How many cups/tablespoons are in x% of a quart?

Question

I'm wanting to find an equation for calculating how many cups or tablespoons are in a certain percentage of a quart.

Example question to be solved with the equation: How many cups are in 10% of a quart?

I know there are four cups in a quart. A cup is 25% of a quart.

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Thoughts after accepting an answer—

I got some good answers. I like the equation 0.01xn (or x% of n), derived from some of those answers, where 0.01x is the percentage, and n is the number of units in the other kind of unit (e.g. the number of cups that can fit in a quart or the number of tablespoons that can fit in a gallon). So, if you want to know how many tablespoons are in 7% of a quart, you can do 0.01*7*64. (There are 64 tablespoons in a quart.) So, there are 4.48 tablespoons in 7% of a quart. So, just convert the quart to tablespoons, cups, etc., and take the percentage of that. Kind of obvious when you really think about it.

Anyway, in practice, this Q/A will be helpful for people wanting to lacto-ferment pickles/etc. with a certain percent salt solution (different people suggest a different percent); also, for people wanting to know how much sugar to add to make vinegar (I read that a 10% sugar solution is good); also, for people wanting to mix fertilizer in the right amounts—and so forth. Some fertilizers won't tell you how many tablespoons to add per quart, but they might tell you a percentage, or some other measure. Anyway, there are broad applications, so I figured I'd ask on the Math StackExchange site instead of Gardening/Landscaping or Cooking.

Answer 1

Since 1 quart equals 64 tablespoons (which I looked up just to be sure), you have that the volume in tablespoons of $x\%$ of 1 quart equals $$64\cdot x\% = \frac{64x}{100} = \frac{16x}{25} = 0.64x$$

Answer 2

Hint:

If $4$ cups equals $100\%$ of a quart, then since they are proportional to each other, you have to divide both sides by $k$ so that the right-hand side is $10\%$.

Can you find what $k$ is?

Answer 3

First off, ditch the $\%$, if you can. They will distract you. $25\%$ of a quart means $0.25$ quarts, and $10\%$ of a quart means $0.1$ quarts.

You say in your post that $$ 1\text{ quart} = 4\text{ cup} $$ is something you already know. Here is a great tip that, if you utilise it correctly, will help you in any converting problem you will ever face: Those units in the equation up there ($\text{quart}$ and $\text{cup}$), and any other unit, behaves algebraically just like a single letter. That means you can manipulate expressions that use those words the same way you can any $x$ or $a$ or $n$ you have ever come across. (Just be careful that if you stop midway through a calculation and try to understand what an intermediary expression says, it's not always easy.)

So, by manipulating our equation above, we get $$ 1 = 4\frac{\text{cup}}{\text{quart}} $$ (Don't simplify away the $\text u$ here; it is the entire word as one that behaves like an algebraic letter and may be simplified if possible.) As you probably know, multiplying by $1$ doesn't change the value of anything. The last equation here says that $1$ and $4\frac{\text{cup}}{\text{quart}}$ are exactly the same thing. That means that multiplying by $4\frac{\text{cup}}{\text{quart}}$ doesn't change the value of anything.

Now you are ready to face your problem. You have $0.1\text{ quart}$. That is the same as $1\cdot 0.1\text{ quart}$. As per the above, this is the same as $4\frac{\text{cup}}{\text{quart}}\cdot 0.1\text{ quart}$. Here we can simplify: $\text{quart}$ disappears, and we're left with $4\text{ cup}\cdot 0.1 = 0.4\text{ cup}$.

So we have gone from $0.1\text{ quart}$ to $0.4\text{ cup}$ by only using operations that do not change the value of the expression (multiply by $1$, substitute one part for something which is equal, fraction simplification and finally carrying out the multiplication $4\cdot 0.1$ to get $0.4$). That means that the two must be equal, and we have our answer.

Now, this might seem like a lot of work, but if you do it like this, most of it will be in your head. The solution, as done here, but written on a test, might look something more like this:

We have $$ 1\text{ quart} = 4\text{ cup}\\ 1 = 4\frac{\text{cup}}{\text{quart}} $$ Therefore $$ 0.1\text{ quart} = 4\frac{\text{cup}}{\text{quart}}\cdot 0.1\text{ quart}\\ = 4\text{ cup}\cdot 0.1 = 0.4\text{ cup} $$