In this section, we're going to solve some basic percent problems that involve translating statements about percent into equations,
and then solving the equations. There's really only three kind of percent problems, and each of them, each one of them is written right here.
The first is, What number is 25% of 32? The second is, What percent of 36 is 9? And the last is, 32 is 50% of what number?
Now, we're going to take a sentence like this and translate it into an equation and solve it. The translations we use are going to be like this:
is will translate into equal; the word of always translates into multiply;
and then, if we say what number or what percent, we're going to let that be n or x or whatever variable you would like.
These are the three basic percent problems. And as you will see when we go on in this chapter and solve some application problems,
the key will be translating our other applications into one of these basic percent problems and then solving it. But I think you're going to see
that the method of solution for these right here is a very straightforward method that's not too difficult to grasp.
Let's look at the first one: What number is 25% of 32? So, we translate:
"what number" translates into n; "is" translates into "="; 25% as a decimal is 0.25; "of" is multiply; and then 32.
So, the translation is really pretty simple.
It's very direct: what number, that stands for n, okay? so, I have n; "is" translates into =; 0.25; "of" is times; and then, 32.
Now, the reason I write 0.25 here instead of 25% is that I'm going to be doing some arithmetic here,
and I need to be able to work with a decimal; I can't do arithmetic with percents. The next thing is simply to multiply.
When I multiply, I end up with 8. So, n = 8. So, 8 is the number that's 25% of 32. And that makes sense, doesn't it,
because 25 percent is the same as 1/4. And isn't 1/4(32) the same as 8? Let's look at our next problem:
What number is 54% of 38? So, again I just do my translation: "what number", that will be n; "is" translates as "="; 54%, 0.54;
and then, "of" translates to multiply (I'll use parentheses this time) 38. So, n = 0.54(38).
When I actually do that multiplication, and I'll do it on a calculator, I end up with n = 20.52. So, a very straightforward process:
a translation, that gives us an equation to solve; we solve the equation, and that's our solution. Let's take a look at our next example.
Problem 3, we have: what percent of 36 is 9? So, let's just continue our sort of mechanical translation: "what percent" is n;
"of" translates as multiply, 36, (Let's write a 36 instead of a 32); and then, "is" translates as "="; and then, the number 9.
So, I have n(36), let's write that as 36n = 9. Here's an equation to solve. (Because I know n times 36 is the same as 36 times n.)
Divide both sides by 36; I end up with n = 9/36; that's the same as 1/4; so, I get 0.25.
And my problem asked for what percent, so I'm going to change this into a percent, and that's 25%.
So, when it says, what percent, I'm going to let that be the variable n; and I have to remember that I'm looking for a percent.
So, when I end up with n itself, in this case as a decimal, I want to convert it into a percent, so I answer the question correctly.
Now, does that seem to make sense? What percent of 36 is 9? Isn't 9, 25% of 36? It's 1/4 of 36. So, it makes sense intuitively.
Let's look at our next example: 65 is what percent of 130? Let's continue to translate mechanically, here: 65 is what percent, so n; times, of; 130?
So, I end up with 65 = n(130). I'll divide both sides by 130. So, I end up with just n right here as 65/130.
Now, that will come out to be, let's see, it's 65/130 = 1/2. So, I end up with n = 1/2, which is 0.50, which is 50%.
So, solving the equation, that's really the easy part, right here. And in this case,
I could either write the answer as a fraction or as a decimal 0.50, and then, change to a percent
because I'm asked for what percent in the problem, right here. So, that's why I changed.
Instead of using a decimal or a fraction, I write the answer as 50%. Let's try a couple of more of these:
problem 5, we have 8 is 2 percent of what number? So, I'm going to translate 8 is 2 percent is 0.02; "of "means multiply; "what number" is n.
So, I'll just multiply times n. Now, to solve this problem, I'm going to divide both sides by 0.02;
0.02; those will divide out; I will have left just n on the right side, right here. And when I divide 8 by 0.02, I'll get 400. So, n = 400.
And let's see, that seems to make sense: 8 = 2% of 400. Sure. 8 is 2% of 400.
So, I just simply translate into an equation, solve that equation, and there's my result: n = 400.
Here's our last problem: 32 is 50 percent of what number? Well, we could do this in our heads if we have an intuitive idea about percent:
32 is going to be 50 percent of 64. But let's do the translation and see how it looks. 32 is 50 percent is 0.50; "of" means multiply;
and then, what number, that will be n. So, 32 = 0.50 n. Divide both sides by .5, or 0.5, whichever one you want,
32 ÷ 1/2 = 64. So, n = 64. So, even if we don't have an intuitive grasp of a percent,
we're still going to come out with the right answers by doing the translations this way. So, there's a look at a basic percent problems.