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:
Multiply both sides by 0.5+x:
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:
Multiply both sides by 0.5+x:
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:
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:
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.
