A baker needs sugar syrup that is 40% sugar. How many gallons of water should he add to 5 gallons of 70% sugar syrup to make the 40% sugar syrup?
Answers
The baker should add 3.75 gallon of water to 5 gallons of 70 % sugar syrup to make the 40 % syrup.
The percentage concentration (rr ), dimensionless, is determined by the following expression:
r = \frac{V_{sugar}}{V_{sugar}+V_{water}}r=
V
sugar
+V
water
V
sugar
(1)
Where:
V_{sugar}V
sugar
- Volume of sugar, measured in gallons.
V_{water}V
water
- 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:
r' = \frac{V_{sugar}}{V_{sugar}+V_{water}+V'}r
′
=
V
sugar
+V
water
+V
′
V
sugar
(2)
Where V'V
′
is the addition of water needed to dilute the solution, measured in gallons.
If we now that V_{sugar} = 3.5\,galV
sugar
=3.5gal , V_{water} = 1.5\,galV
water
=1.5gal and r' = 0.4r
′
=0.4 , then the quantity of additional water is:
\frac{3.5\,gal}{5\,gal+V'} = 0.4
5gal+V
′
3.5gal
=0.4
3.5 = 0.4\cdot (5+V')3.5=0.4⋅(5+V
′
)
3.5 = 2+0.4\cdot V'3.5=2+0.4⋅V
′
0.4\cdot V' = 1.50.4⋅V
′
=1.5
V' = 3.75\,galV
′
=3.75gal
The baker should add 3.75 gallon of water to 5 gallons of 70 % sugar syrup to make the 40 % syrup.