Question

If the absolute price of good \(X\) is $$\$ 10$$ and the absolute price of good \(Y\) is $$\$ 14,$$ then what is \((a)\) the relative price of good \(X\) in terms of good \(Y\) and \((b)\) the relative price of good \(Y\) in terms of good \(X ?\)

Short answer

The relative price of good \(X\) in terms of good \(Y\) is \(\frac{5}{7}\) and the relative price of good \(Y\) in terms of good \(X\) is \(\frac{7}{5}\)

Step-by-step solution

Relative price of good X in terms of good Y

To find the relative price of good \(X\) in terms of good \(Y\), divide the absolute price of good \(X\) by the absolute price of good \(Y\). So using the given values: Relative price of \(X = \frac{10}{14}\)

Simplify the fraction

Simplify the fraction from step 1 to its simplest form by dividing both the numerator and the denominator by the greatest common divisor. In this case, the greatest common divisor is 2, so we get: \(Relative price of X = \frac{10/2}{14/2} = \frac{5}{7}\)

Relative price of good Y in terms of good X

To find the relative price of good \(Y\) in terms of good \(X\), divide the absolute price of good \(Y\) by the absolute price of good \(X\). So using the given values: Relative price of \(Y = \frac{14}{10}\)

Simplify the fraction

Simplify the fraction from step 3 to its simplest form by dividing both the numerator and the denominator by the greatest common divisor. In this case, the greatest common divisor is 2, so we get: \(Relative price of Y = \frac{14/2}{10/2} = \frac{7}{5}\)

Key concepts

Understanding Absolute Prices

When we talk about absolute prices, we are referring to the actual dollar amount or cost assigned to a good or service. It's the number you see on the price tag.
For example, in our exercise, the absolute price of good \(X\) is \(\\( 10\) and that of good \(Y\) is \(\\) 14\). These numbers tell us how much you need to pay to purchase each respective item.Absolute prices are straightforward but don't give a full picture when comparing different goods. They only tell you the standalone cost, without information on how prices relate to each other.
This is why understanding relative prices is crucial.When comparing goods, often the relationship between their prices matters more than their actual price tags. That's where relative pricing comes into play.

The Art of Price Comparison

Price comparison is a method that helps you decide which product offers the best value or suits your needs better. Instead of just looking at the absolute prices, it's helpful to compare these prices against each other.
In this concept, we're looking at how one good's price stacks up next to another.For example, in the exercise, to find the relative price of good \(X\) in terms of good \(Y\), we calculated \(\frac{10}{14}\).
When you do this, you're checking how many units of good \(Y\) have the same value as one unit of good \(X\). Finding the relative price of good \(Y\) in terms of \(X\) follows a similar idea, dividing \(\frac{14}{10}\).
By comparing prices like this, you uncover the trade-off between two goods, making it easier to determine what to purchase based on what you value more or what's more economical.

Simplifying Fractions Made Easy

Simplifying fractions is a useful skill in mathematics, especially when dealing with price comparisons. It helps make calculations easier to understand and more accurate.
The goal is to reduce fractions to their simplest form, making them easier to work with. For the relative price of good \(X\), we simplified \(\frac{10}{14}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2 in this case.
Performing this operation gives us \(\frac{5}{7}\), a cleaner and simpler representation.
Similarly, for good \(Y\), we simplified \(\frac{14}{10}\) to \(\frac{7}{5}\), again dividing by the greatest common divisor, which is also 2.This whole process helps not just with mathematical clarity, but also in gaining insights on actual product value comparison. Knowing how to simplify fractions will save time and reduce errors in your calculations, allowing a clearer view of what's really worth more or less compared to something else.