Question

Use a common denominator to determine whether the ratios form a proportion.

1/8 and 3/20

Answer 1

What is a number which is a multiple of both 8 and 20?

Answer 2

LCM would be 40.

Answer 3

Good. Now what do you have to do to make \[\frac{1}{8} = \frac{x}{40}\]?

Answer 4

You're going to multiply it by a fraction whose value is 1, but is written as \[\frac{n}{n}\]where \(n\) is some number that when multiplied by 8 will give you 40.

\[\frac{1}{8}*\frac{n}{n} = \frac{1*n}{8n} = \frac{n}{40}\]

Answer 5

What is the value of \(n\)?

Answer 6

5? Because 8 x 5 is 40?

Answer 7

Very good. So what is \[\frac{1}{8}\]when expressed as a fraction with \(40\) as the denominator?

Answer 8

Since n equals 5, 5/40? But simplified would also be 1/8.

Answer 9

5/40 is correct. We don't want to simplify, because we need to compare with a common denominator in place to do this problem as requested. (I'll tell you a method where you don't need to have a common denominator when we finish the problem, but I believe in making sure you know how to do the problem as requested).

Now, what do we do to make 3/20 have a denominator of 40, and what will the fraction be?

Answer 10

I think you'd multiply 20 by 2 since that equals 40.

So 20/40 or 2/40?

Answer 11

Right! For the denominator. How about the numerator? 3/20 = ???/40?

Answer 12

\[\frac{3}{20}*\frac{2}{2} = \frac{3*2}{20*2} = \frac{6}{40}\]right?

Answer 13

So our two fractions are \[\frac{5}{40},\,\frac{6}{40}\]Are they a proportion?

Answer 14

Yes, I think.

Answer 15

No, they are not equal, so they are not in proportion. If the ratios are in proportion, they will be equal when compared with a common denominator.

\[\frac{1}{2},\frac{2}{4}\]are a proportion, because if we view them with a common denominator, they are \[\frac{1*2}{2*2},\frac{2}{4}\]and those fractions are equal. Put another way, if you reduce both fractions completely, they will be equal if they are proportional.

3/20 and 1/8 are both fully reduced, and they are not equal, so they are not a proportion.

Does that make sense?

Answer 16

I told you that there was another way to do this. Even if you are required to do the problem a different way, it can serve as a way to check your answer, so I'll provide it here.

Cross-multiplication:

\[\frac{a}{b} = \frac{c}{d}\]If we want to get rid of the fractions, we could multiply both sides of the equation by \(d\) without changing the equality (anything you do to both sides preserves the equality)
\[\frac{a}{b}d = \frac{c}{d}d\]Now we can multiply both sides by \(b\) in the same fashion\[\frac{a}{b}d*b = \frac{c}{d}d*b\]And simplify\[\frac{a}{\cancel{b}}d\cancel{*b} = \frac{c}{\cancel{d}}\cancel{d*}b\]leaving us with \[ad=cb\]as being completely equivalent to our original expression. So, if the two fractions reduce to the same thing, then the numerator of one times the denominator of the other will be equal to the denominator of one times the numerator of the other.

Answer 17

Let's try it on \[\frac{1}{2} = \frac{2}{4}\]\[1*4 = 2*2\]\[4=4\checkmark\]the two fractions are equal, so the ratios are a proportion.

Now let's try it on your problem:
\[\frac{1}{8}=\frac{3}{20}\]\[1*20 = 3*8\]\[20=24\]Uh, no, the fractions are not equal, so the ratios do not form a proportion.

Answer 18

So when cross multiplying if you end up with the same number when multiplying both they are a proportion?

Answer 19

Yes, that is exactly correct. Sorry for the delay, I didn't notice that you had responded! :-(

Answer 20

If they are a proportion, then you would have the following situation:

\[\frac{a}{b} = \frac{k}{k}*\frac{a}{b}\]or\[\frac{a}{b} = \frac{k*a}{k*b}\]if we cross-multiply:
\[a*k*b = b*k*a\]we can cancel out the \(a\) and \(b\) from each side and we are left with \(k=k\) which is always true. Therefore, if the two products are equal, the ratios are a proportion.

Answer 21

That's okay, and thank you so much!

Answer 22

You're welcome!