Question

There are 45 boys and 81 girls in a dance competition. What is the ratio of boys to girls, in the simplest form?

Answer 1

To find the ratio of boys to girls, we start with the numbers of boys and girls: there are 45 boys and 81 girls. The ratio can be expressed as:

\[
\text{Ratio of boys to girls} = \frac{45}{81}
\]

Next, we simplify the fraction. We do this by finding the greatest common divisor (GCD) of the two numbers.

First, we can find the GCD of 45 and 81. We can start by finding the prime factorization:

- The prime factorization of 45 is:
\[
45 = 3^2 \times 5
\]

- The prime factorization of 81 is:
\[
81 = 3^4
\]

The GCD will be the product of the lowest powers of all prime factors present in both numbers. The common prime factor is 3, and the lowest power is \(3^2\).

Thus, the GCD of 45 and 81 is \(9\).

Now, we can divide both the numerator and the denominator of the fraction by their GCD:

\[
\frac{45 \div 9}{81 \div 9} = \frac{5}{9}
\]

So, the ratio of boys to girls in the simplest form is:

\[
\boxed{5:9}
\]