Question

the equivalent ratios are $20: \frac{1}{6}$, $\square:\frac{2}{3}$ , $10:\frac{1}{12}$ , and $90:\frac{3}{4}$ .

Explanation:

Step1: Find base ratio value

First, calculate the value of the given complete ratio:
$20 \div \frac{1}{6} = 20 \times 6 = 120$
This means all equivalent ratios have a value of 120.

Step2: Solve for the missing numerator

Let the missing number be $x$. We know $x \div \frac{2}{3} = 120$. Rearrange to solve for $x$:
$x = 120 \times \frac{2}{3} = 80$

Step3: Verify other ratios (check validity)

Check the third ratio: $10 \div \frac{1}{12} = 10 \times 12 = 120$ (matches the base value)
Check the fourth ratio: $90 \div \frac{3}{4} = 90 \times \frac{4}{3} = 120$ (matches the base value)

Answer:

The missing number is $\boldsymbol{80}$