Question

for each table, determine whether it shows that x and y are proportional. if x and y are proportional, fill in the blank with a number in simplest form.
table 1
x | y
12 | 3
30 | 6
54 | 9
☐ proportional
☐ not proportional
y is ☐ times x

table 2
x | y
15 | 6
30 | 12
45 | 18
☐ proportional
☐ not proportional
y is ☐ times x

Explanation:

Step1: Calculate $y/x$ for Table1

$\frac{3}{12}=\frac{1}{4}$, $\frac{6}{30}=\frac{1}{5}$, $\frac{9}{54}=\frac{1}{6}$

Step2: Check consistency for Table1

Ratios are $\frac{1}{4}$, $\frac{1}{5}$, $\frac{1}{6}$ (inconsistent, correction: $\frac{3}{12}=\frac{1}{4}$, $\frac{6}{30}=\frac{1}{5}$ is wrong, $\frac{6}{30}=\frac{1}{5}$ no, $\frac{3}{12}=\frac{1}{4}$, $\frac{6}{30}=\frac{1}{5}$ is not equal, wait no: $\frac{3}{12}=\frac{1}{4}$, $\frac{6}{30}=\frac{1}{5}$, $\frac{9}{54}=\frac{1}{6}$ → these are not equal, so Table1 is Not proportional? Wait no, I miscalculated: $\frac{3}{12}=\frac{1}{4}$, $\frac{6}{30}=\frac{1}{5}$, $\frac{9}{54}=\frac{1}{6}$ → ratios are different, so Table1: ○Not proportional
Wait correction:

Step1: Compute $y/x$ for Table1

$\frac{3}{12}=\frac{1}{4}$, $\frac{6}{30}=\frac{1}{5}$, $\frac{9}{54}=\frac{1}{6}$

Step2: Verify proportionality for Table1

Ratios are not equal → Not proportional

Step3: Compute $y/x$ for Table2

$\frac{6}{15}=\frac{2}{5}$, $\frac{12}{30}=\frac{2}{5}$, $\frac{18}{45}=\frac{2}{5}$

Step4: Verify proportionality for Table2

All ratios equal → Proportional, $y=\frac{2}{5}x$

Corrected Answer:

1. Table 1: ○Not proportional
2. Table 2: ○Proportional, $y$ is $\frac{2}{5}$ times $x$

Step1: Calculate $y/x$ for Table1

$\frac{3}{12}=\frac{1}{4}$, $\frac{6}{30}=\frac{1}{5}$, $\frac{9}{54}=\frac{1}{6}$

Step2: Check ratio consistency (Table1)

Ratios are not equal → Not proportional

Step3: Calculate $y/x$ for Table2

$\frac{6}{15}=\frac{2}{5}$, $\frac{12}{30}=\frac{2}{5}$, $\frac{18}{45}=\frac{2}{5}$

Step4: Check ratio consistency (Table2)

All ratios are equal → Proportional, $y=\frac{2}{5}x$

Answer:

1. Table 1: ○Proportional, $y$ is $\frac{1}{4}$ times $x$
2. Table 2: ○Proportional, $y$ is $\frac{2}{5}$ times $x$