A ratio can be rewritten as a quotient; do this, and simplify it.

\(\displaystyle 10 \frac{1}{2} : 1 \frac{1}{2}\)

Rewrite as

 \(\displaystyle 10 \frac{1}{2} \div 1 \frac{1}{2} = \frac{21}{2} \div \frac{3}{2} = \frac{21}{2} \cdot \frac{2}{3} = \frac{7}{1} \cdot \frac{1}{1} = \frac{7}{1}\)

or \(\displaystyle 7:1\)

The ratio of wins to losses requires knowing the number of wins and losses.  The question says that there are 5 wins.  That means there must have been

\(\displaystyle 20-5=15\) losses. 

The ratio of wins to losses is thus 5 to 15 or 1 to 3.

Rewrite in fraction form for the sake of simplicity, then divide each number by \(\displaystyle GCF (90,63) = 9\):

\(\displaystyle \frac{90}{63} = \frac{90\div 9}{63\div 9} = \frac{10}{7}\)

The ratio, in simplest form, is \(\displaystyle 10:7\)

A ratio involving fractions can be simplified by rewriting it as a complex fraction, and simplifying it by division:

\(\displaystyle \frac{\frac{2}{3}}{\frac{5}{12}} = \frac{2}{3} \div \frac{5}{12}\)

Write as a product by taking the reciprocal of the divisor, cross-cancel, then multiply it out:

\(\displaystyle \frac{2}{3} \div \frac{5}{12} = \frac{2}{3} \times \frac{12}{5} = \frac{2}{1} \times \frac{4}{5} =\frac{8}{5}\)

The ratio simplifies to \(\displaystyle 8:5\)

A ratio involving fractions can be simplified by rewriting it as a complex fraction, and simplifying it by division:

\(\displaystyle \frac{\frac{4}{5}}{\frac{3}{10}} = \frac{4}{5} \div \frac{3}{10}\)

Write as a product by taking the reciprocal of the divisor, cross-cancel, then multiply it out:

\(\displaystyle \frac{4}{5} \div \frac{3}{10} = \frac{4}{5} \times \frac{10}{3} = \frac{4}{1} \times \frac{2}{3} =\frac{8}{3}\)

The ratio simplifies to \(\displaystyle 8:3\)

Rewrite in fraction form for the sake of simplicity, then divide each number by \(\displaystyle GCF (72,42) = 6\):

\(\displaystyle \frac{72}{42}= \frac{72\div 6}{42\div 6}= \frac{12}{7}\)

In simplest form, the ratio is \(\displaystyle 12:7\)

Since the answer to this question does not depend on the actual lengths of the sides, we will assume for simplicity that the larger square has sidelength 5; if this is the case, the smaller square has sidelength 3. The areas of the large and small squares are, respectively, \(\displaystyle 5^{2} = 25\) and \(\displaystyle 3^{2} = 9\). 

The white region is the small square and has area 9. The grey region is the small square cut out of the large square and has area \(\displaystyle 25-9=16\). Therefore, the ratio of the area of the gray region to that of the white region is 16 to 9.

Since the answer to this question does not depend on the actual lengths of the sides, we will assume for simplicity that the larger square has sidelength \(\displaystyle 4\); if this is the case, the smaller square has sidelength \(\displaystyle 3\). The areas of the large and small squares are, respectively, \(\displaystyle 4^{2} = 16\) and \(\displaystyle 3^{2} = 9\). 

The white region is the small square and has area \(\displaystyle 9\). The grey region is the small square cut out of the large square and has area \(\displaystyle 16-9=7\). Therefore, the ratio of the area of the gray region to that of the white region is \(\displaystyle 7\) to \(\displaystyle 9\).

Rewrite this in fraction form for the sake of simplicity, and divide both numbers by \(\displaystyle GCF (150,60) = 30\):

\(\displaystyle \frac{150}{60}=\frac{150\div 30}{60\div 30} = \frac{5}{2}\)

The ratio, simplified, is \(\displaystyle 5:2\).

A ratio of fractions can best be solved by dividing the first number by the second. Rewrite the mixed fraction as an improper fraction, rewrite the problem as a multiplication by taking the reciprocal of the second fraction, and corss-cancel: 

\(\displaystyle 6 \frac{2}{5} \div \frac{4}{5} = \frac{32}{5} \div \frac{4}{5} = \frac{32}{5} \cdot \frac{5} {4} =\frac{8}{1} \cdot \frac{1} {1} =\frac{8}{1}\)

The ratio, simplified, is \(\displaystyle 8:1\).