Answer:
The baker should add 3.75 gallon of water to 5 gallons of 70 % sugar syrup to make the 40 % syrup.
Step-by-step explanation:
The percentage concentration ([tex]r[/tex]), dimensionless, is determined by the following expression:
[tex]r = \frac{V_{sugar}}{V_{sugar}+V_{water}}[/tex] (1)
Where:
[tex]V_{sugar}[/tex] - Volume of sugar, measured in gallons.
[tex]V_{water}[/tex] - Volume of water, measured in gallons.
A 70 % sugar syrup means that there are 7 parts of sugar for each 10 parts of solution, that is, 7 parts of sugar and 3 parts of water. We need to add more water to dilute the solution to 40 %. Then, the new concentration must be equal to:
[tex]r' = \frac{V_{sugar}}{V_{sugar}+V_{water}+V'}[/tex] (2)
Where [tex]V'[/tex] is the addition of water needed to dilute the solution, measured in gallons.
If we now that [tex]V_{sugar} = 3.5\,gal[/tex], [tex]V_{water} = 1.5\,gal[/tex] and [tex]r' = 0.4[/tex], then the quantity of additional water is:
[tex]\frac{3.5\,gal}{5\,gal+V'} = 0.4[/tex]
[tex]3.5 = 0.4\cdot (5+V')[/tex]
[tex]3.5 = 2+0.4\cdot V'[/tex]
[tex]0.4\cdot V' = 1.5[/tex]
[tex]V' = 3.75\,gal[/tex]
The baker should add 3.75 gallon of water to 5 gallons of 70 % sugar syrup to make the 40 % syrup.
