Twice a number, n, subtracted from 36 is less than 15 more than one third the number.

Question

 Twice a number, n,  subtracted from 36 less than 15 more than 1/3 the number.

find all possible values for the number.

 

Answer 1

Hi Morgan! Let's start by breaking down the sentence and "translating" it from English into math symbols.

 

I like to start with the "middle" of the statement -- the equals sign, less than sign, greater than sign, or whatever is going to separate our left side from our right. These are pretty easy to find; we just have to look for phrases like "is" or "is equal to", "is less than," or "is greater than." This can be a little tricky sometimes depending on how it's structured, but this problem looks pretty simple:

 

Twice a number, n, subtracted from 36 is less than 15 more than one third the number.

 

Great! So now all we have to do is replace that with the corresponding math symbol:

 

Twice a number, n, subtracted from 36 < 15 more than one third the number.

 

The next step is to find our "action words" so that later we can separate out the pieces based on what we're doing to each piece. "Action words" are anything that implies multiplying, dividing, subtracting, adding, or any other operation.

 

I've found a few here:

 

[Twice] a number, n, [subtracted from] 36 < 15 [more than] [ one third] the number.

 

So I'm seeing it looks like we'll be multiplying ("twice,") subtracting ("subtracted from,"), adding ("more than,") and probably multiplying by a fraction or dividing ("one-third") -- with this last one, note that they didn't just give us the quantity one-third; they told us it would be "one third the number," meaning that there's a secret "of" hanging around in there. This phrase is usually read as "one-third of the number."

 

 

Let's start on the left side:

 

[Twice] a number, n, [subtracted from] 36

 

With these word problems, there's no reason we can't go left to right. Keep in mind, though, that the expression we build might change in ways that aren't strictly left to right; we might add things at the beginning or end, based on what the instructions say to do. So let's start with the first "action" there: "Twice."

 

They're telling us to multiply something by 2. Multiply what by 2? The instructions say "twice a number n", so it looks like we'll multiply that number "n" by 2. Let's write this out in math language:

 

2n subtracted from 36

 

("2n" can also be written 2*n or however you like; I usually use the first format because it visually "glues" the two pieces together more tightly.)

 

Okay, so our first action was to multiply the number n by 2. What's next? "Subtracted from."

 

Note that "from"! It's important, because it tells us which way around our pieces go. What are we subtracting from what? The instructions tell us to subtract something FROM 36, so we know we'll be writing "36 - something."

 

What's the something that we are subtracting? We already saw that we're taking 2n and subtracting it from 36, so let's make sure we have that "2n" where the "something" was:

 

36 - 2n

 

Great! There's the left side translated. We now have:

 

36 - 2n < 15 [more than] [one third] the number

 

Alright, let's tackle the right side. Again, no reason we can't go left-to-right, we just have to be conscious of where things are ending up and making sure we're maintaining our order of operations as instructed.

 

The first action here is "more than." This tells us we'll be adding, and we already know one of the things we'll be adding: 15! So to start off our right side, we can write: 

 

15 + [one third] the number

 

Now we just need to know what we're adding 15 to. Since it's the first thing we did, we'll probably be adding it to the rest of the entire expression. So to make sure I remember this, I'm going to put parentheses around everything else:

 

15 + ( [one third] the number )

 

 

Alright, now we just have to tackle that last operation. It looks a little tricky; if I were to literally just replace English terms with math symbols, though, I just have to make sure that I'm using the same operation (multiplication) that we originally determined we'd need here.

 

First, the "one third":

 

15 + ( [one third] the number )

turns into

15 + ( 1/3 the number )

 

And then, the "the number":

 

15 + ( 1/3 the number )

turns into

15 + (1/3 * n) (remember to keep the multiplication in mind!)

 

 

I find that since I'm writing this in computer text, and don't have the ability to create nicely formatted fractions, it's probably easier to express "1/3 * n" as "n / 3". Does that make sense? We just took multiplication by a fraction, and turned it into a pretty simple division. So then, our right side is:

 

15 + (n/3)

 

 

Alright! And our last step is to put all our pieces together -- the left side, the middle piece that defines the relationship, and the right side:

 

36 - 2n < 15 + (n/3)

 

I think the interpretation and translation from English to math is the hardest part for most people, so I'll leave it there for now. If you need help following through and completing the problem to solve for all values of x, let me know and I'll help out!