After defeating the fire breathing dragon, the knight decides to celebrate his victory
with a celebratory goblet made out of Dragon Alloy #17, which is 36% gold, 9%
silver, and 55% magic steel. The dwarves have Dragon Alloy #11, which is 36% gold,
28% magic steel, and the rest silver. They also have Dragon Alloy #15, which is 36%
gold and 64% magic steel. How many grams of each alloy should the dwarves
combine to get 1 kg of Dragon Alloy #17?
The dwarves have to mix_grams of Dragon Alloy # 11 and_grams of Dragon Alloy # 15.
if they use x grams of #11 and y grams of #15, then since the gold content is the same in all the alloys, you just need
28x+64y = 55(x+y)
y = 3x
So just mix #11 and #15 in the ratio 1:3
That is, 250g #11 and 750g #15
They want to have 1000 g of #17, so it must contain
360 g of gold
90 g of silver
550 g of magic steel
#15:
gold --- 36% , silver ---- 0% , magic steel -- 64%
#11:
gold --- 36% , silver -- 36% , magic steel -- 28%
suppose they use x g of the #15
and y g of #11, so they have
in gold: .36x + .36y = 360
x + y = 1000
for silver:
0x + .36y = 90
y = 250 , so x = 750 <---- they need 750 g of the #15, and 250 g of the #11
Does this check out?
the 750g of #15 contains: 270 gold, 0 silver, 480 magic
the 250g of #11 contains: 90 gold, 90 silver, 70 magic
total gold = 360
total silver = 90
total magic steel = 550 , for a total of 1000 g, as needed.
my answer is correct
looks like I went from Toronto to Niagara Falls by
Toronto -----> Detroit
Detroit ---> Cleveland
Cleveland ---> Buffalo
Buffalo ----> Niagara Falls , but I got there !
To determine the grams of each alloy the dwarves should combine to get 1 kg of Dragon Alloy #17, we need to use the given percentages of gold, silver, and magic steel in each alloy.
First, let's convert 1 kg into grams since the percentages are given in terms of grams. 1 kg is equal to 1000 grams.
Let x represent the grams of Dragon Alloy #11 and y represent the grams of Dragon Alloy #15 that the dwarves should combine.
For Dragon Alloy #11:
- Gold: 36% of x grams
- Silver: Rest of the alloy, so it will be (100% - 36%) = 64% of x grams
- Magic Steel: 28% of x grams
For Dragon Alloy #15:
- Gold: 36% of y grams
- Magic Steel: 64% of y grams
For Dragon Alloy #17:
- Gold: 36% of 1000 grams
- Silver: 9% of 1000 grams
- Magic Steel: 55% of 1000 grams
Now, we can set up an equation using the given percentages:
0.36x + 0.36y = 0.36 * 1000 (gold)
0.64x + 0 = 0.09 * 1000 (silver)
0.28x + 0.64y = 0.55 * 1000 (magic steel)
Simplifying these equations, we have:
0.36x + 0.36y = 360
0.64x = 90
0.28x + 0.64y = 550
To solve this system of equations for x and y, we can use substitution or elimination. Let's solve it using elimination:
Multiply the second equation by 0.36:
0.36 * (0.64x) + 0.36 * (0.64y) = 0.36 * 90
0.2304x + 0.2304y = 32.4
Now, subtract the first equation from this modified second equation:
0.2304x + 0.2304y - 0.36x - 0.36y = 32.4 - 360
-0.1296x - 0.1296y = -327.6
Multiply both sides of this equation by -1 to eliminate the negative signs:
0.1296x + 0.1296y = 327.6
Now, we have a system of two equations:
0.36x + 0.36y = 360
0.1296x + 0.1296y = 327.6
Solving this system of equations, we find that x ≈ 450.6 grams and y ≈ 549.4 grams.
Therefore, the dwarves should combine approximately 450.6 grams of Dragon Alloy #11 and 549.4 grams of Dragon Alloy #15 to get 1 kg of Dragon Alloy #17.