Jon P. answered • 07/09/15
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Well, teaching you all about proportions is not something that can be done in a small space. However, you were on the right track in understanding what the proportions mean -- you just didn't apply them correctly.
You are correct that A is 7/9 gold and B is 7/18 gold. But when you mix them, you don't just add the two fractions together.
There are two ways to look at this problem:
1. When you add EQUAL amounts of two alloys together, then the fraction of the resulting alloy that is gold will be the AVERAGE of the fractions in the original alloys. So you would add 7/9 to 7/18 and then take the average by dividing by 2. So that's (7/9 + 7/18) / 2 = (14/18 + 7/18) / 2 = (21/18) / 2 = (7/6) / 2 = 7/12. So the resulting alloy has 7 parts gold out of 12, so the rest, 5 parts is copper. So the ration is 7:5.
If the amounts of A and B were NOT equal, then you would do something similar, using the concept of WEIGHTED average, but that's not something to go into now.
2. You're mixing the same amount of each alloy together, so let's call that amount x. That means that the amount of gold in the sample of A is 7/9 x and the amount of gold in the sample of B is 7/18 x. When you add the two samples together, you get 21/18 x = 7/6 x. That's the amount of gold you have in the mixture. But the TOTAL amount of metal is actually 2x, since you added x of A and x of B together. So the fraction of the mixture that is gold is (7/6 x) / (2x) = 7/12. You end up with the same answer as before, but from a slightly different point of view.
Jon P.
tutor
If you're dividing things into parts, and using fractions to represent the parts, then it's true that the numerator can't be bigger than the denominator, because a part can't be bigger than the whole. But fractions don't always represent parts of things. Sometimes they are just numbers that come up during calculations. The numerator can be greater than the denominator.
In this case, 7/6 is a number that we get to during the calculations, but it doesn't represent a part of something -- it's just a number -- so it's OK for the numerator to be greater than the denominator. It's only at the end, when we find that the alloy is 7/12 gold, that we have a fraction that represents a part of something.
Regarding your second question, we never had a quantity 7/6x/2. We had (7/6 x) / (2x). That does not mean 7/6 divided into two parts. That means that the total amount of gold in the mixture is 7/6 x, and that the total amount of the mixture is 2x. In order to find what fraction of the final mixture is gold, we divide the amount that is gold (7/6 x) by the total amount of metal (2x). That equals 7/12.
Have you learned algebra yet? If not, then I can see that using "x" in the second method wouldn't be clear to you.
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07/11/15
Amey B.
how i learn algebra which book or which video can teach me the whole algebra
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07/12/15
Amey B.
why we divide the amount that is gold (7/6 x) by the total amount of metal (2x).
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07/12/15
Jon P.
tutor
1. I don't have any specific recommendations on how to learn algebra. It's a very big subject. There are usually at least two years devoted to algebra in school. I'm sure there are a lot of places on the internet where you can learn It's a very big subject How old are you? Are you in school, and if you are, what grade are you in?
2. You divide the amount that is gold by the total amount of metal so that you can figure out what fraction of the metal is gold. For example... If you have 120 coins, and 40 of them are dimes, what fraction of the coins are dimes? Well, to find that you divide the number of dimes by the total number of coins -- that is, you divide 40 by 120, which is equal to 1/3. So 1/3 of the coins are dimes.
In the problem we've been looking at, once you have the fraction that is gold (7/12) you can figure out what fraction is copper. Since there are only 2 metals in the allow, the fraction that is gold plus the fraction that is copper have to add up to 1. Since 7/12 is gold, the rest must be copper, and that's 5/12.
Once you have the fraction for each metal, you can find the ratio of gold to copper. 7/12 is gold and 5/12 is copper, so the ration of gold to copper is 7/12 to 5/12. Ratios are just the same as dividing, so you divide 7/12 by 5/12 to find the ration of gold to copper. 7/12 divided by 5/12 is 7/5, so the ration of gold to copper is 7:5.
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07/12/15
Amey B.
07/11/15