Substitution Word Problem

**JayRich88** · February 28th 2010, 08:49 AM

I'm fine with regular substitution problems, but this word problem is really bothering me:

A writing workshop enrolls novelists and poets in a ratio of 5 to 3. There are 24 people at the workshop. How many novelists are there? How many poets are there?

I've tried to write down some equations, but they were all wrong. Does n + p = 24?

**NRS** · February 28th 2010, 09:54 AM

n + p = 24 is correct.

to get the others:

(3/5)n = p

(5/3)p = n

**masters** · February 28th 2010, 10:23 AM

Originally Posted by JayRich88

I'm fine with regular substitution problems, but this word problem is really bothering me:

A writing workshop enrolls novelists and poets in a ratio of 5 to 3. There are 24 people at the workshop. How many novelists are there? How many poets are there?

I've tried to write down some equations, but they were all wrong. Does n + p = 24?

Hi JayRich88,

Here's another way to look at this.

The ratio of the number of Novelists (N) to the number of Poets (P) is $\frac{N}{P}=\frac{5}{3}$

The Total (T) in this is 5 + 3 = 8 Novelists and Poets.

Now set up a couple of proportions where Total (T) goes from 8 to 24:

[1] $\frac{N}{T}\Longrightarrow\frac{5}{8}=\frac{N}{24}$

[2] $\frac{P}{T}\Longrightarrow\frac{3}{8}=\frac{P}{24}$

**Soroban** · February 28th 2010, 11:27 AM

Hello, JayRich88!

Yet another way . . . (not too different).

A writing workshop enrolls Novelists and Poets in a ratio of 5 to 3.
There are 24 people at the workshop.
How many novelists are there? How many poets are there?

I've tried to write down some equations, but they were all wrong.

Does $N + P \:=\: 24$ ? . . Yes!

Use the "ratio" information.

We are told: . $N:P \:=\:5:3 \quad\Rightarrow\quad \frac{N}{P} \:=\:\frac{5}{3} \quad\Rightarrow\quad N \:=\:\frac{5}{3}P$

Now substitute that into your equation.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Another approach to Ratio problems . . .

Let $5n$ = number of Novelists.
Let $3n$ = number of Poets.

Then: . $5n+3n \:=\:24 \quad\Rightarrow\quad n = 3$

Therefore: . $\begin{Bmatrix}5n &=&15 & \text{ Novelists} \\ 3n &=& 9 & \text{ Poets} \end{Bmatrix}$

**JayRich88** · February 28th 2010, 11:42 AM

Thank you all. I knew that there would be 15 novelists and 9 poets! I just didn't know I would use only one variable! Thanks

**JayRich88** · February 28th 2010, 11:56 AM

Just so I could get a better understanding, may someone explain the fact that N= 5/3P?

**masters** · February 28th 2010, 12:03 PM

Originally Posted by JayRich88

Just so I could get a better understanding, may someone explain the fact that N= 5/3P?

Sure, JayRich88.

$\frac{N}{P}=\frac{5}{3}$ , right?

Now, just use the cross product rule to get:

$3N=5P$

Finally, divide both sides by 3 and you get:

$N=\frac{5}{3}P$

**JayRich88** · February 28th 2010, 12:31 PM

Ah, I see! Thanks very much.

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