Joint Variation | Solving Joint Variation Problems and Application (2024)

If more than two variables are related directlyor one variable changes with the change product of two or more variables it iscalled as joint variation.

IfX is in joint variation with Y and Z, it can be symbolically written as X α YZ.If Y is constant also then X is in direct variation with Z. So for jointvariation two or more variables are separately in direct variation. So jointvariation is similar to direct variation but the variables for joint variationare more than two.

Equationfor a joint variation is X = KYZ where K is constant.

One variable quantity is said to vary jointly as a number of other variable quantities, when it varies directly as their product. If the variable A varies directly as the product of the variables B, C and D, i.e., if.A ∝ BCD or A = kBCD (k = constant ), then A varies jointly as B, C and D.

For solving a problems related to joint variation first we need to build the correct equation by adding a constant and relate the variables. After that we need determine the value of the constant. Then substitute the value of the constant in the equation and by putting the values of variables for required situation we determine the answer.

If x men take y days to plough z acres of land, then x varies directly as z and inversely as y.

1.The variable xis in jointvariation with yand z. When the values of y and z are 4 and 6, x is 16.What is the value of x when y= 8and z=12?

Solution:

Theequation for the given problem of joint variation is

x =Kyz where K is the constant.

Forthe given data

16 =K×4×6

or, K = \(\frac{4}{6}\).

Sosubstituting the value of K the equation becomes

x =\(\frac{4yz}{6}\)

Nowfor the required condition

x = \(\frac{4 × 8 × 12}{6}\)

= 64

Hencethe value of x will be 64.

2.A is in joint variation with Band square of C. When A = 144, B = 4 and C= 3. Then what is the value ofAwhen B= 6 and C = 4?

Solution:

Fromthe given problem equation for the joint variation is

A = KBC²

From the givendata value of the constant K is

K =\(\frac{BC^{2}}{A}\)

K = \(\frac{4 × 3^{2}}{144}\)

=\(\frac{36}{144}\)

= \(\frac{1}{4}\).

Substitutingthe value of K in the equation

A =\(\frac{BC^{2}}{4}\)

A = \(\frac{6 × 4^{2}}{4}\)

= 24

3. The area of a triangle is jointly related to the height and the base of the triangle. If the base is increased 10%and the height is decreased by10%, what will be the percentage change of the area?

Solution:

We know the area of triangle is half the product of base and height. So the joint variation equation for area of triangle is A = \(\frac{bh}{2}\)where A is the area, b is the base and h is the height.

Here \(\frac{1}{2}\)is the constant for the equation.

Base is increased by 10%, so it will be b x \(\frac{110}{100}\)= \(\frac{11b}{10}\).

Height is decreased by 10%, so it will be h x \(\frac{90}{100}\)= \(\frac{9h}{10}\).

So the new area after the changes of base and height is

\(\frac{\frac{11b}{10} \times \frac{9h}{10}}{2}\)

= (\(\frac{99}{100}\))\(\frac{bh}{2}\)= \(\frac{99}{100}\)A.

So the area of the triangle is decreased by 1%.

4. A rectangle’s length is 6 m and width is 4 m. If length is doubled and width is halved, how much the perimeter will increase or decrease?

Solution:

Formula for the perimeter of rectangle is P = 2(l + w) where P is perimeter, l is length and w is width.

This is joint variation equation where 2 is constant.

So P = 2(6 + 4) = 20 m

If length is doubled, it will become 2l.

And width is halved, so it will become \(\frac{w}{2}\).

So the new perimeter will be P = 2(2l +\(\frac{w}{2}\)) = 2(2 x 6 +\(\frac{4}{2}\)) = 28 m.

So the perimeter will increase by (28 - 20) = 8 m.

●Variation

• What is Variation?
  
• Direct Variation
  
• InverseorIndirect Variation
  
• Joint Variation
  
• Theorem of Joint Variation
  
• Worked out Examples on Variation
  
• Problems on Variation

11 and 12 Grade Math

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