M40-27

Official Solution:


At a market, Emily bought oranges and apples. Each orange cost $1 and each apple cost $1.50. What was the ratio of the number of oranges to the number of apples?

Let the number of apples and oranges bought be \(x\) and \(y\), respectively. The question asks for the value of \(\frac{x}{y}\).

(1) The ratio of the total cost of all the oranges to the total cost of all the apples was 2 to 3.

This implies that \(\frac{1x}{1.5y} = \frac{2}{3}\). Multiplying this by 1.5 gives \(\frac{x}{y} = 1\). Sufficient.

(2) The combined cost of all the oranges and apples was $10.

This implies that \(x + 1.5*y = 10\). Multiplying by 2 gives \(2x + 3y = 20\). Don't rush to conclude that this is automatically insufficient, since we have one equation and two variables. We should still check because \(x\) and \(y\) must be integers here, so it's possible for the equation to have only one set of \((x, y)\) satisfying it. However, this is not the case for this specific equation. By trial and error, we can find that three sets of \((x, y)\) satisfy \(2x + 3y = 20\): (7, 2), (4, 4), and (1, 6). Not sufficient.


Answer: A