What is direct and Inverse Variation?
in maths
please give some examples also
Answers
Answer:
Direct variation is a linear function defined by an equation of the form y = kx when x is not equal to zero. Inverse variation is a nonlinear function defined by an equation of the form xy = k when x is not equal to zero and k is a nonzero real number constant.
examples
exactly the opposite:
Answer:
In direct variation, as one number increases, so does the other. This is also called direct proportion: they're the same thing. ... In inverse variation, it's exactly the opposite: as one number increases, the other decreases.
Example 1: A and B can do a particular work in 72 days. B and C in 120 days. A and C in 90 days. In how many days can A alone do the work?
Solution:
Let us say A, B, C can respectively do work alone in x, y, z days
Therefore, In 1 day A, B, C alone can work in 1 / x, 1 / y, 1 / z days
⇒ (A + B) in 1 day can do 1 / x + 1 / y work
∴ (A + B) can do full work in 1 / ([1 / x] + [1 / y]) day
⇒1 / ([1 / x] + [1 / y]) = 72 i.e.,
1 / x + 1 / y = 1 / 72 ———— (1)
similarly, 1 / y + 1 / z =1 / 120 ———- (2)
1 / z + 1 / x = 1 / 90 ——– (3)
From (1) – (2), 1 / x − 1 / z = 1 / 72 − 1 / 120 ——– (4)
1 / x + 1 / z = 1 / 92 ———– (5)
Adding IV & V, 2 / x = 1 / 72 − 1 / 120 + 1 / 90
= [5 − 3 + 4] / [360] = 6 / 360 = 1 / 60
∴ x = 120
Example 2: A and B undertake to do a piece of work for Rs. 600. A alone can do it in 6 days, while B alone can do it in 8 days but with the help of C, they finish it in 3 days. Find the share of C.
Solution:
A, B, C can do (alone) work in 6 days, 8 days, & x days (assume) respectively.
∴ Together, they will do it in 1 / ([1 / 6] + [1 / 8] + [1 / x ]) days.
Now, 1 / [(1 / 6) + (1 / 8) + (1 / x)] = 3
⇒ (1 / 6) + (1 / 8) + (1 / x) = 1 / 3
⇒ 1 / x = 1 / 3 − 1 / 6 − 1 / 8 = 1 / 24
⇒ x = 24 days
Efficiency ratio of A : B : C = [1 / 6] : [1 / 8] : [1 / 24] = 4 : 3 : 1
Share of C = (1 / [4 + 3 + 1]) × 600 = Rs 75
Example 3: 45 men can complete a work in 16 days. Six days after they started working, 30 more men joined them. How many days will they now take to complete the remaining work?
Solution:
45 men, 16 days ⇒ 1 work.
1 man, 1 day ⇒ 1 / [45 × 16] work
For the first 6 days: 45 men, 6 days ⇒ [45 × 16] / [45 × 16] work = 3/8 work
Work left = 1 − ⅜ = ⅝
Now, 45 + 30 = 75 men, 1 man, 1 day = 1 / [45 × 16] = work 75 men,
1 day = 75 / [45 × 16] work 75 men, x days = 75x / [45 × 16] work
But = 75x / [45 × 16] work = 5 / 8
⇒ x = 5 / 8 × [(45 × 16) / 75] = 6 days.
Example 4: Two pipes A and B can fill a tank in 24 min. and 32 minutes respectively. If both the pipes are opened together, after how much time should B be closed so that the tank is full in 18 minutes?
Solution:
Let after ‘t’ minutes ‘B’ be closed
For first ‘t’ minutes part filled= t ∗ [1 / 24 + 1 / 32] = t ∗ 7 / 96
Part left = (1 − 7t / 96) ‘A’ filled 1 /24 part in 1 min ⇒ A fills = (1 − [7t / 96]) part in 24
= (1 − 7t / 96) min
Now, [t + 24] (1 − 7t / 96) = 18
⇒t + 24 − [7 t / 4] = 8
⇒6 = 3t / 4
⇒ t = 18
Example 5: Twenty women can do work in sixteen days. Sixteen men can complete the same work in fifteen days. What is the ratio between the capacity of a man and a woman?
Solution:
One woman can do work in 20 × 16 = 320 days
In 1 day, 1 woman can do (1 / 320) part of work
Similarly, 1 man can do work in 16 × 15 = 240 days
∴ In 1 day, one man can do 1 / 240 part of work
Capacity of man: woman = 1 / 240 : 1 / 320 = 1 / 3 : 1 / 4 = 4 : 3