If you could exactly afford either 4 units of x and 24 units of y, or 9 units of x and 4 units of y, then if you spent all of your income on y, how many units of y could you buy?
A)40
B)20
C)60
D)13
E)None of the above.

Question

Quizplus · Accepted Answer

The problem can be solved using the concept of linear equations. Let the price of one unit of x be $p_x$ and the price of one unit of y be $p_y$. According to the problem, we have two equations based on the given scenarios:1. $4p_x + 24p_y = I$ (where $I$ represents the total income)2. $9p_x + 4p_y = I$We need to find the value of $y$ when all income is spent on $y$, which means finding $I/p_y$.By solving the two equations, we can eliminate $p_x$ to find a relation between $p_y$ and $I$. Subtracting the second equation from the first gives us:$4p_x + 24p_y - (9p_x + 4p_y) = 0$$-5p_x + 20p_y = 0$However, to directly find the number of units of $y$ that can be bought with all income without explicitly solving for $p_x$ and $p_y$, we observe the ratios of $x$ and $y$ in both purchasing options. The key is to find a common expenditure in terms of $y$ that could be derived from both options, which simplifies to finding the total units of $y$ that could be bought with the total income.Given the linear nature of the problem and the fact that the total expenditure in both cases is equal (since it equals the total income), we can set up a proportion based on the given options to find how many units of $y$ could be bought if all income is spent on $y$. However, the direct approach involves understanding that the total income can buy either combination of $x$ and $y$, and we need to find an equivalent value in $y$ only.By analyzing the given options and understanding that the total income is fixed, we can deduce that the total income can buy 40 units of $y$ without explicitly solving the equations, as it's the only option that fits the given conditions when considering the linear relationship between the price of $x$ and $y$ and their quantities. This is a more intuitive approach rather than a detailed algebraic solution, which would involve solving for $p_x$ and $p_y$ explicitly.

If You Could Exactly Afford Either 4 Units of X