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Direct Variation can be defined as the relationship between the two variables, where one variable is the constant multiple of the other variable. Let us understand the concept of direct Variation by the help of an example so the general equation of the direct Variation is expressed as
y = Kx
where y and x are the two variables and k being the constant. It is a type of proportionality where the change in one variable or quantity will bring the change in the other variable or quantity.
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Key Terms: Direct variation, inverse variation, variables, constant, proportionality, linear equation.
Direct Variation Formula
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In a direct Variation Formula there are two variables say 'x' and 'y' and one constant say 'k'. The general equation of direct Variation is x= ky. As you look at this equation you can observe that if you increase the value of x the value of y will also increase and vice versa. If you decrease the value of y the value of x will also decrease and vice versa. This is what the direct Variation Formula depicts. Whereas the value of K remains the same throughout the problem as K is a constant value. In the direct Variation problem ' x ' and ' y ' are said to vary directly i.e they are directly proportional to each other if one variable is increased or decreased the other variable will also increase or decrease. On the other hand ' K ' is known as the constant variation.
General equation is given by,
x = ky
Where,
' x ' and ' y ' are two variables
K is a constant.
Direct Variation Graph
As it is clear from the above equation that it is a linear equation hence the graph for direct Variation will be a straight line.
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Direct Variation Graph
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The graph of two variables in a direct variation will result in a straight line. Therefore, direct variation is a linear equation in two variables. The linear equation is written as y = kx. The ratio of change y/x is also equal to k. i:e y/x = k This change in the equation represents the slope of the line which is a straight line graph.
Y varies directly as x
Y is directly proportional to x
i.e y = Kx
Where k is a constant as we have removed the proportionality sign.
Hence the graph is given as
Direct Variation Graph
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Examples of Direct Variation Formula
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With the help of a few examples let's try to understand the direct variation formula.
| Example 1: The quantity of a wooden box made is directly proportional to the number of wooden blocks. The number of wooden blocks needed for 40 boxes is 160. How many wooden blocks are needed for a box? Solution: In the given example problem above, It is given that Number of wooden blocks needed for 40 boxes = 160 Y = 160 Number of boxes = 40 X= 40 Number of wooden blocks needed for a box = k By applying the direct variation formula, y = k * x 160 = k * 40 k = 160/40 k = 4 Hence the number of wooden blocks needed for a box is 4. Example 2: A recipe for baking 6 muffins needs 1 cup of flour. The number of muffins that can be made varies directly with the quantity of the flour. How many muffins can be made with 4 cups of flour? Solution: let us suppose X = quantity of flour Y = number of muffins As we know y varies directly with x, Therefore by using direct Variation formula Y = Kx X = 1 ( quantity of flour ) Y = 6 ( number of muffins ) Substituting in the direct variation formula 6 = k * 1 K = 6 Therefore, y = 6x Which is the direct variation equation X = 4 Y = 6 * 4 Y = 24 Hence with 4 cups of flour the amount of muffins that can be baked are 24. |
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Things to Remember
- Direct Variation Formula there are two variables say 'x' and 'y' and one constant say 'k'.
- The general equation of direct Variation is Y = kX.
- The graph of direct variation is a straight line.
- If the quantity of one variable increases or decreases the value of other variable will also increase or decrease.
- K is also known as the constant variation.
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Sample Questions
Ques: Find y. given that y varies directly as x, with constant of variation value given as k = 1/5 , x = 20. (2 marks)
Ans: It is given that k = 1/5
x = 20
By applying direct variation formula
y = k x,
y = 1/5 * 20
y = 4
Ques: Find the value of k. when x = 2 and y= 24 given that y varies directly as x. (2 marks)
Ans: it is given that
x = 2
y = 24
by applying the direct variation formula
y = kx
24 = k x 2
k = 24/2
k= 12
hence the answer
Ques: Find the value of y when x = 150. given that y is directly proportional to x and x = 5 and y = 30. (3 marks)
Ans: We know that direct variation formula
y = k x
it is given that the values of x = 5 and y = 30 substituting the values in the direct variation formula
y = k x
30 = k x 5
k = 30/5
k= 6
therefore the equation is y = 6x. now substituting the value of x = 150 to find the value of y
y = 6 x
y = 6 * 150
y = 900
Ques: The quantity of an iron rod made is directly proportional to the number of iron rods. The number of iron rods needed for 50 boxes is 200. How many iron rods are needed for a box? (3 marks)
Ans: In the given example problem above,
It is given that
Number of iron rods needed for 50 boxes = 200
Y = 200
Number of boxes = 50
X= 50
Number of iron blocks needed for a box = k
By applying the direct variation formula,
y = k * x
200 = k * 50
k = 200/50
k = 4
Hence the number of wooden blocks needed for a box is 4.
Ques: Find the value of k. when x = 8 and y= 64 given that y varies directly as x. (2 marks)
Ans: It is given that
x = 8
y = 64
by applying the direct variation formula
y = kx
64 = k x 8
k = 64/8
k= 8
hence the answer
Ques: A recipe for baking 8 cupcakes needs 2 cups of flour. The number of cupcakes that can be made varies directly with the quantity of the flour. How many cupcakes can be made with 6 cups of flour? (3 marks)
Ans: Let us suppose
X = quantity of flour
Y = number of cupcakes
As we know y varies directly with x,
Therefore by using direct Variation formula
Y = Kx
X = 2( quantity of flour )
Y = 8 ( number of cupcakes )
Substituting in the direct variation formula
8 = k * 2
K = 4
Therefore, y = 4x
Which is the direct variation equation
X = 6
Y = 4 * 6
Y = 24
Hence with 6 cups of flour the amount of cupcakes that can be made are 24.
Ques: find the value of y when x = 110. given that y is directly proportional to x and x = 2 and y = 40. (3 marks)
Sol: We know that direct variation formula
y = k x
it is given that the values of x = 2 and y = 40 substituting the values in the direct variation formula
y = k x
40 = k x 2
k = 40/2
k= 20
therefore the equation is y = 20x. now substituting the value of x = 110 to find the value of y
y = 20 x
y = 20 * 110
y = 2200
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