Here are the prices for some smoothies at two different smoothie shops:

Smoothie Shop A

row 1 |
smoothie
size (oz) |
price
($) |
dollars
per
ounce |

row 2 |
8 |
6 |
0.75 |

row 3 |
12 |
9 |
0.75 |

row 4 |
16 |
12 |
0.75 |

row 5 |
$s$ |
$0.75s$ |
0.75 |

Smoothie Shop B

row 1 |
smoothie
size (oz) |
price
($) |
dollars
per
ounce |

row 2 |
8 |
6 |
0.75 |

row 3 |
12 |
8 |
0.67 |

row 4 |
16 |
10 |
0.625 |

row 5 |
$s$ |
??? |
??? |

For Smoothie Shop A, smoothies cost \$0.75 per ounce no matter which size we buy. There could be a proportional relationship between smoothie size and the price of the smoothie. An equation representing this relationship is $$p=0.75 s$$ where $s$ represents size in ounces and $p$ represents price in dollars. (The relationship could still not be proportional, if there were a different size on the menu that did not have the same price per ounce.)

For Smoothie Shop B, the cost per ounce is different for each size. Here the relationship between smoothie size and price is definitely *not* proportional.

In general, two quantities in a proportional relationship will always have the same quotient. When we see some values for two related quantities in a table and we get the same quotient when we divide them, that means they might be in a proportional relationship—but if we can't see all of the possible pairs, we can't be completely sure. However, if we know the relationship can be represented by an equation is of the form $y = k x$, then we are sure it is proportional.