Ratio Tables - Additive and Multiplicative Structure

All summer long we've been baffled by the frozen brews Basti, the barista, has blended. He's just about to add a new concoction to his board called the Berry Brew Blend and we want to know how he creates it. Basti's refreshing brew mixes coffee with caramel, strawberries, and cream. To be able to recreate Basti's blend, we should understand the additive and multiplicative structure of ratio tables. A ratio table is a structured set of equivalent ratios. Let's examine this ratio table that Basti already has showing a 1 to 3 ratio of cream to coffee. This means, for every 1 part cream, he needs 3 parts coffee. What does that mean if he has 4 parts cream? According to the table, he needs 12 parts coffee. Can you describe any patterns you see in this ratio table? Moving across the table, we always multiply by 3. 1 times 3 is 3, 2 times 3 is 6, and so on. Notice that the ratios are equivalent and can be reduced to one third. Let's move vertically down the table. Every time we add one to the cream column, we add 3 to the coffee column. Notice that ratio appears again: add 1, add 3. That's 1 to 3. We're seeing that pattern a lot in this table. So, what happens when a large order comes in for the Berry Brew Blend? We'll need to help Basti build out his table. Remember that 1 to 3 ratio Adding the one vertically along this column and adding the 3 along the coffee column helps us complete the table. We decided to use the additive structure along the columns, but using the multiplicative structure along the rows would still result in an equvailant ratio table. The way you create your ratio tables is up to you just as long as the ratios are equivalent. So now we can see for this large order if Basti uses 9 parts cream he needs 27 parts coffee. That's a lot of coffee, but his concoction doesn't stop there.
It also includes caramels and strawberries. For every 2.5 cubes of caramel Basti adds to his coffee creation, he blends in 10 frozen strawberries. Yum! Using the equivalent ratios 2.5 to 10, or 4 to 16, let's fill in the rest of the table. The reduced ratio is 1 to 4. Again, there are many ways to fill in this table. We can use addition or multiplication, or both. So how do we get from 2.5 caramels to 10 strawberries using multiplication? We multiply by 4. We can see that it works here: 2.5 times 4 is 10, and here: 4 times 4 is 16 We can use this multiplicative structure to complete the table. 5.5 times 4 is 22, 7 times 4 is 28, and 8.5 times 4 is 34. We decided to use the multiplicative structure, but we could also use the additive structure.
To get from 2.5 to 4 we add 1.5 and from 10 to 16 we add 6. We can see the addition of 1.5 works along the caramel column, and adding 6 along the strawberry column also matches our earlier information. Also notice that this ratio of 1.5 to 6 simplifies to 1 to 4 as well. Interesting. To summarize what we've learned. Ratio tables are a structured set of equivalent ratios. There are lots of patterns in a ratio table. We always add the same numbers down each column or multiply across the rows by the same multiplier. You may have noticed the multiplier is the reciprocal of the simplified ratio. Neat, huh? It's really up to you to decide which structure to use! Let's reward ourselves for persevering through ratio tables. Fortunately, Basti's just whipped up a new batch of his Berry Brew Blend for us to try. Oh, be careful, Basti!