Question

If \(y\) varies jointly as \(w\) and \(x,\) by what percent will \(y\) change if \(w\) is increased by \(12 \%\) and \(x\) is decreased by \(7.0 \% ?\)

Short answer

The value of 'y' will change by approximately 4.16%.

Step-by-step solution

Understanding Joint Variation

Joint variation means that one variable is directly proportional to the product of two or more other variables. In this case, since 'y' varies jointly as 'w' and 'x', the relationship can be expressed as the equation 'y = kw x', where 'k' is the constant of proportionality.

Calculating the New Values of 'w' and 'x'

If 'w' is increased by 12%, the new value of 'w' becomes 'w * (1 + 0.12)'. Similarly, if 'x' is decreased by 7%, the new value of 'x' becomes 'x * (1 - 0.07)'.

Finding the New Value of 'y'

Substitute the new values of 'w' and 'x' into the initial equation to find the new 'y' value: 'y_new = k * w * (1 + 0.12) * x * (1 - 0.07)'.

Comparing the Original 'y' to the New 'y'

To find the percent change in 'y', we compare the new 'y' to the original 'y'. Since 'y = kw x', we can see that the change comes only from the factors '(1 + 0.12)' and '(1 - 0.07)', thus simplifying to 'y_new/y = (1 + 0.12) * (1 - 0.07)'.

Calculating the Percent Change in 'y'

Calculate the change in 'y' by evaluating 'Percent Change = ((y_new/y) - 1) * 100%'. Substitute the expression from Step 4 to find 'Percent Change = (((1 + 0.12) * (1 - 0.07)) - 1) * 100%'.

Perform the Calculations

Now perform the multiplication and subtraction to find the percent change: 'Percent Change = ((1 + 0.12) * (1 - 0.07) - 1) * 100% = (1.12 * 0.93 - 1) * 100% = (1.0416 - 1) * 100% = 0.0416 * 100% = 4.16%'.

Key concepts

Proportional Relationships

In mathematics, a proportional relationship is one where two quantities maintain a consistent ratio to one another. If one quantity increases, the other increases at the same rate, and the same goes for decreases.

This foundational concept is best observed in simple scenarios, such as recipes or conversion rates. For instance, if a cake recipe is designed for four people and we want to adjust it for eight, we simply double all ingredients, keeping a proportional relationship between the number of servings and the quantity of each ingredient.

In the context of our exercise, the relationship between the variables 'y', 'w', and 'x' is jointly proportional, which means as 'w' and 'x' change, 'y' changes in a way that keeps the product 'w*x' proportional to 'y'.

Percent Change Calculation

Percent change is a useful statistic that describes the degree of change over time. It helps us understand growth or decline in various contexts, such as population studies, price changes, or in this case, changes in a mathematical function.

To calculate percent change, we use the formula:
\( \text{Percent Change} = \left(\frac{{\text{New Value} - \text{Original Value}}}{{\text{Original Value}}}\right) \times 100\% \)

Applying this formula indicates how much something has increased or decreased in percentage terms. When dealing with joint variation, calculating the percent change allows us to predict how a dependent variable will respond to simultaneous changes in two or more independent variables.

Direct Variation

Direct variation describes a linear relationship between two variables in which they increase or decrease at the same rate. The general equation for direct variation is:
\( y = kx \)
where 'y' is the dependent variable, 'x' is the independent variable, and 'k' is the constant of proportionality.

If 'k' is positive, 'y' increases when 'x' increases; if 'k' is negative, 'y' decreases when 'x' increases. The concept of direct variation extends into joint variation, where a variable varies directly as the product of two or more variables. For instance, if 'y' varies directly with both 'w' and 'x', as in our exercise, this can be expressed as 'y = kwx', indicating the proportionality constant 'k' relates to both 'w' and 'x' simultaneously.