Solution - Explanation

[Guest]

Here is a problem with solution. I don't understand parts of the solution:

In a quantity of metal, the brass is 10 pounds more than 2/3 of the total, the iron 4 1/2 pounds less than 1/6 of the total, and nickel 2 pounds less than 1/7 of the brass. Calculate amount of metal.

Let x = total

2x/3 + 10 = brass

x/6 - 4 1/2 = iron

2x/21 + 10/7 - 2 = nickel *

2x/3 + x/6 + 2x/21 + 10/7 + 3 1/2 = x *

4x + x + 4x/7 + 60/7 + 21 = 6x

4x/7 + 60/7 + 21 = x

4x + 60 + 21 * 7 = 7x

60 + 21 * 7 = 3x

20 + 7 * 7 = x

69 = x

* I don't understand what the 2x/21 or 3 1/2 is. Thanks.

[Guest]

The question says (brass)/7 - 2 = nickel,
but you've already said that brass = 2x/3 + 10,
substitute this in to the nickel formula to get
(2x/3+10)/7 - 2 = nickel
expand the bracket to get
2x/21 + 10/7 - 2 = nickel
(I would personally simplify "10/7 - 2" to -4/7 and rewrite all the mixed fractions as top heavy fractions but the person who wrote the solution hasn't bothered.)

We know that
(brass)+(iron)+(nickel) = total
Substituting the formulas you've just given gives:
(2x/3 + 10) + (x/6 - 4 1/2) + (2x/21 + 10/7 - 2) = x
Expanding the brackets and reordering the terms gives:
2x/3 + x/6 + 2x/21 + 10/7 + 10 - 4 1/2 - 2 = x
The last three terms on the left hand side are 10 - 4 1/2 -2 which simplify to 3 1/2.
(Why the person who wrote the solution didn't include 10/7 is a bit of a mystery, perhaps they thought because of the 7 denominator it would take too much effort to work out 10/7 + 3 1/2 = 69/14. In any case the way it was solved is valid, though not how I would have simplified things.)

Hope this helped,

R. Baber.

[Guest]

"Expanding the brackets and reordering the terms gives:
2x/3 + x/6 + 2x/21 + 10/7 + 10 - 4 1/2 - 2 = x"

I am not clear on this part of your answer.

[Guest]

(2x/3 + 10) + (x/6 - 4 1/2) + (2x/21 + 10/7 - 2) = x
is the same as:
2x/3 + 10 + (x/6 - 4 1/2) + (2x/21 + 10/7 - 2) = x
is the same as:
2x/3 + 10 + x/6 - 4 1/2 + (2x/21 + 10/7 - 2) = x
is the same as:
2x/3 + 10 + x/6 - 4 1/2 + 2x/21 + 10/7 - 2 = x
(I've simply just removed the brackets.)

The terms in a sum can be reordered, for example a+b+c is the same as b+a+c or c+b+a or b+c+a.
Doing this on the left hand side of the equation tells us that
2x/3 + 10 + x/6 - 4 1/2 + 2x/21 + 10/7 - 2 = x
is the same as
2x/3 + x/6 + 2x/21 + 10 - 4 1/2 + 10/7 - 2 = x
(I've reordered things so the x terms are next to each other and the constant terms are next to each other.)
Furthermore we can reorder the constant terms to get:
2x/3 + x/6 + 2x/21 + 10/7 + 10 - 4 1/2 - 2 = x
(I've reordered the constant terms so the simpler fractions (i.e. the terms that are not 10/7) are together.)
Adding the last three terms on the left hand side gives 3 1/2 and so the equation becomes
2x/3 + x/6 + 2x/21 + 10/7 + 3 1/2 = x
which matches the second starred line that you wrote in your first post.

Hope this helped,

R. Baber.

[Guest]

"(2x/3+10)/7 - 2 = nickel
expand the bracket to get
2x/21 + 10/7 - 2 = nickel"

I still don't understand the 2x/21.

[Guest]

Forget the above posts and do it again.....

Let T = Total
Let B = Brass
Let I = Iron
Let N = Nickle

2/3T + 10 = B
1/6T - 4 1/2 = I same as 1/6 T - 9/2 = I
1/7B - 2 = N

B + I + N = T

2/3 T + 10 + 1/6 T - 4 1/2 + 1/7 B - 2 = T

collect like terms

5/6 T + 3 1/2 + 1/7 B = T

1/7 B = 1/6 T - 3 1/2

B = 7/6 T - 24 1/2

So... 7/6 T - 24 1/2 = 2/3T + 10 ... from B above

3/6 T = 34 1/2

or 1/2 T = 34 1/2

T = 69

[Guest]

"B = 7/6 T - 24 1/2"

I don't understand this step.

[Guest]

"(2x/3+10)/7 - 2 = nickel
expand the bracket to get
2x/21 + 10/7 - 2 = nickel"

I still don't understand the 2x/21.



You said to forget the posts but PLEASE explain also. I am trying to understand both solutions.

[Guest]

There are two people trying to help you now. To distinguish myself I end my posts with "R. Baber". So it was not my suggestion to "Forget the above posts and do it again" but someone else's. I will let them explain the step "B = 7/6 T - 24 1/2" from their post.

When I wrote:

(2x/3+10)/7 - 2 = nickel
expand the bracket to get
2x/21 + 10/7 - 2 = nickel

I was simply using the fact that (a + b)/c = a/c + b/c. When you remove the brackets you apply the division to every term inside the bracket. For example (10+4)/2 is 7 (the terms inside the bracket add up to 14, and when we divide by 2 we get 7) but we can remove the bracket using the rule I just outlined to get
(10+4)/2 = 10/2 + 4/2
which of course simplifies to 5+2 which again is 7.

Applying this rule to (2x/3+10)/7 we get (2x/3)/7 +10/7. The term (2x/3)/7 can be simplified further to 2x/21, using the rule for divisions by fractions. The rule is
[tex]\frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c} = \frac{a\times d}{b\times c}[/tex]
We can apply this to (2x/3)/7 (using the fact that 7 is the same as 7/1)
[tex]\frac{2x}{3}\div\frac{7}{1} = \frac{2x}{3}\times\frac{1}{7} = \frac{2x\times 1}{3\times 7} = \frac{2x}{21}.[/tex]

Hope this helped,

R. Baber.

[Guest]

I didn't mean forget the above posts completely......I meant in relation to my post forget the above and I would do it again
I meant rather than me reading through the earlier posts I decided I would do it again......
What I did is not a follow on from the above posts, but it is a new complete solution set out slightly differently.
.......................................................................
You say you don't understand the following bit........
"B = 7/6 T - 24 1/2"
I don't understand this step.
.....................................................
Well if we start from the line above it......
Simplifying the terms in the lines above it we end up with this.....
1/7 B = 1/6 T - 3 1/2
And in words this reads " one seventh of "B" equals one sixth of "T" minus three and a half."
So if one seventh of "B" equals that then seven sevenths of "B must equal seven times it. So multiply across by 7.
So....
7 times 1/7 B = 7/7 B = just B
and.......
7 times 1/6 T = 7/6 T and I didn't do any more with it.
and.......
7 times three and a half equals twenty four and a half. 7 x 3 1/2 = 24 1/2
You could change the 3 1/2 to 7/2 then you would have 7 x 7/2 = 49/2 = 24 1/2.
and that is the 3 terms multiplied by 7 to give the line below.....
B = 7/6 T - 24 1/2
Then I substituted in B from above....

So... 7/6 T - 24 1/2 = 2/3T + 10 ... from B above
and simplified.......
3/6 T = 34 1/2

or 1/2 T = 34 1/2
ending up with this.................
T = 69

.................................

[Guest]

Thanks for both solutions. I understand it now.

[Guest]

To the other person who gave an answer:

My apologies if I made it sound like your advice was wrong, I just wanted to distinguish my posts from yours and I thought there might be some confusion.

R. Baber.