Weight

[Guest]

Gunpowder has the following components:

Niter- 10 pounds more than 3/4 of the weight.

Sulphur- 3 pounds more than 1/12. (unknown if 1/12 of weight, niter, or charcoal.)

Charcoal- 3 pounds less than 1/10 of the niter.

Calculate weight of the gunpowder.

Let W = Weight
Let N = Niter
Let S = Sulphur
Let C = Charcoal

N = .75W + 10
S = .833 + 3
C = .1N - 3

.75W + 10 + .833 + 3 +.1N - 3
.75W + 13.883 + .1N
.75W + 10.833 + .1N

How do I proceed from this step ?

[Guest]

OK.... most of your reasoning is OK.....

I would changes as follows.....

Let W = Weight (pounds) of the Gunpowder mix to be made in this batch
Let N = pounds of Niter used
Let S = pounds of Sulphur used
Let C = pounds of Charcoal used

I would have kept working in fractions because the numbers are simple....
but will continue your way.......

N = .75W + 10 .....OK

S = .833 + 3 ....1/12 = 0.0833 and you forgot about W.....
So should be.....

S = 0.0833W + 3

And for Charcoal......
C = .1N - 3

This is OK for charcoal but you need it in terms of W ....substitute for N.....

C = 0.1(0.75W + 10) - 3

========================================
As a result of the above mistakes your next 3 lines are not complete and incorrect...
.75W + 10 + .833 + 3 +.1N - 3
.75W + 13.883 + .1N
.75W + 10.833 + .1N
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So continuing from above.....
Add the above amounts together to make up the batch W of gunpowder...

W = N + S + C

W = 0.75W + 10 + 0.0833W + 3 +0.1(0.75W + 10) - 3 .....this is the whole batch to be made...

W = 0.75W + 0.0833W + 0.075W + 10 + 3 + 1 - 3 .....collecting like terms....

W = 0.9083W + 11

0.0917W = 11

W = 119.956

So W = 120 pounds approx. gunpowder made in this batch....

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You can see the numbers did not work out exactly because you used decimals...
Due to roundings in the calculator used etc.....but the answer is near enough...

We will do it again using fractions and hopefully get an exact answer....

N = 3/4W + 10
S = 1/12W + 3
C = 1/10(3/4W + 10) -3

W = (3/4W + 10) + (1/12W + 3) + (1/10(3/4W + 10)) - 3

W = 3/4W + 1/12W + 3/40W + 10 + 3 + 1 - 3

W = ((90 + 10 + 9)/120)W + 11 .....changing all to 120th LCM...

W = (109/120)W + 11

120W = 109W + 1320 .....multiply across by 120...gives this...

11W = 1320

W = 120 Pounds exactly of gunpowder mage in this batch

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[Guest]

"W = 3/4W + 1/12W + 3/40W + 10 + 3 + 1 - 3"

I can't recall how to calculate the fractions. Thanks.

[Guest]

as stated in the answer above the LCM of 4, 12 and 40 is 120 so take common denominator of 120 or change all to 120ths
3/4 = 90/120
1/12 = 10/120
3/40 = 9/120

that is 109/120 alltogether