Ella has 0.5 lbs of sugar. How much water should she add to make the following concentrations? Tell Ella how much syrup she will have in each case. 50% syrup? 5% syrup? 75% syrup? 1.5% syrup?

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 have0.5 = 0.5(0.5)+0.5(x)0.5 = 0.25+0.5xSubtract 0.25 from each side:0.5-0.25 = 0.25+0.5x-0.250.25 = 0.5xDivide both sides by 0.5:0.25/0.5 = 0.5x/0.5x = 0.5For 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 + 0.05xSubtract 0.025 from both sides:0.5-0.025 = 0.025+0.05x-0.0250.475 = 0.05xDivide both sides by 0.05:0.475/0.05 = 0.05x/0.05x = 9.5For 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 + 0.75xSubtract 0.375 from each side:0.5 - 0.375 = 0.375 + 0.75x - 0.3750.125 = 0.75xDivide both sides by 0.75:0.125/0.75 = 0.75x/0.750.17 = xFor 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 + 0.015xSubtract 0.0075 from each side:0.5-0.0075 = 0.0075 + 0.015x - 0.00750.4925 = 0.015xDivide both sides by 0.015:0.4925/0.015 = 0.015x/0.01532.8 = xFor 1.5% syrup, use 32.8 lb of water; the total mass is 32.8+0.5 = 33.3 lb.