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% syrup?
The correct answer and explanation is:
Let’s solve the problem step-by-step:
Problem Restatement:
A baker has 5 gallons of syrup that is 70% sugar. He wants to dilute it with water (which has 0% sugar) to get syrup that is 40% sugar. We need to find out how many gallons of water to add.
Step 1: Understand what is given
- Initial syrup volume = 5 gallons
- Initial sugar concentration = 70%
- Final sugar concentration desired = 40%
- Water added = xx gallons (this is what we want to find)
- Water contains 0% sugar
Step 2: Calculate the amount of sugar in the initial syrup
Since the syrup is 70% sugar, the amount of sugar in the initial syrup is: sugar amount=5×0.70=3.5 gallons of sugar\text{sugar amount} = 5 \times 0.70 = 3.5 \text{ gallons of sugar}
Step 3: After adding xx gallons of water, total volume is:
5+x gallons5 + x \text{ gallons}
Step 4: Set up the equation for final concentration
After dilution, the sugar concentration should be 40%. The amount of sugar remains the same (3.5 gallons), but total volume changes to 5+x5 + x.
So, 3.55+x=0.40\frac{3.5}{5 + x} = 0.40
Step 5: Solve for xx
Multiply both sides by 5+x5 + x: 3.5=0.40(5+x)3.5 = 0.40 (5 + x)
Distribute 0.40: 3.5=2+0.40×3.5 = 2 + 0.40x
Subtract 2 from both sides: 3.5−2=0.40×3.5 – 2 = 0.40x 1.5=0.40×1.5 = 0.40x
Divide both sides by 0.40: x=1.50.40=3.75x = \frac{1.5}{0.40} = 3.75
Final answer:
The baker needs to add 3.75 gallons of water to the 5 gallons of 70% syrup to get syrup that is 40% sugar.
Explanation in 300 words:
This problem involves dilution, where a concentrated solution (70% sugar syrup) is mixed with a diluent (water) that contains no sugar to achieve a less concentrated solution (40% sugar syrup). The key is understanding that the total amount of sugar in the solution does not change during the mixing process—only the total volume changes.
Initially, you have 5 gallons of syrup with 70% sugar. Multiplying the volume by concentration gives the total sugar content: 5×0.70=3.55 \times 0.70 = 3.5 gallons of sugar. When you add water, which contains zero sugar, the sugar content stays at 3.5 gallons but is spread out over a larger volume (the original 5 gallons plus however much water you add, xx gallons).
To achieve a syrup with 40% sugar, the ratio of sugar to total solution must be 0.40. Using the formula: sugar amount=final concentration×total volume\text{sugar amount} = \text{final concentration} \times \text{total volume}
and substituting the known sugar amount (3.5 gallons) and volume (5 + xx) gives: 3.5=0.40×(5+x)3.5 = 0.40 \times (5 + x)
Solving this for xx, the amount of water to add, results in 3.75 gallons. This means that to lower the sugar concentration from 70% to 40%, the baker must add 3.75 gallons of water to the original 5 gallons of syrup. This process is typical in many practical applications where concentrations need adjustment by dilution.