To solve the equation
\[ \frac{5}{b} = \frac{36}{45} \]
we can cross-multiply:
\[ 5 \cdot 45 = 36 \cdot b \]
Calculating the left side:
\[ 5 \cdot 45 = 225 \]
So we have:
\[ 225 = 36b \]
Next, to isolate \(b\), divide both sides by 36:
\[ b = \frac{225}{36} \]
Now we can simplify the fraction. First, we find the greatest common divisor (GCD) of 225 and 36.
225 can be factored as:
\[ 225 = 15 \cdot 15 = 3^2 \cdot 5^2 \]
36 can be factored as:
\[ 36 = 6 \cdot 6 = 2^2 \cdot 3^2 \]
The GCD here is \(9\) (which is \(3^2\)). Now, let's simplify:
\[ b = \frac{225 \div 9}{36 \div 9} = \frac{25}{4} \]
Thus, the solution for \(b\) is:
\[ b = \frac{25}{4} \]
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