A colony of ants set out in search of food. As they search, they keep splitting equally. Class 7

A colony of ants set out in search of food. As they search, they keep splitting equally. Class 7

Question 1.

A colony of ants set out in search of food. As they search, they keep splitting equally at each point (as shown in the Fig.) and reach two food sources, one near a mango tree and another near a sugarcane field. What fraction of the original group reached each food source? Class 7

Solution:

Fraction for mango tree

= [latex]\frac {1}{2}[/latex] + [latex]\frac {1}{4}[/latex] + [latex]\frac {1}{16}[/latex] + [latex]\frac {1}{16}[/latex] + [latex]\frac {1}{32}[/latex]

= [latex]\frac {16}{32}[/latex] + [latex]\frac {8}{32}[/latex] + [latex]\frac {2}{32}[/latex] + [latex]\frac {2}{32}[/latex] + [latex]\frac {1}{32}[/latex]

= [latex]\frac {16+8+2+2+1}{32}[/latex] = [latex]\frac {29}{32}[/latex]

Fraction for sugarcane field

= [latex]\frac {1}{16}[/latex] + [latex]\frac {1}{32}[/latex] = [latex]\frac {2}{32}[/latex] + [latex]\frac {1}{32}[/latex] = [latex]\frac {3}{32}[/latex]

Another method for sugarcane field :

Fraction = 1 - [latex]\frac {29}{32}[/latex] = [latex]\frac {3}{32}[/latex]

Question 2.

What is 1 - [latex]\frac {1}{2}[/latex]? Class 7

(1 - [latex]\frac {1}{2}[/latex]) × (1 - [latex]\frac {1}{3}[/latex]) ?

(1 - [latex]\frac {1}{2}[/latex]) × (1 - [latex]\frac {1}{3}[/latex]) × (1 - [latex]\frac {1}{4}[/latex]) × (1 - [latex]\frac {1}{5}[/latex]) ?

(1 - [latex]\frac {1}{2}[/latex]) × (1 - [latex]\frac {1}{3}[/latex]) × (1 - [latex]\frac {1}{4}[/latex]) × (1 - [latex]\frac {1}{5}[/latex]) × (1 - [latex]\frac {1}{6}[/latex]) × (1 - [latex]\frac {1}{7}[/latex]) × (1 - [latex]\frac {1}{8}[/latex]) × (1 - [latex]\frac {1}{9}[/latex]) × (1 - [latex]\frac {1}{10}[/latex]) ?

Make a general statement and explain.

Solution:

1 - [latex]\frac {1}{2}[/latex] = [latex]\frac {2}{2}[/latex] - [latex]\frac {1}{2}[/latex] = [latex]\frac {2-1}{2}[/latex] = [latex]\frac {1}{2}[/latex],

(1 - [latex]\frac {1}{2}[/latex]) × (1 - [latex]\frac {1}{3}[/latex]) = [latex]\frac {1}{2}[/latex] × [latex]\frac {2}{3}[/latex] = [latex]\frac {1}{3}[/latex],

(1 - [latex]\frac {1}{2}[/latex]) × (1 - [latex]\frac {1}{3}[/latex]) × (1 - [latex]\frac {1}{4}[/latex]) = [latex]\frac {1}{2}[/latex] × [latex]\frac {2}{3}[/latex] × [latex]\frac {3}{4}[/latex] = [latex]\frac {1}{4}[/latex]

(1 - [latex]\frac {1}{2}[/latex] [latex]\frac {1}{2}[/latex]) × (1 - [latex]\frac {1}{3}[/latex]) × (1 - [latex]\frac {1}{4}[/latex]) × (1 - [latex]\frac {1}{5}[/latex])

= [latex]\frac {1}{2}[/latex] × [latex]\frac {2}{3}[/latex] × [latex]\frac {3}{4}[/latex] × [latex]\frac {4}{5}[/latex] = [latex]\frac {1}{5}[/latex] and so on

(1 - [latex]\frac {1}{2}[/latex]) × (1 - [latex]\frac {1}{3}[/latex]) × (1 - [latex]\frac {1}{4}[/latex]) × (1 - [latex]\frac {1}{5}[/latex]) × (1 - [latex]\frac {1}{6}[/latex]) × (1 - [latex]\frac {1}{7}[/latex]) × (1 - [latex]\frac {1}{8}[/latex]) × (1 - [latex]\frac {1}{9}[/latex]) × (1 - [latex]\frac {1}{10}[/latex])

= [latex]\frac {1}{2}[/latex] × [latex]\frac {2}{3}[/latex] × [latex]\frac {3}{4}[/latex] × [latex]\frac {4}{5}[/latex] × [latex]\frac {5}{6}[/latex] × [latex]\frac {6}{7}[/latex] × [latex]\frac {7}{8}[/latex] × [latex]\frac {8}{9}[/latex] × [latex]\frac {9}{10}[/latex] = [latex]\frac {1}{10}[/latex]

In general,

(1 - [latex]\frac {1}{2}[/latex]) × (1 - [latex]\frac {1}{3}[/latex]) × (1 - [latex]\frac {1}{4}[/latex]) × ....... × (1 - [latex]\frac {1}{n}[/latex]) = [latex]\frac {1}{n}[/latex].