two values remain constant, also we say that the two quantities vary directly.**

**In other words, x and y are said to vary directly if x/y=k, where k is a positive constant.**

Here are some more **Practice Sums in Direct Variation** which you can try. The answers are given at the bottom of the post.

**Q1)** Traveling 900 miles by rail costs $ 140. What would be the fare for a journey of 139.5 miles, when a person travels by the same class?

**Q2)** In a hospital, the monthly consumption of milk of 60 patients is 1350 litres. How many patients can be accommodated in the hospital if the monthly ratio of milk is raised to 1710 litres, assuming the quota per head remains the same?

**Q3)** A worker is paid $ 639 for working 6 days. If his total wages during a month are $ 2779, for how many days did he work?

**Q4)** The extension in an elastic string varies directly as the weight hung on it. If a weight of 150 g produces an extension of 2.8 cm, what weight would produce an extension of 19.6 cm?

**Solutions:**

Q1) $ 21.70

Q2) 76

Q3) 26 days

Q4) 1kg 50 g

You can even try the really cool Ratio and Proportion Practice Test to assess yourself more.

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