Work out the positive value of x for which the ratio x : 45 is equivalent to the ratio 20 : x.

Step-by-step solution

1. 1

   Write both ratios as fractions. Equivalent ratios mean equal fractions:

   $\frac{x}{45}=\frac{20}{x}.$

2. 2

   Cross-multiply. The unknown appears on both sides, which is what makes this a quadratic rather than a one-step problem:

   $x\cdot x=45\cdot 20\ \Longrightarrow\ x^{2}=900.$

3. 3

   Take the square root and select the positive root.

   $x=\pm 30,$

   and the question asks for the positive value, so $x=30$. (The value $-30$ satisfies the algebra but does not describe a sensible ratio.)

4. 4

   Recognise the structure. $x^{2}=45\times 20$ says $x$ is the geometric mean of the two outer terms. Whenever a quantity appears as the second term of one ratio and the first term of an equal ratio, it is the geometric mean of the other two — worth spotting, because it turns the problem into one multiplication and a square root.

5. 5

   Check both ratios reduce to the same thing.

   $30:45=2:3\qquad\text{and}\qquad 20:30=2:3\ \checkmark$

   They match, confirming $x=30$.