Activities/Abacus/Worksheet/Answersheet

Answers to the Abacus Worksheet

Ways to make 1/2

Ways to make 1/2 just using one string

To make 1/2 just using one string, the fraction denominator must be divisible by 2. The string must contain an even number of beads, 2 4 6 8 10 12

Ways to make 1/2 with more than one string

1/3 + 1/6 = (2+1)/6 = 3/6 = 1/2

1/3 + 2/12 = (4+2)/12 = 6/12 = 1/2

1/4 + 1/6 + 1/12 = (3+2+1)/12 = 6/12 = 1/2

1/4 + 2/8 = (2+2)/8 = 4/8 = 1/2

1/4 + 3/12 = (3+3)/12 = 6/12 = 1/2

3/9 + 2/12 = (12+6)/36 = 18/36 = 1/2

2/8 + 3/12 = 1/4 + 1/4 = 1/2

1/6 + 4/12 = (2+4)/12 = 6/12 = 1/2

1/6 + 3/9 = (3+6)/18 = 9/18 = 1/2

1/6 + 2/8 + 1/12 = (4+6+2)/24 = 12/24 = 1/2

2/6 + 2/12 = (2+1)/6 = 3/6 = 1/2

1/5 + 3/10 = (2+3)/10 = 5/10 =1/2

2/5 + 1/10 = (4+1)/10 = 5/10 =1/2

Is this all of the possibilities?

See Diophantine equations, computer programs including ones written in TurtleArt can find all the solutions by trial and error.

You can paste File:Turtle Art Activity fraction diophantine.ta into Turtle Art to calculate all the possibilities

Ways to make 1/3

This program can be modified for 1/3 and 2/3 (and with more work it could be used for the Caacupé abacus). Circled below, the program above has 0.5 replaced with 1/3 to find the ways of making 1/3

4/12

3/9

2/8 + 1/12

1/6 + 2/12

2/6

1/4 + 1/12

1/3