Respuesta :
Answer:
They should add [tex]4\frac23[/tex] gallons of the store-bought lemonade.
Step-by-step explanation:
Given that,
Two friends make 2 gallons of lemonade containing 30% lemon juice.
The amount of lemon juice in 2 gallons of lemonade is
= 30% of 2 gallons
[tex]=\frac{30}{100}\times 2 \ gallons[/tex]
[tex]=\frac{60}{100}[/tex] gallons
Let they added x gallons of store-bought lemonade that contains 20% lemon juice.
The amount of lemon juice in x gallons of lemonade is
= 20% of x gallons
[tex]=\frac{20}{100}\times x \ gallons[/tex]
[tex]=\frac {20x}{100}[/tex] gallons
Now the total amount of lemon juice in the mixture is
[tex]=\frac{60}{100}+\frac{20x}{100}[/tex] gallons
[tex]=\frac{60+20x}{100}[/tex] gallons
Total amount of lemonade is = (2+x) gallons.
The percentage of lemon juice in the lemonade is
[tex]=\frac{\textrm{The amount of lemon juice}}{\textrm{The amount of lemonade}}\times 100[/tex]
[tex]=\frac{\frac{60+20x}{100}}{2+x}\times 100[/tex]
[tex]=\frac{60+20x}{2+x}\%[/tex]
According to the problem,
[tex]\frac{60+20x}{2+x}=23[/tex]
⇒ 60+20x= 23(2+x)
⇒60+20x= 46+23x
⇒23x-20x= 60-46
⇒3x=14
[tex]\Rightarrow x=\frac{14}{3}[/tex]
[tex]\Rightarrow x=4\frac{2}{3}[/tex] gallon
They should add [tex]4\frac23[/tex] gallons of the store-bought lemonade.
