Negative numbers with fraction problem: simplify 3/4 [ 5/6 ( -18/25 ) + 1/2 ]

[TheDoctor]

Hey guys, I am seriously confused by this problem.

3/4 [ 5/6 ( -18/25 ) + 1/2 ]

I would appreciate it if someone shows me the step by step process.

Thanks in advance :)

 

[MarkFL]

Hey guys, I am seriously confused by this problem.

3/4 [ 5/6 ( -18/25 ) + 1/2 ]

I would appreciate it if someone shows me the step by step process.

Thanks in advance :)

We are given to simplify:

\(\displaystyle \frac{3}{4}\left(\frac{5}{6}\left(-\frac{18}{25}\right)+\frac{1}{2}\right)\)

The first thing I would do is use the commutative property of addition within the outer brackets:

\(\displaystyle \frac{3}{4}\left(\frac{1}{2}-\frac{5}{6}\cdot\frac{18}{25}\right)\)

Now, reduce the product within the brackets by dividing out common factors (cancellation):

\(\displaystyle \frac{3}{4}\left(\frac{1}{2}-\frac{\cancel{5}}{\cancel{6}}\cdot\frac{3\cdot\cancel{6}}{5\cdot\cancel{5}}\right)\)

\(\displaystyle \frac{3}{4}\left(\frac{1}{2}-\frac{3}{5}\right)\)

Now, we need a common denominator to find the difference within the brackets:

\(\displaystyle \frac{3}{4}\left(\frac{1}{2}\cdot\frac{5}{5}-\frac{3}{5}\cdot\frac{2}{2}\right)\)

\(\displaystyle \frac{3}{4}\left(\frac{5}{10}-\frac{6}{10}\right)\)

Can you continue?

 

[TheDoctor]

I flunk on the cancellation part a whole lot of times. When do you actually need to cancel out something and are there rules for cancellation?

 

[MarkFL]

I flunk on the cancellation part a whole lot of times. When do you actually need to cancel out something and are there rules for cancellation?

Canceling is the process of simplifying a fraction by dividing out any factors common to the numerator and denominator. A fraction in its simplest form has a numerator and denominator that have no common factors, that is, they are co-prime. This makes the numbers (or expressions) in the numerator/denominator as small as possible, and easier with which to work.

The way I was taught to do this is to factor both the numerator and denominator into their prime factors, and then divide out any factors they have in common.

 

[TheDoctor]

Canceling is the process of simplifying a fraction by dividing out any factors common to the numerator and denominator. A fraction in its simplest form has a numerator and denominator that have no common factors, that is, they are co-prime. This makes the numbers (or expressions) in the numerator/denominator as small as possible, and easier with which to work.

The way I was taught to do this is to factor both the numerator and denominator into their prime factors, and then divide out any factors they have in common.

This is just going over my head right now because cancellation is kind of a new thing for me. For some reason, we didn't really discuss it in class, or rather, we didn't really put our time into it. Can you give a basic example or a problem that you could use cancellation on? And earlier when you enumerated the steps, you said to use the commutative property of addition right? But what doesn't get me is that why did the plus sign turned minus?

Thanks again, it's just that I forgot some of the key terms when it comes to fraction problems.

 

[Joppy]

This is just going over my head right now because cancellation is kind of a new thing for me. For some reason, we didn't really discuss it in class, or rather, we didn't really put our time into it. Can you give a basic example or a problem that you could use cancellation on?

Think about some situations were you need to divide something up into portions. Maybe you and two friends want to share a chocolate bar. How much should each of you get?

There are 3 people, and only 1 chocolate bar to go around! You will need to divide the chocolate bar up into smaller pieces. How many pieces? 3 of course! One piece for each person. i.e., you divide the bar up into thirds - each person gets $\dfrac{1}{3}$ of a chocolate bar.

Now to cancellation. Suppose you and your friends start getting worried about your teeth, so you decide to have some apples. You have 3 apples, and there are 3 people! Perfect! One apple per person right? $\dfrac{3}{3} = 1$. This is essentially what's happening when we 'cancel' terms (no apples though).

So for the chocolate bar, we couldn't cancel any terms (there are no common factors in the numerator and denominator), but with the apples, we could cancel the terms. These aren't the only situations though. Sometimes we have a fraction that can be made simpler, but will still appear as a fraction.

Does it help if i write MarkFL's cancellation line like this instead?

\(\displaystyle \frac{3}{4}\left(\frac{1}{2}-\frac{\cancel{\color{blue}5}}{\cancel{\color{red}6}}\cdot\frac{3\cdot\cancel{\color{red}6}}{5\cdot\cancel{\color{blue}5}}\right)\)

 

[TheDoctor]

Ah, so

\(\displaystyle \frac{18}{25}\)

were just rewritten with their greatest common factors which are 3 and 5?

Also, I still don't get why the plus in

3/4 (5/6(-18/25)+1/2)

would turn into a minus. Am I missing a rule here?

 

[I like Serena]

Also, I still don't get why the plus in

3/4 (5/6(-18/25)+1/2)

would turn into a minus. Am I missing a rule here?

The plus is not turning into a minus.
Let's break it up a bit more:
$$
\frac 34 \left(\frac 56\left(-\frac{18}{25}\right)+\frac 12\right)
= \frac 34 \left(\frac 56\cdot -1 \cdot\frac{18}{25}+\frac 12\right)
= \frac 34 \left(\frac 12 + \frac 56\cdot -1 \cdot\frac{18}{25}\right)
= \frac 34 \left(\frac 12 + -1 \cdot\frac 56\cdot \frac{18}{25}\right)
= \frac 34 \left(\frac 12 - \frac 56\cdot \frac{18}{25}\right)
$$
See where the minus comes from?

 

[TheDoctor]

The plus is not turning into a minus.
Let's break it up a bit more:
$$
\frac 34 \left(\frac 56\left(-\frac{18}{25}\right)+\frac 12\right)
= \frac 34 \left(\frac 56\cdot -1 \cdot\frac{18}{25}+\frac 12\right)
= \frac 34 \left(\frac 12 + \frac 56\cdot -1 \cdot\frac{18}{25}\right)
= \frac 34 \left(\frac 12 + -1 \cdot\frac 56\cdot \frac{18}{25}\right)
= \frac 34 \left(\frac 12 - \frac 56\cdot \frac{18}{25}\right)
$$
See where the minus comes from?

I have done some reading on my math book and there are some problems which I couldn't figure out where the minus symbol comes from. There were some instances where you'd have to use the communicative property when adding something inside parentheses, and that usually turns the plus symbol into a minus. Why is this?

I'm ignoring that right now and the continuation to @MarkFL's reply should be

\(\displaystyle \frac{3}{4}\cdot\frac{-1}{10}\)

then to simplify,

\(\displaystyle -\frac{3}{40}\)

 

[Greg]

An example:

(-1) + (+2) = 1

Commutative property:

(+2) + (-1) = 1

2 + (-1) = 1

2 - 1 = 1

See? Adding a negative is the same as subtracting a positive. So you could also write

(-1) + (+2) = 1

(+2) + (-1) = 1

(+2) - (+1) = 1

2 - 1 = 1

In both examples we change the order of operations but the result is the same - these are examples of the commutative property of addition.