QuestionTwo friends mix $31\%$ and $26\%$ lemon juice to create $24$ liters at $27\%$ . Find the liters of each type needed.

Studdy Solution

STEP 1

Assumptions1. The total amount of lemonade to be made is24 liters. . The final lemonade mixture contains27% lemon juice.
3. The homemade lemonade contains31% lemon juice.
4. The store-bought lemonade contains26% lemon juice.
5. The percentage of lemon juice in the final mixture is a weighted average of the percentages in the homemade and store-bought lemonades.

STEP 2

Let's denote the amount of homemade lemonade used as $x$ liters and the amount of store-bought lemonade as $y$ liters. We can set up the following system of equations based on the problem1. The total amount of lemonade is24 liters $x + y =24$
2. The final mixture contains27% lemon juice $0.31x +0.26y =0.27 \times24$

STEP 3

We can rewrite the first equation to express $y$ in terms of $x$ : $y =24 - x$

STEP 4

Now, we substitute $y$ from the first equation into the second equation $0.31x +0.26(24 - x) =0.27 \times24$

STEP 5

implify the equation $0.31x +.24 -0.26x =.48$

STEP 6

Combine like terms $0.05x =0.24$

STEP 7

olve for $x$ $x =0.24 /0.05 =4.$

STEP 8

Now, we substitute $x =4.8$ into the first equation to solve for $y$ $y =24 -4.8 =19.2$ So, the friends should use4.8 liters of homemade lemonade and19.2 liters of store-bought lemonade.
However, the problem asks for the answers as whole numbers or as decimals rounded to the nearest tenth. So, we round the answersRound the values of $x$ and $y$ to the nearest tenth $x \approx4.8$ $y \approx19.2$ The friends should use approximately4.8 liters of homemade lemonade and approximately19.2 liters of store-bought lemonade.

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QuestionTwo friends mix $31\%$ and $26\%$ lemon juice to create $24$ liters at $27\%$ . Find the liters of each type needed.

Studdy Solution

STEP 1

STEP 2

STEP 3

We can rewrite the first equation to express $y$ in terms of $x$ : $y =24 - x$

STEP 4

Now, we substitute $y$ from the first equation into the second equation $0.31x +0.26(24 - x) =0.27 \times24$

STEP 5

implify the equation $0.31x +.24 -0.26x =.48$

STEP 6

Combine like terms $0.05x =0.24$

STEP 7

olve for $x$ $x =0.24 /0.05 =4.$

STEP 8

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QuestionTwo friends mix 31%31\%31% and 26%26\%26% lemon juice to create 242424 liters at 27%27\%27%. Find the liters of each type needed.

Studdy Solution

STEP 1

STEP 2

STEP 3

We can rewrite the first equation to express yyy in terms of xxx: y=24−xy =24 - xy=24−x

STEP 4

Now, we substitute yyy from the first equation into the second equation0.31x+0.26(24−x)=0.27×240.31x +0.26(24 - x) =0.27 \times240.31x+0.26(24−x)=0.27×24

STEP 5

implify the equation0.31x+.24−0.26x=.480.31x +.24 -0.26x =.480.31x+.24−0.26x=.48

STEP 6

Combine like terms0.05x=0.240.05x =0.240.05x=0.24

STEP 7

olve for xxxx=0.24/0.05=4.x =0.24 /0.05 =4.x=0.24/0.05=4.

STEP 8

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QuestionTwo friends mix $31\%$ and $26\%$ lemon juice to create $24$ liters at $27\%$ . Find the liters of each type needed.

We can rewrite the first equation to express $y$ in terms of $x$ : $y =24 - x$

Now, we substitute $y$ from the first equation into the second equation $0.31x +0.26(24 - x) =0.27 \times24$

implify the equation $0.31x +.24 -0.26x =.48$

Combine like terms $0.05x =0.24$

olve for $x$ $x =0.24 /0.05 =4.$