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Answer:
For 50% syrup, add 0.5 lb of water for 1 lb of syrup; for 5% syrup, add 9.5 lb of water for 10 lb of syrup; for 75% syrup, add 0.17 lb of water for 0.67 lb of syrup; for 1.5% syrup, add 32.8 lb of water for 33.3 lb of syrup.
Step-by-step explanation:
Let x represent the amount of water added. This means the total mass of the syrup is represented by 0.5+x.
To find percentages, we divide the part by the whole. For a 50% syrup, we need the part (sugar) compared to the whole (sugar + water) to equal 50%, or 0.50:
[tex]\frac{0.5}{0.5+x}=0.5[/tex]
Multiply both sides by 0.5+x:
[tex]\frac{0.5}{0.5+x}\times (0.5+x)=0.5\times (0.5+x)\\\\0.5=0.5(0.5+x)[/tex]
Using the distributive property, we have
0.5 = 0.5(0.5)+0.5(x)
0.5 = 0.25+0.5x
Subtract 0.25 from each side:
0.5-0.25 = 0.25+0.5x-0.25
0.25 = 0.5x
Divide both sides by 0.5:
0.25/0.5 = 0.5x/0.5
x = 0.5
For 50% syrup, add 0.5 lb of water. This makes the total mass 0.5+0.5 = 1 lb.
For 5% syrup, we change the equation to equal 0.05 (5% = 5/100 = 0.05) and leave everything else the same:
[tex]\frac{0.5}{0.5+x}=0.05[/tex]
Multiply both sides by 0.5+x:
[tex]\frac{0.5}{0.5+x}\times (0.5+x)=0.05(0.5+x)\\\\0.5=0.05(0.5+x)[/tex]
Using the distributive property,
0.5 = 0.05(0.5)+0.05(x)
0.5 = 0.025 + 0.05x
Subtract 0.025 from both sides:
0.5-0.025 = 0.025+0.05x-0.025
0.475 = 0.05x
Divide both sides by 0.05:
0.475/0.05 = 0.05x/0.05
x = 9.5
For 5% syrup, add 9.5 lb of water; the total mass will be 9.5+0.5 = 10 lb.
For 75% syrup, set the equation equal to 0.75:
[tex]\frac{0.5}{0.5+x}=0.75\\\\\frac{0.5}{0.5+x}\times (0.5+x)=0.75(0.5+x)\\\\0.5=0.75(0.5+x)[/tex]
Using the distributive property,
0.5 = 0.75(0.5)+0.75(x)
0.5 = 0.375 + 0.75x
Subtract 0.375 from each side:
0.5 - 0.375 = 0.375 + 0.75x - 0.375
0.125 = 0.75x
Divide both sides by 0.75:
0.125/0.75 = 0.75x/0.75
0.17 = x
For 75% syrup, add 0.17 lb of water; the total mass is 0.17+0.5 = 0.67 lb.
For 1.5% syrup, set the equation equal to 0.015:
[tex]\frac{0.5}{0.5+x}=0.015\\\\\frac{0.5}{0.5+x}\times (0.5+x)=0.015(0.5+x)\\\\0.5=0.015(0.5+x)[/tex]
Using the distributive property,
0.5 = 0.015(0.5) + 0.015(x)
0.5 = 0.0075 + 0.015x
Subtract 0.0075 from each side:
0.5-0.0075 = 0.0075 + 0.015x - 0.0075
0.4925 = 0.015x
Divide both sides by 0.015:
0.4925/0.015 = 0.015x/0.015
32.8 = x
For 1.5% syrup, use 32.8 lb of water; the total mass is 32.8+0.5 = 33.3 lb.
We need to find how much water Ella needs to make the different concentrations of syrups. The solutions are:
- 50% → we need 0.5 lbs of water.
- 5% → we need 9.5 lbs of water.
- 75% → we need 0.167 lbs of water.
- 1.5% → we need 32.83 lbs of water.
Understanding concentrations:
The percentage in the concentration tells us how much in the mixture is sugar.
So for example, in the 50% syrup, we have a 50% of sugar, meaning that half of it is sugar, and we have 0.5 lbs of sugar, so we need to add the same amount of water, 0.5 lbs of water.
In the case of the 5% syrup we have:
If we add a mass M of water, the total mass of the mixture is M + 0.5lbs
And we have 0.5lbs of sugar, that is a 5% of the mixture, then we have:
[tex](\frac{0.5 lbs}{M + 0.5lbs} )*100\% = 5\%[/tex]
Now we can solve this for M, the mass of water:
[tex]\frac{0.5 lbs}{M + 0.5lbs} = 5\%/100\% = 0.05\\\\0.5 lbs = (M + 0.5 lbs)*0.05 = M*0.05 + 0.025 lbs\\\\\frac{0.5 lbs - 0.025lbs}{0.05} = 9.5 lbs[/tex]
So in this case we need to add 9.5lbs of water.
With the same procedure, we can find the mass of water needed for the 75% case and the 1.5% case, we will get:
for the 75% syrup:
[tex]\frac{0.5 lbs - 0.5lbs*0.75}{0.75} = 0.167 lbs[/tex]
For the 1.5% syrup:
[tex]\frac{0.5 lbs - 0.5lbs*0.015}{0.015} = 32.83 lbs[/tex]
If you want to learn more about concentrations, you can read:
https://brainly.com/question/10725862
