Where do they get this number? Recognizing Proportional Relationships

[JJarami4]

I'm having trouble figuring out where this tutorial is getting the fraction 1/3 used to multiply to find if the ratio is equivalent.

Please refer to the attachment. Thank you

 

[Denis]

Hard to tell what you're asking....

Anyhoooo: 3/2/36
= (3/2) / (36/1)
= (3/2) * (1/36)
= (3*1) / (2*36)
= 3 / 72
= 1 / 24

 

[mmm4444bot]

I'm having trouble figuring out where this tutorial is getting the fraction 1/3 used to multiply to find if the ratio is equivalent.

They probably expect students to determine it through number sense (multiplication table, multiplying fractions, mental effort). Yet, this method of determining whether those ratios are equivalent seems a bit like overkill, to me. I think it's easier to simplify each ratio to lowest terms, instead, because then it's obvious whether two ratios are identical or not. (I'll describe that method, at the end.)

To answer your question, their approach is to determine whether or not the first ratio can be converted into the second ratio.

The first numerator is 3/2, and it needs to become 1/2 (the second numerator).

The first denominator is 36, and it needs to become 24 (the second denominator).

The idea, therefore, is to brainstorm (guess-and-check) using knowledge of the multiplication table and multiplying fractions to see whether you can find a single number ?? such that:

3/2 × ?? = 1/2

36 × ?? = 24

I suppose it's easier for most students to think about the second conversion, first. In other words, ask yourself: what number do I multiply 36 by to get 24? Then check to see whether that same number yields 1/2 when you multiply 3/2 by it.

Again, this method entails extra work; maybe they presented it to help students develop number sense.

---

Here's the other method (reducing each ratio to lowest terms). Use the rule for simplifying a compound fraction -- a fraction whose numerator and/or denominator are fractions themselves. That is, we multiply the numerator by the reciprocal of the denominator.

\(\displaystyle \displaystyle \frac{\frac{3}{2}}{36} = \frac{\frac{3}{2}}{\frac{36}{1}} = \frac{3}{2} × \frac{1}{36}\)

This is what Denis showed above, although he multiplied first and reduced at the end. I would note that 36 is a multiple of 3 and do the cancellation first, then multiply. I do cancellations with fractions before multiplying because often it's less work to reduce sooner (while numbers are smaller) than after multiplication (when numbers become larger).

\(\displaystyle \displaystyle \frac{3}{2} × \frac{1}{36} = \frac{1}{2} × \frac{1}{12} = \frac{1}{24}\)

Simplify the second compound fraction the same way, and it will become obvious that the two ratios are equivalent. :cool:

 

[Dr.Peterson]

I'm having trouble figuring out where this tutorial is getting the fraction 1/3 used to multiply to find if the ratio is equivalent.

Please refer to the attachment. Thank you

The way I would describe what they are doing is that they saw that 36 is 3 times 12, or equivalently, that 12 is 1/3 of 36. To make an equivalent fraction, you have to multiply or divide the numerator and denominator by the same quantity; so we would multiply both 3/2 and 36 by 1/3 (that is, divide them both by 3), and the result is 1/2 and 12. That shows that the second ratio is equivalent to the first.

For many people, fractions are hard to work with; well, actually, that's true for everyone. If you think of it as dividing by 3 (or simply as taking 1/3 of the number), rather than multiplying by 1/3, that may be clearer. It's all the same thing. Similarly, if you were focusing your attention on the numerators (fractions), you might want to focus instead on the denominators (as I did above), which are whole numbers and easier to work with.