Bunuel wrote:

If x is a number satisfying 2 < x < 3 and y is a number satisfying 7 < y < 8, which of the following expressions will have the largest value?

A. \(x^2y\)

B. \(xy^2\)

C. 5xy

D. \(\frac{4x^2y}{3}\)

E. \(\frac{x^2}{y}\)

You can plug in values. There are three choices:

1) mixed (improper) fractions, e.g. for x, \(\frac{5}{2}\);

2) decimals, e.g. for x, 2.5 (IMO easier than fractions); or

3) whole numbers, by far the easiest choice. This choice violates the stated conditions.

But all the operations in the answer choices involve multiplication and/or division.

In such cases, whole numbers and improper fractions greater than one all behave the same way (e.g., when squared, their value is greater).

Pick x and y values both on the high end or low end. (It might not matter, but it's safer.)

Let x = 2

Let y = 7

A. \(x^2y = (4 * 7) = 28\)

B. \(xy^2 = (2 * 49) = 98\)

C. \(5xy = (5)(14) = 70\)

D. \(\frac{4x^2y}{3} =

\frac{(4)(4)(7)}{3}\) = 37 + a little

E. \(\frac{x^2}{y} =

\frac{4}{7}\)

These answers are not close. The greatest value is

Answer B

II. Decimal values - in case there's doubt: answers for x = 2.1, y = 7.1

A. \(x^2y\) = 28 + a little

B. \(xy^2\) = 100 + a little

C. 5xy = about 75

D. \(\frac{4x^2y}{3}\) = = a little less than 40

E. \(\frac{x^2}{y}\)= about \(\frac{4}{7}\)

The whole numbers and the decimals behaved exactly the same way. These numbers are not close either.

Answer B

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