If x is a number satisfying 2 < x < 3 and y is a number satisfying 7 <

To get the largest product, each variable that is multiplied should be largest and the one that divides should be smallest. The largest value x can take is slightly less than 3 and the largest value that y can take is slightly less than 8

A. \(x^2y\)

The largest value of this is about 3*3*8

B. \(xy^2\)

The largest value of this is about 3*8*8

C. 5xy

The largest value of this is about 5*3*8

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

The largest value of this is about 4*3*3*8/3 = 4*3*8

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

The largest value of this is about 3*3/7

A simple comparison shows that (B) is the greatest.

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

First, notice that y is greater than x, and y^2 is significantly greater than x^2. This observation allows us to immediately suspect that B is the correct answer because it is the only choice that contains y^2. We can immediately rule out choices A, C, and E without doing any calculations because each of them contains the relatively small value of x^2, and none of them contain the large value y^2.

Let’s consider choice D, which multiplies the relatively small value of (x^2)(y) by 4/3, which makes it larger. But multiplication by 4/3 is not enough to make (x^2)(y)(4/3) greater than choice B.

Answer: B

I got B, but perhaps I was totally inefficient which might be a cause for my propensity to make lots of stupid mistakes.
-Even though x cannot equal 3 and y cannot equal 8, they can be infinitely close to each number. So I used the outer bounds of each range and plugged them into each number. B produced the largest result.
-For A, I got 72.
-For B, I got 192.
-For C, I got 120
-For D, I got 96,
-E, I didn't calculate because it's some small fraction.

-This problem could have been made a lot harder with the inclusion of negatives as we would have to determine whether certain exponents produce a negative or positive.

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