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Answer :
Sure, let's solve this step-by-step. We need to determine how many kilograms of butter worth 140 per kg must be mixed with 20 kg of butter worth 150 per kg to produce a mixture worth 144 per kg.
### Step-by-Step Solution:
1. Define Variables:
- Let [tex]\( x \)[/tex] be the kilograms of butter worth 140 per kg that we need to find.
- We know we have 20 kg of butter worth 150 per kg.
2. Set Up the Equation:
- The total weight of the mixture will be [tex]\( x + 20 \)[/tex] kg.
- The total cost of the butter worth 140 per kg will be [tex]\( 140x \)[/tex].
- The total cost of the butter worth 150 per kg will be [tex]\( 150 \times 20 = 3000 \)[/tex].
- The combined total cost of the mixture will be [tex]\( 140x + 3000 \)[/tex].
3. Equation for the Value of Mixture:
- The worth per kg of the mixture is given as 144 per kg.
- The total cost of the mixture should be distributed over the total weight of the mixture:
[tex]\[
\frac{140x + 3000}{x + 20} = 144
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- To find [tex]\( x \)[/tex], we solve the equation:
[tex]\[
\frac{140x + 3000}{x + 20} = 144
\][/tex]
- Multiply both sides by [tex]\( x + 20 \)[/tex] to clear the fraction:
[tex]\[
140x + 3000 = 144(x + 20)
\][/tex]
- Expand and simplify:
[tex]\[
140x + 3000 = 144x + 2880
\][/tex]
- Subtract [tex]\( 140x \)[/tex] from both sides:
[tex]\[
3000 = 4x + 2880
\][/tex]
- Subtract 2880 from both sides:
[tex]\[
120 = 4x
\][/tex]
- Divide both sides by 4:
[tex]\[
x = 30
\][/tex]
Therefore, 30 kilograms of butter worth 140 per kg must be mixed with 20 kg of butter worth 150 per kg to produce a mixture worth 144 per kg.
### Step-by-Step Solution:
1. Define Variables:
- Let [tex]\( x \)[/tex] be the kilograms of butter worth 140 per kg that we need to find.
- We know we have 20 kg of butter worth 150 per kg.
2. Set Up the Equation:
- The total weight of the mixture will be [tex]\( x + 20 \)[/tex] kg.
- The total cost of the butter worth 140 per kg will be [tex]\( 140x \)[/tex].
- The total cost of the butter worth 150 per kg will be [tex]\( 150 \times 20 = 3000 \)[/tex].
- The combined total cost of the mixture will be [tex]\( 140x + 3000 \)[/tex].
3. Equation for the Value of Mixture:
- The worth per kg of the mixture is given as 144 per kg.
- The total cost of the mixture should be distributed over the total weight of the mixture:
[tex]\[
\frac{140x + 3000}{x + 20} = 144
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- To find [tex]\( x \)[/tex], we solve the equation:
[tex]\[
\frac{140x + 3000}{x + 20} = 144
\][/tex]
- Multiply both sides by [tex]\( x + 20 \)[/tex] to clear the fraction:
[tex]\[
140x + 3000 = 144(x + 20)
\][/tex]
- Expand and simplify:
[tex]\[
140x + 3000 = 144x + 2880
\][/tex]
- Subtract [tex]\( 140x \)[/tex] from both sides:
[tex]\[
3000 = 4x + 2880
\][/tex]
- Subtract 2880 from both sides:
[tex]\[
120 = 4x
\][/tex]
- Divide both sides by 4:
[tex]\[
x = 30
\][/tex]
Therefore, 30 kilograms of butter worth 140 per kg must be mixed with 20 kg of butter worth 150 per kg to produce a mixture worth 144 per kg.
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