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Mathematics Magazine for Grades 1-12  

 

10/2004
 

 

  

Grade 8

Theory: 

Dividing Fractions by Whole Numbers

To Divide Fractions by Whole Numbers: Treat the integer as a fraction (i.e. place it over the denominator 1) Invert (i.e. turn over) the denominator fraction and multiply the fractions. Multiply the numerators of the fractions. Multiply the denominators of the fractions. Place the product of the numerators over the product of the denominators. Simplify the Fraction

Example: Divide 2/9 by 2. The integer divisor (2) can be considered to be a fraction (2/1). Invert the denominator fraction and multiply (2/9 ÷ 2/1 = 2/9 x  1/2). Multiply the numerators (2 x 1=2). Multiply the denominators (9 x 2=18). Place the product of the numerators over the product of the denominators (2/18). Simplify the Fraction if possible (2/18 = 1/9)

The Easy Way.  After inverting, it is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.

For example: 2/9 ÷ 2 = 2/9 ÷ 2/1 = 2/9 x 1/2 = (2 x 1)/(9 x 2) = (1 x 1)/(9 x 1) = 1/9

Solutions from the Previous Issue:

1.        The sum of two numbers is 825. When you divide the bigger number by the smaller you get 8 and the remainder is 15. Find these two numbers.

Solution:

x + y = 825

x = 8y +15

825 – y = 8y + 15

810 = 9y

90 = y

x = 825 – 90 = 735

1.        (5y - 1)2

Solution:

(5y - 1)2 =(5y –1)(5y –1)=5y(5y –1) –(5y- 1) =25y2 –5y – 5y +1 = 25y2 –10y +1

Proposed Exercises:

1.        An ancient mathematician wrote a puzzle for his tombstone. It went like this: I was a boy for 1/6 of my life, and then a youth for another 1/12. After another 1/7 of my life I married. Five years later I had a son; but alas he died at when his age was half the length of my life. I lived another four years after that. How old was I when I died?

2.        A rectangular parking lot is 80 m by 50 m. How much less, in square meters, is its area than that of a square parking lot with the same perimeter?

Washington State Mathematics Council -2001 Middle School Mathematics Olympiad