Understanding equivalent ratios

We're told that Burger Barn makes dipping sauce by mixing two spoonfuls of honey with one half spoonful of mustard. Sandwich Town makes dipping sauce by mixing four spoonfuls of honey with one spoonful of mustard. Which dipping sauce has a stronger mustard flavor? So, pause this video and see if you can work through that on your own.

All right, now let's think about the ratios of honey to mustard at each of these restaurants. So first, let's think about the scenario with Burger Barn. So I'll just say BB for short for Burger Barn. They have two spoonfuls of honey for every one half spoonful of mustard. So the ratio of honey to mustard in terms of spoonfuls is two spoonfuls of honey for every one half spoonful of mustard.

So this is the ratio of honey to mustard. Let me write this: this is honey, and this right over here is mustard. Now let's look at Sandwich Town, so I'll call that ST. Sandwich Town makes dipping sauce by having four spoonfuls of honey for every one spoonful of mustard. So the ratio of honey to mustard is four spoonfuls to one spoonful. So once again, that is honey, and that is mustard.

Now, can we make these equivalent ratios, or can we compare them somehow? Well, let's see. We have one half spoonful of mustard here; we have one spoon of mustard here. So what if we multiplied both the mustard and the honey spoonfuls by two? That still would be an equivalent ratio because we're multiplying by the same amount. So if we multiply by two in both situations, you have four spoonfuls of honey for every one spoonful of mustard. Well, that's the exact same ratio that we have at Sandwich Town.

So it actually turns out that they have the same concentration of mustard. They have the same ratio of honey to mustard: four spoonfuls of honey for every spoonful of mustard in either situation.

Let's do another example. Here we are told Patrick's favorite shade of purple paint is made with four ounces of blue paint. So let me underline that in blue: four ounces of blue paint for every three ounces of red paint. The ratio of blue paint to red paint is four ounces of blue for every three ounces of red. So four to three. Which of the following paint mixtures will create the same shade of purple?

All right, pause this video and see if you can figure it out on your own. This is three ounces of blue paint mixed with four ounces of red paint. Well, this is a ratio here of three to four, and even though it's dealing with the same numbers, this is a different ratio. The order matters. This is four ounces of blue for every three ounces of red; this is saying three ounces of blue for every four ounces of red. So we could rule this one out.

Eight ounces of blue paint mixed with six ounces of red paint. So here this ratio is eight ounces of blue for every six ounces of red. Well, are these equivalent ratios? Well, the difference, or you can go if you multiply by two in either case, you will get to eight to six. Four times two is eight; three times two is six. So this is indeed an equivalent ratio, so we would select this one.

All right, here they say six ounces of blue paint mixed with eight ounces of red paint. So this is once they've swapped the blues and the red relative to this one. This is a ratio of six to eight. So let me write this down. This is a ratio of six ounces of blue paint for every eight ounces of red paint. So just like we ruled out that first one, this is dealing with the same numbers but in a different order, and the order matters. We'll rule that out.

Twenty ounces of blue paint for every fifteen ounces of red paint. So are these equivalent? Well, let's think about it. To go from four to twenty, you could multiply by five, and to go from three to fifteen you could multiply by five. So we can multiply by the same factor to go from four to three to twenty to fifteen. So this is indeed an equivalent ratio.

Twelve ounces of blue paint mixed with sixteen ounces of red paint. All right, so this is a ratio here of twelve ounces of blue for every sixteen ounces of red. So let's think about this: to go from four to twelve, you would multiply by three. Now, if you multiplied three by three, you would have a nine here, not a sixteen. So this is definitely not an equivalent ratio.

Another way of thinking about it: in terms of ounces, you have more ounces of blue than you have of red for any of the equivalent ratios, but here you have more ounces of red than blue. So once again, another way of realizing that that is not equivalent.

So only B and D are the equivalent mixtures that will provide the same shade of purple. To have that same shade, you need the same ratio of blue to red.