\(\frac { 1 }{ 2 } =\frac { 1\times 4 }{ 2\times 4 } =\frac { 4 }{ 8 } \)
\(\frac { 1 }{ 4 } =\frac { 1\times 2 }{ 4\times 2 } =\frac { 2 }{ 8 } \)
Comparing \(\frac { 4 }{ 8 } ,\frac { 2 }{ 8 } ,\frac { 6 }{ 8 } and\frac { 1 }{ 8 } \). We have
\(\frac { 1 }{ 8 } <\frac { 2 }{ 8 } <\frac { 4 }{ 8 } <\frac { 6 }{ 8 } \).   i.e \(\frac { 1 }{ 8 } <\frac { 1 }{ 4 } <\frac { 1 }{ 2 } <\frac { 6 }{ 8 } \)

\(\frac { 9 }{ 12 } =\frac { 3 }{ 4 } \)
\(\frac { 3 }{ 4 } =\frac { 3\times 3 }{ 4\times 3 } =\frac { 9 }{ 12 } \)
Here \(\frac { 9 }{ 12 } =\frac { 9 }{ 12 } \)
\(\therefore \frac { 9 }{ 12 } =\frac { 3 }{ 4 } \)

\(\frac { 8 }{ 5 } =\frac { 5\times 10 }{ 8\times 10 } =\frac { 50 }{ 80 } \)
\(\frac { 1 }{ 10 } =\frac { 1\times 8 }{ 10\times 8 } =\frac { 8 }{ 80 } \)
Here \(\frac { 50 }{ 80 } >\frac { 8 }{ 80 } \)  
\(\therefore \frac { 5 }{ 8 } >\frac { 1 }{ 10 } \)

Number of working days in an academic year = 220
\(\therefore\) Number of holidays = 365 - 220 = 145
\({Number \ of \ holidays\over Number \ of \ working \ days}={145\over220}={29\over44}\)
Number of holidays: No of working days = 29 : 44

Cost of 15 chairs = Rs. 7500
Cost of 1 chair = \(\frac { 7500 }{ 15 } \) = Rs. 500
For Rs. 500 number of chair purchases = 1
For Rs.12,000 number of chair purchased = \(\frac { 12000 }{ 500 } \) = 24
∴ 24 Chairs can be purchased for Rs.12,000.

75 : 100 
Fraction = \(\frac{75}{100} \div \frac{25}{25}=\frac{3}{4}\)
Simplest form = 3 : 4

32 : 24
Fraction = \(\frac { 32 }{ 24 } =\frac { 32\div 8 }{ 24\div 8 } =\frac { 4 }{ 3 } \)
Simplest form = 4 : 3 

Yes, 12 : 24 : : 18: 36
Because product of extremes = 12 x 36 = 432
Product of means = 24 x 18 = 432
∴ 12 : 24 : : 18 : 36.

\(3:4=\frac{3}{4}\times\frac{2}{2}=\frac{6}{8}\)
\(7:8=\frac{7}{8}\)
Here \(\frac{7}{8}\) larger than \(\frac{6}{8}\)
\(\therefore\) 7 : 8 larger than 3 : 4

\(4:5=\frac{4}{5}\times\frac{3}{3}=\frac{12}{15}\)
\(8:15=\frac{8}{15}\)
Here \(\frac{12}{15}\) larger than \(\frac{8}{15}\) and 4: 5 is larger than 8: 15

Ratio of red beads: blue beads = 35: 30
Fraction of 35 : 30  \(=\frac{35}{30}\)
= 7:6
Also given that the bracelet made is smaller in size
∴ The possible ways are red beads: blue beads
The equivalent ways are
\( \frac{7 \times 2}{6 \times 2}=\frac{14}{12}=14: 12 \)
\( \frac{7 \times 3}{6 \times 3}=\frac{21}{18}=21: 18 \)
\( \frac{7 \times 4}{6 \times 4}=\frac{28}{24}=28: 24\)
= 7 : 6 (or) 14: 12 (or) 21 : 18 (or) 28 : 24
In 4 different way can she made.

Total length 63 cm, on dividing 3 + 4 = 7 equal parts, we get the line segments with lengths
\(\frac{3}{7}\times63=27\ cm\)
\(\frac{4}{7}\times63=36\ cm\)
Length of each line segment is 27 cm & 36 cm