Question

a chemical company makes two brands of antifreeze the first brand is 20% pure antifreeze and the second brand is 45% pure antifreeze in order to obtain 70 gallons of the mixture that contains 30% antifreeze how many gallons of each brand of antifreeze must be used ​

Answer 1

first brand: 42 gallons

second brand: 28 gallons

Explanation:

So you know that the first brand is 20% antifreeze, this means that if you have x gallons then 0.20x is pure antifreeze. The same thing can be applied for the second brand except the coefficient is 0.45 and I'll just say the variable is y. If you haven't noticed, the coefficients are simply the percentages of antifreeze, but in decimal form. Anyways, since it's asking for a mixture that's 30% antifreeze and is 70 gallons that means 0.30(70) gallons is pure antifreeze. This simplifies to 21 gallons. So if we take the pure antifreeze of both mixtures we get the following equation:

`21 = 0.2x + 0.45y`

the x represents how many gallons of the first brand, and the y represents how many gallons of the second brand. Since that's what it represents we also get the equation:

`70=x+y`

In this equation there aren't any coefficients, since x and y represent the whole amount. the 0.2x and 0.45 simply represent how much is pure antifreeze.

So to solve this systems of equation we can solve for y or x, either works, and then substitute it into the other equation. So in this example I'll solve for y

Original equation:

`70=x+y`

Subtract x from both sides

`70-x=y`

So now we can substitute this into the pure antifreeze question

`21 = 0.2x + 0.45y \implies21=0.2x+0.45(70-x)`

So now we can solve for x since it's the only variable in this equation

`21=0.2x+31.5-0.45x`

Simplify

`21=-0.25x+31.5`

Subtract 31.5 from both sides

`-10.5=-0.25x`

Now divide both sides by -0.25

`42=x`

So this means we have to use 42 gallons of the first brand, now we can substitute this into either equation to solve for y, but it's much easier to use the total gallon equation:

`x+y=70\implies 42+y=70`

Subtract 42 from both sides

`y=28`

So this means we need to use 42 gallons of the first brand and 28 gallons of the second brand