Answers:
- He needs <u> 310 </u> grams of the 99% cocoa brand
- He needs <u> 590 </u>grams of the 70% cocoa brand
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Explanation:
Let's label the two brands of chocolate as A and B
- Brand A is 99% cocoa
- Brand B is 70% cocoa
Furthermore, define these variables
- x = amount of brand A
- y = amount of brand B
The amounts are in grams.
The two amounts must add to 900 grams, so x+y = 900. This can be solved to y = 900-x which we'll use later.
Since we have x grams of brand A, and brand A has 99% cocoa, this means 0.99x represents the amount of pure cocoa from brand A.
Similarly, brand B contributes another 0.70y grams of pure cocoa
Overall, we're dealing with 0.99x+0.70y grams of pure cocoa from the two brands mixed together.
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Rolf needs 900 grams of final product. He wants it to be 80% cocoa, which means 80% of the 900 is going to be pure cocoa
80% of 900 = 0.80*900 = 720
Rolf wants 720 grams of pure cocoa after mixing the two brands together.
So this leads to the second equation: 0.99x+0.70y = 720
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Let's apply substitution to replace y with 900-x
0.99x+0.70y = 720
0.99x+0.70( y ) = 720
0.99x + 0.70( 900-x ) = 720
At this point, we have a single variable in which we can isolate x fully.
0.99x + 0.70( 900-x ) = 720
0.99x + 0.70( 900) + 0.70(-x) = 720
0.99x + 630 - 0.70x = 720
0.29x = 720-630
0.29x = 90
x = 90/(0.29)
x = 310.344827586 which is approximate
Rounding to the nearest full gram, we get x = 310
Use this x value to find y
y = 900-x
y = 900-310.344827586
y = 589.655172414 which is also approximate
y = 590 when rounding to the nearest whole number
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In summary, we have
Meaning that Rolf will need <u>310 grams</u> of brand A to mix with <u>590 grams</u> of brand B.
If we plugged those values into x+y = 900, then that equation works out.
If we tried the second equation, then,
0.99x+0.70y = 720
0.99*310+0.70*590 = 720
719.9 = 720
The slight discrepancy is due to the fact we rounded those approximate x and y values to the nearest whole number. We would get better accuracy if we rounded to say the nearest tenth or hundredth.
