If a varies inversely as b, and b is increased by 50%, what is the percentage of change in a?
a = k/b
k/(1.5b) = k/(3/2 b) = 2/3 k/b = 2/3 a
So, the amount has been reduced by 1/3, or 33.3%
To find the percentage of change in a, given that a varies inversely as b and b is increased by 50%, we need to understand the concept of inverse variation and how to work with it.
Inverse variation, also known as inverse proportion, is a relationship between two variables in which they change in opposite directions. Mathematically, this can be represented as:
a = k/b
where 'a' and 'b' are the variables, and 'k' is the constant of variation.
In this case, since we are considering an inverse variation between 'a' and 'b', we can express the relationship as:
a = k/b
To find the percentage of change in 'a', we need to compare the initial value of 'a' with the final value of 'a' after 'b' has increased by 50%.
Let's assume the initial value of 'a' is 'a₀' and the final value is 'a₁'. Similarly, let the initial value of 'b' be 'b₀' and the final value after the increase be 'b₁'.
Since a varies inversely with b, we have:
a₀ = k/b₀ (Equation 1)
After b is increased by 50% (which means b₁ = b₀ + 0.5b₀ = 1.5b₀), we can find the new value of 'a':
a₁ = k/(1.5b₀) (Equation 2)
To calculate the percentage change in 'a', we can use the formula:
Percentage Change = (Final Value - Initial Value) / Initial Value * 100
Substituting the values into the formula, we have:
Percentage Change = (a₁ - a₀) / a₀ * 100
From Equation 1, we can substitute a₀ = k/b₀ into the percentage change formula:
Percentage Change = ([k/(1.5b₀)] - [k/b₀]) / [k/b₀] * 100
Simplifying further:
Percentage Change = [(k(b₀ - 1.5b₀))/(1.5b₀²)] * 100
Percentage Change = [-0.5k/1.5b₀] * 100
Since 'k' and 'b₀' are constants, we can simplify the equation to:
Percentage Change = (-0.5/1.5) * 100
Percentage Change = -33.33...%
Hence, the percentage of change in 'a' when 'b' is increased by 50% is approximately -33.33...%.