Expert: Josh - 4/1/2010

Question
Do you understand decimals? Such as .01 = 1 oz and what .5 is?
This is not homework. I am long past high school. I just won't say how long. lol
but I was never good with this part of math so I need help if you understand it.

Answer
Hi Joyce,

I read your comments. If what I said had confused you, I am sorry. The last thing I want is for  you to walk away feeling low, that would be the worst possible outcome. Why don't you give me a chance, we will look at this problem again from another angle.

I'm not sure if your main difficulty has to do with understanding decimals, i.e., what they represent, and how they relate to other numbers such as fractions. OR, perhaps, you want to focus on a word problem on proportions, which is not exclusively about decimal figures.

I think it is important that we get this clear first up. Otherwise, I can type six paragraphs, going into every detail without actually answering your concern.

Just to be sure, in case you are not confident about decimals, what you have mentioned in the question, and in particular, 0.01 equals one part in a hundred. So, we can say 0.01=1/100. Similarly, 0.5 equals a half, so we can say 0.5=1/2. This explains the relationship between decimals (left of the = sign) and fractions (right of the = sign).

In regard to the word problem on proportions, we cannot actually say "0.01 = 1 oz". There is something missing in this statement, because 0.01 is clearly not the same as the number 1. Correct me if I am wrong, I believe what you have in mind, is 0.01 times of "something" corresponds to 1 ounce. Note that I have been careful in using the word "correspond" (not "equal"). When we say 0.01 of "something" corresponds to 1 oz, we are in fact specifying a ratio. Another word for a ratio is "proportion". Since 0.01 is same as 1/100 (which in turn is same as 1% of something), we can summarize everything we have discussed so far -- that 0.01 of something corresponds to 1 ounce -- by a fraction 0.01/1. This represents a known ratio.

Next, recall that 0.5 = 1/2 represents 50% of something.
Knowing "1% of something is 1 oz", we want to find out "what 50% of something corresponds to"? So, on the other side of the equation, we have 0.5/Y, a proportion that is same as 0.01/1, but Y is unknown. (Just go over this in your head, 0.01 of something is 1 oz, what is 0.5 of something in oz. The answer is Y, something that we need to find out). Remember, these two proportions (0.01/1 and 0.5/Y) are the same. So, we can say 0.01/1 = 0.5/Y.

We can solve this using the "cross multiplication" technique or using basic reasoning.
I tried explaining the cross-multiplication technique last time, without much success. So, this time I will not bother with the reasons behind the trick, but simply say what needs to be done.

Finding Y in 0.01/1 = 0.5/Y    ...Line 1
is SAME AS
solving 0.01*Y = 0.5*1.        ...Line 2

| What have we done going from Line 1 to Line 2?
| Answer: The left hand side (LHS) in Line 2 is formed by taking the product of
| the numerator on the LHS of Line 1 (that is, 0.01) with the denominator on
| the right hand side (RHS) of Line 1 (that is, Y).
| The RHS in Line 2 is formed by taking the product of
| the numerator on the RHS of Line 1 (that is, 0.5) with the denominator on
| the LHS of Line 1 (that is, 1).

In Line 2, dividing both sides of the equation by 0.01 gives Y = 0.5/0.01 = 50 ounce.

An alternate method uses reasoning.
If 1% is to 1 ounce, then 50% is to Y ounce.  Notice that there is a pattern here.
Since 50% is 50 times greater than 1% (just as 0.50 is 50 times larger than 0.01),
we conclude that Y must be 50 times 1 (ounce).
So, the answer is 50 oz.

I hope you can get something useful from this.
Please talk to someone and see what they think.
May be they can explain this better making use of diagrams.

========================


Hi Joyce,

I can sort of understand what you are saying.
If 0.01 of something equals 1 oz, what does 0.5 represent in ounce?

With all things equal (i.e., provided the ratio or proportion does not change), we can use cross multiplication to answer this.

First, let us represent "something" as X.
Then, one hundredth of something is translated as 0.01*X (in words, 0.01 times X).
Comparing 0.01 of something to 1 oz, is same as forming a ratio of 0.01*X/1 (just a fraction).
Let this sit on the left hand side of a scale.

On the right hand side of the scale, we have 0.5 of "something" to Y as a proportion, where Y is the unknown.

Here comes the "with all things equal" part, placing the first ratio 0.01*X/1 on the left hand side (LHS) of the scale, and balancing this against the second ratio 0.5*X/Y on the right hand side (RHS) of the scale, mathematically we arrive at (0.01*X)/1 = (0.5*X)/Y.

Since the "something" (which we called X) appears on both sides of the scale (equation), we can cancel them out without affecting anything. We simplify the problem as 0.01/1 = 0.5/Y.

To find out the amount Y (in ounce),
Step 1: We multiply both sides of the equation by the denominator on the LHS (bottom part of fraction on the left hand side).
 This gives 0.01 = 0.5/Y  (note: actually 0.01*1/1 appears on the LHS, but 1/1 is just one)
Step 2: We multiply both sides of the equation by the denominator on the RHS, namely, Y.
 This gives 0.01*Y = 0.5  (note: again, Y/Y on the RHS cancels to 1)
Step 3: We divide both sides by 0.01 to get Y.
 Y = 0.5/0.01 = 50 (ounce).

The steps I just outlined is the long way of doing things. It provides some insight into the short cut, which combines step 1 and step 2.

Starting from the beginning, 0.01/1 = 0.5/Y can be turned right away into 0.01*Y = 0.5*1. This process is called cross multiplication, because when we are given A/B = C/D, we proceed by forming a product between A and D on the LHS (so the denominator crosses over from right to left), and we form a product between B and C on the RHS (pretty much the denominator crosses over from left to right).

Consider a new example.

Example 1: If 2 lots of "something" corresponds to 10 oz, how many ounce does 5 lots of "something" correspond to?

We translate this to 2/10 = 5/Y.
Using cross multiplication (combining steps 1 and 2), we get an equivalent expression
2*Y = 5*10. Dividing both sides by 2 yield Y = 5*10/2 = 25 (ounce).

It makes no difference whether we are dealing with integers or decimal.