Compare Fractions

In her will your late Aunt Charlotte made a provision allowing you to choose the size of her estate you inherit. Your choices are given in percentages represented by the fractions:  5/17, 11/37, 17/68.  The will requires you to make your choice at the reading of the will, without help from anyone and without foreknowledge of the choices. Pencil and paper are the only tools you may use. If you wish to maximize your inheritance, which fraction should you choose? Do you have a strategy?

Comparing familiar numbers seldom present problems. Unfamiliar fractions can stump some. But a calculator easirly eliminates these traumatic decisions. Convert the fractions to decimals then choose the larger decimal. But sometimes this method will not work.  There are fractions a typical calculator renders as equals, when in fact, they are not. Okay, these are not the trusty fractions we see everyday but they do exist.  the post, UNDERSTAND EQUIVALENT FRACTIONS, discusses a method that determines when fractions are equivalent. That same method also determines the larger of two fractions - it compares fractions.

This is a review of the method to demonstrated how  it can be used to compare fractions.

Given fractions 3 / 7 and   8 / 19.

Placed them side-by- side.       3 / 7      8 / 19

Form Products

The first product is called the downward product. It is made with the numerator of the first fraction (3) and the denominator of the second fraction (19).

It is  3 × 19 = 57.

The second product is called the upward product. It is made with the denominator of the first fraction (7) and the numerator of the second fraction (8).

It is  7 × 8 = 56.

Place the downward product under the first fractio and the upper product under the second fraction

3 / 7          8 / 19

 57             56

Put the correct sign  ( < , =, >)  between the products.  In this case put, 57 >  56 .

That is the correct sign to be place between the fractions.

So 3 / 7    >     8 / 19.

Now determine your inheritance choice.

CLOSING NOTE

Given real numbers a, and b. Only one of the statements is true: a  <  b; a = b; or a  >  b. This is called the trichotomy law.

We demonstragted the cross-multiplication procedure can be used to determine which of the trichotomy statements is true when fractions are given.