Question

find the missing values in the ratio table. then write the equivalent ratios in the order they appear in the table.
calories 20 \square 10 90
miles \frac{1}{6} \frac{2}{3} \square \square
\square : \square
\square : \square
\square : \square
\square : \square

Explanation:

Step1: Find the constant of proportionality

The ratio of Calories to Miles should be constant. For the first pair, Calories = 20, Miles = $\frac{1}{6}$. So the constant $k = \frac{20}{\frac{1}{6}} = 20\times6 = 120$. So the ratio is Calories : Miles = 120 : 1 (since $\frac{Calories}{Miles}=120$).

Step2: Find the missing Calories when Miles = $\frac{2}{3}$

Using $\frac{Calories}{Miles}=120$, Calories = $120\times\frac{2}{3}=80$.

Step3: Find the missing Miles when Calories = 10

Miles = $\frac{10}{120}=\frac{1}{12}$.

Step4: Find the missing Miles when Calories = 90

Miles = $\frac{90}{120}=\frac{3}{4}$.

Step5: Write the equivalent ratios

First ratio: 20 : $\frac{1}{6}$ (simplify by multiplying both by 6: 120 : 1, but as per table order, 20 : $\frac{1}{6}$)
Second ratio: 80 : $\frac{2}{3}$ (multiply both by 3: 240 : 2 = 120 : 1, table order: 80 : $\frac{2}{3}$)
Third ratio: 10 : $\frac{1}{12}$ (multiply both by 12: 120 : 1, table order: 10 : $\frac{1}{12}$)
Fourth ratio: 90 : $\frac{3}{4}$ (multiply both by 4: 360 : 3 = 120 : 1, table order: 90 : $\frac{3}{4}$)

But in simplified equivalent ratios (using the constant 120:1), the ratios are 20 : $\frac{1}{6}$, 80 : $\frac{2}{3}$, 10 : $\frac{1}{12}$, 90 : $\frac{3}{4}$. Or simplifying each ratio:

20 : $\frac{1}{6}$ = (20×6) : 1 = 120 : 1

80 : $\frac{2}{3}$ = (80×3) : 2 = 240 : 2 = 120 : 1

10 : $\frac{1}{12}$ = (10×12) : 1 = 120 : 1

90 : $\frac{3}{4}$ = (90×4) : 3 = 360 : 3 = 120 : 1

So the equivalent ratios (in table order) are:

20 : $\frac{1}{6}$, 80 : $\frac{2}{3}$, 10 : $\frac{1}{12}$, 90 : $\frac{3}{4}$

Answer:

Missing Calories: 80; Missing Miles: $\frac{1}{12}$, $\frac{3}{4}$

Equivalent ratios:
20 : $\frac{1}{6}$
80 : $\frac{2}{3}$
10 : $\frac{1}{12}$
90 : $\frac{3}{4}$

(Or simplified as 120 : 1 for all, but as per table values, the ratios are 20 : $\frac{1}{6}$, 80 : $\frac{2}{3}$, 10 : $\frac{1}{12}$, 90 : $\frac{3}{4}$)